Type  Label  Description 
Statement 

Theorem  clwwlkextfrlem1 27101 
Lemma for numclwwlk2lem1 27124. (Contributed by Alexander van der Vekens,
3Oct2018.) (Revised by AV, 27May2021.)

⊢ (((𝑁 ∈ ℕ_{0} ∧ 𝑍 ∈ (Vtx‘𝐺)) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0))) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ (𝑊‘0))) 

Theorem  extwwlkfablem2 27102 
Lemma 2 for extwwlkfab 27112. (Contributed by Alexander van der Vekens,
15Sep2018.) (Revised by AV, 28May2021.)

⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ (ℤ_{≥}‘3))
∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0)) → (𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ ((𝑁 − 2) ClWWalksN 𝐺)) 

Theorem  numclwwlkovf 27103* 
Value of operation 𝐹, mapping a vertex 𝑣 and a
positive integer
𝑛 to the "(For a fixed vertex v,
let f(n) be the number of) walks
from v to v of length n" according to definition 5 in [Huneke] p. 2.
(Contributed by Alexander van der Vekens, 14Sep2018.) (Revised by AV,
28May2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) 

Theorem  numclwwlkffin 27104* 
In a finite graph, the value of operation 𝐹 is also finite.
(Contributed by Alexander van der Vekens, 26Sep2018.) (Revised by AV,
28May2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐹𝑁) ∈ Fin) 

Theorem  numclwwlkffin0 27105* 
In a finite graph, the value of operation 𝐹 is also finite.
(Contributed by Alexander van der Vekens, 26Sep2018.) (Revised by AV,
2Jun2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ_{0}) → (𝑋𝐹𝑁) ∈ Fin) 

Theorem  numclwwlkovfel2 27106* 
Properties of an element of the value of operation 𝐹. (Contributed
by Alexander van der Vekens, 20Sep2018.) (Revised by AV,
28May2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋))) 

Theorem  numclwwlkovf2 27107* 
Value of operation 𝐹 for argument 2. (Contributed by
Alexander van
der Vekens, 19Sep2018.) (Revised by AV, 28May2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}) 

Theorem  numclwwlkovf2num 27108* 
In a 𝐾regular graph, therere are 𝐾 closed
walks of length 2
starting at a fixed vertex. (Contributed by Alexander van der Vekens,
19Sep2018.) (Revised by AV, 28May2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (#‘(𝑋𝐹2)) = 𝐾) 

Theorem  numclwwlkovf2ex 27109* 
Extending a closed walk starting at a fixed vertex by an additional edge
(forth and back). (Contributed by AV, 22Sep2018.) (Revised by AV,
28May2021.)

⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
∧ 𝑄 ∈ (𝐺 NeighbVtx 𝑋) ∧ 𝑃 ∈ (𝑋𝐹(𝑁 − 2))) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝑁 ClWWalksN 𝐺)) 

Theorem  numclwwlkovg 27110* 
Value of operation 𝐶, mapping a vertex v and an integer n
greater
than 1 to the "closed nwalks v(0) ... v(n2) v(n1) v(n) from v =
v(0)
= v(n) with v(n2) = v" according to definition 6 in [Huneke] p. 2.
(Contributed by Alexander van der Vekens, 14Sep2018.) (Revised by AV,
29May2021.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))}) 

Theorem  numclwwlkovgel 27111* 
Properties of an element of the value of operation 𝐶. (Contributed
by Alexander van der Vekens, 24Sep2018.) (Revised by AV,
29May2021.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑃 ∈ (𝑋𝐶𝑁) ↔ (𝑃 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))) 

Theorem  extwwlkfab 27112* 
The set (𝑋𝐶𝑁) of closed walks (having a fixed
length greater
than 1 and starting at a fixed vertex) with the last but 2 vertex is
identical with the first (and therefore last) vertex can be constructed
from the set (𝑋𝐹(𝑁 − 2)) of closed walks with
length smaller
by 2 than the fixed length appending a neighbor of the last vertex and
afterwards the last vertex (which is the first vertex) itself
("walking
forth and back" from the last vertex). 3 ≤
𝑁 is required since
for
𝑁 =
2: (𝑋𝐹(𝑁 − 2)) = (𝑋𝐹0) = ∅, see
umgrclwwlksge2 26812 stating that a closed walk of length 0 is
not
represented as word, at least not for an undirected simple graph.
(Contributed by Alexander van der Vekens, 18Sep2018.) (Revised by AV,
29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝑋𝐹(𝑁 − 2)) ∧ (𝑤‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)}) 

Theorem  numclwlk1lem2foa 27113* 
Going forth and back form the end of a (closed) walk: 𝑃 represents
the closed walk p_{0}, ..., p_{n}3, p_{0}. With 𝑋 =
p_{0} and 𝑄 =
p_{n}1, ((𝑃 ++ ⟨“𝑋”⟩) ++
⟨“𝑄”⟩) represents the closed
walk
p_{0}, ..., p_{n}3, p_{0}, p_{n}1, p_{0}. (Contributed
by Alexander van der
Vekens, 22Sep2018.) (Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ((𝑃 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑄 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝑋𝐶𝑁))) 

Theorem  numclwlk1lem2f 27114* 
𝑇
is a function, mapping a closed walk having a fixed length and
starting at a fixed vertex) with the last but 2 vertex is identical
with the first (and therefore last) vertex to the pair of the shorter
closed walk and its successor in the longer closed walk, which must be
a neighbor of the first vertex. (Contributed by Alexander van der
Vekens, 19Sep2018.) (Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwlk1lem2fv 27115* 
Value of the function 𝑇. (Contributed by Alexander van der
Vekens, 20Sep2018.) (Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ (𝑃 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩) 

Theorem  numclwlk1lem2f1 27116* 
𝑇
is a 11 function. (Contributed by AV, 26Sep2018.) (Revised
by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)–11→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwlk1lem2fo 27117* 
𝑇
is an onto function. (Contributed by Alexander van der Vekens,
20Sep2018.) (Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwlk1lem2f1o 27118* 
𝑇
is a 11 onto function. (Contributed by Alexander van der
Vekens, 26Sep2018.) (Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) & ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)–11onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwlk1lem2 27119* 
There is a bijection between the set of closed walks (having a fixed
length greater than 2 and starting at a fixed vertex) with the last but
2 vertex identical with the first (and therefore last) vertex and the
set of closed walks (having a fixed length less by 2 and starting at the
same vertex) and the neighbors of this vertex. (Contributed by
Alexander van der Vekens, 6Jul2018.) (Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ∃𝑓 𝑓:(𝑋𝐶𝑁)–11onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwwlk1 27120* 
Statement 9 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) with v(n2) = v
is kf(n2)". Since 𝐺 is kregular, the vertex v(n2) = v
has k
neighbors v(n1), so there are k walks from v(n2) = v to v(n) = v (via
each of v's neighbors) completing each of the f(n2) walks from v=v(0)
to v(n2)=v. This theorem holds even for k=0, but only for finite
graphs! (Contributed by Alexander van der Vekens, 26Sep2018.)
(Revised by AV, 29May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (#‘(𝑋𝐶𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2))))) 

Theorem  numclwwlkovq 27121* 
Value of operation 𝑄, mapping a vertex 𝑣 and a
positive integer
𝑛 to the not closed walks v(0) ... v(n)
of length 𝑛 from a fixed
vertex 𝑣 = v(0). "Not closed" means
v(n) =/= v(0). (Contributed by
Alexander van der Vekens, 27Sep2018.) (Revised by AV,
30May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}) 

Theorem  numclwwlkqhash 27122* 
In a 𝐾regular graph, the size of the set
of walks of length 𝑛
starting with a fixed vertex 𝑣 and ending not at this vertex is the
difference between 𝐾 to the power of 𝑛 and the
size of the set
of closed walks of length 𝑛 starting and ending at this vertex
𝑣. (Contributed by Alexander van der
Vekens, 30Sep2018.)
(Revised by AV, 30May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (#‘(𝑋𝐹𝑁)))) 

Theorem  numclwwlkovh 27123* 
Value of operation 𝐻, mapping a vertex 𝑣 and a
positive integer
𝑛 to the "closed nwalks v(0) ...
v(n2) v(n1) v(n) from v = v(0) =
v(n) ... with v(n2) =/= v" according to definition 7 in [Huneke] p. 2.
(Contributed by Alexander van der Vekens, 26Aug2018.) (Revised by AV,
30May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))}) 

Theorem  numclwwlk2lem1 27124* 
In a friendship graph, for each walk of length 𝑛 starting at a fixed
vertex 𝑣 and ending not at this vertex, there
is a unique vertex so
that the walk extended by an edge to this vertex and an edge from this
vertex to the first vertex of the walk is a value of operation 𝐻.
If the walk is represented as a word, it is sufficient to add one vertex
to the word to obtain the closed walk contained in the value of
operation 𝐻, since in a word representing a
closed walk the
starting vertex is not repeated at the end. This theorem generally
holds only for Friendship Graphs, because these guarantee that for the
first and last vertex there is a (unique) third vertex "in
between".
(Contributed by Alexander van der Vekens, 3Oct2018.) (Revised by AV,
30May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))) 

Theorem  numclwlk2lem2f 27125* 
𝑅
is a function mapping the "closed (n+2)walks v(0) ... v(n2)
v(n1) v(n) v(n+1) v(n+2) starting at 𝑋 = v(0) = v(n+2) with
v(n)
=/= X" to the words representing the prefix v(0) ... v(n2)
v(n1)
v(n) of the walk. (Contributed by Alexander van der Vekens,
5Oct2018.) (Revised by AV, 31May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁)) 

Theorem  numclwlk2lem2fv 27126* 
Value of the function R. (Contributed by Alexander van der Vekens,
6Oct2018.) (Revised by AV, 31May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))) 

Theorem  numclwlk2lem2f1o 27127* 
R is a 11 onto function. (Contributed by Alexander van der Vekens,
6Oct2018.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–11onto→(𝑋𝑄𝑁)) 

Theorem  numclwwlk2lem3 27128* 
In a friendship graph, the size of the set of walks of length 𝑁
starting with a fixed vertex 𝑋 and ending not at this vertex equals
the size of the set of all closed walks of length (𝑁 + 2)
starting
at this vertex 𝑋 and not having this vertex as last
but 2 vertex.
(Contributed by Alexander van der Vekens, 6Oct2018.) (Revised by AV,
31May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2)))) 

Theorem  numclwwlk2 27129* 
Statement 10 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) ... with v(n2)
=/= v is k^(n2)  f(n2)." According to rusgrnumwlkg 26773, we have
k^(n2) different walks of length (n2): v(0) ... v(n2). From this
number, the number of closed walks of length (n2), which is f(n2) per
definition, must be subtracted, because for these walks v(n2) =/= v(0)
= v would hold. Because of the friendship condition, there is exactly
one vertex v(n1) which is a neighbor of v(n2) as well as of
v(n)=v=v(0), because v(n2) and v(n)=v are different, so the number of
walks v(0) ... v(n2) is identical with the number of walks v(0) ...
v(n), that means each (not closed) walk v(0) ... v(n2) can be extended
by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way.
(Contributed by Alexander van der Vekens, 6Oct2018.) (Revised by AV,
31May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (#‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (#‘(𝑋𝐹(𝑁 − 2))))) 

Theorem  numclwwlk3lem 27130* 
Lemma for numclwwlk3 27131. (Contributed by Alexander van der Vekens,
6Oct2018.) (Revised by AV, 1Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (#‘(𝑋𝐹𝑁)) = ((#‘(𝑋𝐻𝑁)) + (#‘(𝑋𝐶𝑁)))) 

Theorem  numclwwlk3 27131* 
Statement 12 in [Huneke] p. 2: "Thus f(n)
= (k  1)f(n  2) + k^(n2)."
 the number of the closed walks v(0) ... v(n2) v(n1) v(n) is the sum
of the number of the closed walks v(0) ... v(n2) v(n1) v(n) with
v(n2) = v(n) (see numclwwlk1 27120) and with v(n2) =/= v(n) ( see
numclwwlk2 27129): f(n) = kf(n2) + k^(n2)  f(n2) = (k 
1)f(n  2) +
k^(n2). (Contributed by Alexander van der Vekens, 26Aug2018.)
(Revised by AV, 1Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (#‘(𝑋𝐹𝑁)) = (((𝐾 − 1) · (#‘(𝑋𝐹(𝑁 − 2)))) + (𝐾↑(𝑁 − 2)))) 

Theorem  numclwwlk4 27132* 
The total number of closed walks in a finite simple graph is the sum of
the numbers of closed walks starting at each of its vertices.
(Contributed by Alexander van der Vekens, 7Oct2018.) (Revised by AV,
2Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ) → (#‘(𝑁 ClWWalksN 𝐺)) = Σ𝑥 ∈ 𝑉 (#‘(𝑥𝐹𝑁))) 

Theorem  numclwwlk5lem 27133* 
Lemma for numclwwlk5 27134. (Contributed by Alexander van der Vekens,
7Oct2018.) (Revised by AV, 2Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ_{0}) → (2
∥ (𝐾 − 1)
→ ((#‘(𝑋𝐹2)) mod 2) =
1)) 

Theorem  numclwwlk5 27134* 
Statement 13 in [Huneke] p. 2: "Let p be
a prime divisor of k1; then
f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander
van
der Vekens, 7Oct2018.) (Revised by AV, 2Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑋𝐹𝑃)) mod 𝑃) = 1) 

Theorem  numclwwlk7lem 27135 
Lemma for numclwwlk7 27137, frgrreggt1 27139 and frgrreg 27140: If a finite,
nonempty friendship graph is 𝐾regular, the 𝐾 is a nonnegative
integer. (Contributed by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈
ℕ_{0}) 

Theorem  numclwwlk6 27136 
For a prime divisor 𝑃 of 𝐾 − 1, the total
number of closed
walks of length 𝑃 in a 𝐾regular friendship graph
is equal
modulo 𝑃 to the number of vertices.
(Contributed by Alexander van
der Vekens, 7Oct2018.) (Revised by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((#‘𝑉) mod 𝑃)) 

Theorem  numclwwlk7 27137 
Statement 14 in [Huneke] p. 2: "The total
number of closed walks of
length p [in a friendship graph] is (k(k1)+1)f(p)=1 (mod p)",
since the
number of vertices in a friendship graph is (k(k1)+1), see
frrusgrord0 27095 or frrusgrord 27096, and p divides (k1), i.e. (k1) mod p =
0 => k(k1) mod p = 0 => k(k1)+1 mod p = 1. Since the null graph
is a
friendship graph, see frgr0 27028, as well as kregular (for any k), see
0vtxrgr 26376, but has no closed walk, see 0clwlk0 26892, this theorem would
be false for a null graph: ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0
≠ 1, so this case must be excluded (by
assuming 𝑉
≠ ∅).
(Contributed by Alexander van der Vekens, 1Sep2018.) (Revised by AV,
3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 1) 

Theorem  numclwwlk8 27138 
The size of the set of closed walks of length 𝑃, 𝑃 prime, is
divisible by 𝑃. This corresponds to statement 9 in
[Huneke] p. 2:
"It follows that, if p is a prime number, then the number of closed
walks
of length p is divisible by p", see also clwlksndivn 26872. (Contributed by
Alexander van der Vekens, 7Oct2018.) (Revised by AV, 3Jun2021.)

⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0) 

Theorem  frgrreggt1 27139 
If a finite nonempty friendship graph is 𝐾regular with 𝐾 > 1,
then 𝐾 must be 2.
(Contributed by Alexander van der Vekens,
7Oct2018.) (Revised by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2)) 

Theorem  frgrreg 27140 
If a finite nonempty friendship graph is 𝐾regular, then 𝐾 must
be 2 (or 0).
(Contributed by Alexander van der Vekens,
9Oct2018.) (Revised by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))) 

Theorem  frgrregord013 27141 
If a finite friendship graph is 𝐾regular, then it must have order
0, 1 or 3. (Contributed by Alexander van der Vekens, 9Oct2018.)
(Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3)) 

Theorem  frgrregord13 27142 
If a nonempty finite friendship graph is 𝐾regular, then it must
have order 1 or 3. Special case of frgrregord013 27141. (Contributed by
Alexander van der Vekens, 9Oct2018.) (Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((#‘𝑉) = 1 ∨ (#‘𝑉) = 3)) 

Theorem  frgrogt3nreg 27143* 
If a finite friendship graph has an order greater than 3, it cannot be
𝑘regular for any 𝑘.
(Contributed by Alexander van der Vekens,
9Oct2018.) (Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑘 ∈ ℕ_{0}
¬ 𝐺 RegUSGraph 𝑘) 

Theorem  friendshipgt3 27144* 
The friendship theorem for big graphs: In every finite friendship graph
with order greater than 3 there is a vertex which is adjacent to all
other vertices. (Contributed by Alexander van der Vekens, 9Oct2018.)
(Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) 

Theorem  friendship 27145* 
The friendship theorem: In every finite (nonempty) friendship graph
there is a vertex which is adjacent to all other vertices. This is
Metamath 100 proof #83. (Contributed by Alexander van der Vekens,
9Oct2018.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) 

PART 17 GUIDES AND
MISCELLANEA


17.1 Guides (conventions, explanations, and
examples)


17.1.1 Conventions
This section describes the conventions we use. These conventions often refer
to existing mathematical practices, which are discussed in more detail in
other references.
For the general conventions, see conventions 27146, and for conventions related
to labels, see conventionslabel 27147.
Logic and set theory provide a foundation for all of mathematics. To learn
about them, you should study one or more of the references listed below. We
indicate references using square brackets. The textbooks provide a
motivation for what we are doing, whereas Metamath lets you see in detail all
hidden and implicit steps. Most standard theorems are accompanied by
citations. Some closely followed texts include the following:
 Axioms of propositional calculus  [Margaris].
 Axioms of predicate calculus  [Megill] (System S3' in the article
referenced).
 Theorems of propositional calculus  [WhiteheadRussell].
 Theorems of pure predicate calculus  [Margaris].
 Theorems of equality and substitution  [Monk2], [Tarski], [Megill].
 Axioms of set theory  [BellMachover].
 Development of set theory  [TakeutiZaring]. (The first part of [Quine]
has a good explanation of the powerful device of "virtual" or
class abstractions, which is essential to our development.)
 Construction of real and complex numbers  [Gleason]
 Theorems about real numbers  [Apostol]


Theorem  conventions 27146 
Here are some of the conventions we use in the
Metamath Proof Explorer (aka "set.mm"), and how they correspond to
typical textbook language (skipping the many cases
where they are identical).
For conventions related to labels, see conventionslabel 27147.
 Notation.
Where possible, the notation attempts to conform to modern
conventions, with variations due to our choice of the axiom system
or to make proofs shorter. However, our notation is strictly
sequential (lefttoright). For example, summation is written in the
form Σ𝑘 ∈ 𝐴𝐵 (dfsum 14367) which denotes that index
variable 𝑘 ranges over 𝐴 when evaluating 𝐵. Thus,
Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 14558).
The notation is usually explained in more detail when first introduced.
 Axiomatic assertions ($a).
All axiomatic assertions ($a statements)
starting with " ⊢ " have labels starting
with "ax" (axioms) or "df" (definitions). A statement with a
label starting with "ax" corresponds to what is traditionally
called an axiom. A statement with a label starting with "df"
introduces new symbols or a new relationship among symbols
that can be eliminated; they always extend the definition of
a wff or class. Metamath blindly treats $a statements as new
given facts but does not try to justify them. The mmj2 program
will justify the definitions as sound as discussed below,
except for 4 definitions (dfbi 197, dfcleq 2614, dfclel 2617, dfclab 2608)
that require a more complex metalogical justification by hand.
 Proven axioms.
In some cases we wish to treat an expression as an axiom in
later theorems, even though it can be proved. For example,
we derive the postulates or axioms of complex arithmetic as
theorems of ZFC set theory. For convenience, after deriving
the postulates, we reintroduce them as new axioms on
top of set theory. This lets us easily identify which axioms
are needed for a particular complex number proof, without the
obfuscation of the set theory used to derive them. For more, see
mmcomplex.html. When we wish
to use a previouslyproven assertion as an axiom, our convention
is that we use the
regular "axNAME" label naming convention to define the axiom,
but we precede it with a proof of the same statement with the label
"axNAME" . An example is complex arithmetic axiom ax1cn 9954,
proven by the preceding theorem ax1cn 9930.
The metamath.exe program will warn if an axiom does not match the
preceding theorem that justifies it if the names match in this way.
 Definitions (df...).
We encourage definitions to include hypertext links to proven examples.
 Statements with hypotheses. Many theorems and some axioms,
such as axmp 5, have hypotheses that must be satisfied in order for
the conclusion to hold, in this case min and maj. When presented in
summarized form such as in the Theorem List (click on "Nearby theorems"
on the axmp 5 page), the hypotheses are connected with an ampersand and
separated from the conclusion with a big arrow, such as in " ⊢ 𝜑
& ⊢ (𝜑 → 𝜓) => ⊢ 𝜓". These symbols are _not_
part of the Metamath language but are just informal notation meaning
"and" and "implies".
 Discouraged use and modification.
If something should only be used in limited ways, it is marked with
"(New usage is discouraged.)". This is used, for example, when something
can be constructed in more than one way, and we do not want later
theorems to depend on that specific construction.
This marking is also used if we want later proofs to use proven axioms.
For example, we want later proofs to
use ax1cn 9954 (not ax1cn 9930) and ax1ne0 9965 (not ax1ne0 9941), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 9930 and ax1ne0 9941 are marked as "(New usage is discouraged.)".
In some cases a proof should not normally be changed, e.g., when it
demonstrates some specific technique.
These are marked with "(Proof modification is discouraged.)".
 New definitions infrequent.
Typically, we are minimalist when introducing new definitions; they are
introduced only when a clear advantage becomes apparent for reducing
the number of symbols, shortening proofs, etc. We generally avoid
the introduction of gratuitous definitions because each one requires
associated theorems and additional elimination steps in proofs.
For example, we use < and ≤ for inequality expressions, and
use ((sin‘(i · 𝐴)) / i) instead of (sinh‘𝐴)
for the hyperbolic sine.
 Minimizing axioms and the axiom of choice.
We prefer proofs that depend on fewer and/or weaker axioms,
even if the proofs are longer. In particular, we prefer proofs that do
not use the axiom of choice (dfac 8899) where such proofs can be found.
The axiom of choice is widely accepted, and ZFC is the most
commonlyaccepted fundamental set of axioms for mathematics.
However, there have been and still are some lingering controversies
about the Axiom of Choice. Therefore, where a proof
does not require the axiom of choice, we prefer that proof instead.
E.g., our proof of the SchroederBernstein Theorem (sbth 8040)
does not use the axiom of choice.
In some cases, the weaker axiom of countable choice (axcc 9217)
or axiom of dependent choice (axdc 9228) can be used instead.
Similarly, any theorem in first order logic (FOL) that
contains only set variables that are all mutually distinct,
and has no wff variables, can be proved *without* using
ax10 2016 through ax13 2245, by invoking ax10w 2003 through ax13w 2010.
We encourage proving theorems *without* ax10 2016 through ax13 2245
and moving them up to the ax4 1734 through ax9 1996 section.
 Alternative (ALT) proofs.
If a different proof is significantly shorter or clearer but
uses more or stronger axioms, we prefer to make that proof an
"alternative" proof (marked with an ALT label suffix), even if
this alternative proof was formalized first.
We then make the proof that requires fewer axioms the main proof.
This has the effect of reducing (over time)
the number and strength of axioms used by any particular proof.
There can be multiple alternatives if it makes sense to do so.
Alternative (*ALT) theorems should have "(Proof modification is
discouraged.) (New usage is discouraged.)" in their comment and should
follow the main statement, so that people reading the text in order will
see the main statement first. The alternative and main statement
comments should use hyperlinks to refer to each other (so that a reader
of one will become easily aware of the other).
 Alternative (ALTV) versions.
If a theorem or definition is an alternative/variant of an already
existing theorem resp. definition, its label should have the same name
with suffix ALTV. Such alternatives should be temporary only, until it
is decided which alternative should be used in the future. Alternative
(*ALTV) theorems or definitions are usually contained in mathboxes.
Their comments need not to contain "(Proof modification is discouraged.)
(New usage is discouraged.)". Alternative statements should follow the
main statement, so that people reading the text in order will see the
main statement first.
 Old (OLD) versions or proofs.
If a proof, definition, axiom, or theorem is going to be removed,
we often stage that change by first renaming its
label with an OLD suffix (to make it clear that it is going to
be removed). Old (*OLD) statements should have "(Proof modification is
discouraged.) (New usage is discouraged.)" and "Obsolete version of
~ xxx as of ddmmmyyyy." (not enclosed in parentheses) in the comment.
An old statement should follow the main statement, so that people
reading the text in order will see the main statement first.
This typically happens when a shorter proof to an existing theorem is
found: the existing theorem is kept as an *OLD statement for one year.
When a proof is shortened automatically (using Metamath's minimize_with
command), then it is not necessary to keep the old proof, nor to add
credit for the shortening.
 Variables.
Propositional variables (variables for wellformed formulas or wffs) are
represented with lowercase Greek letters and are normally used
in this order:
𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta,
𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma.
Individual setvar variables are represented with lowercase Latin letters
and are normally used in this order:
𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡.
Variables that represent classes are often represented by
uppercase Latin letters:
𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on.
There are other symbols that also represent class variables and suggest
specific purposes, e.g., 0 for poset zero (see p0val 16981) and
connective symbols such as + for some group addition operation.
(See prdsplusgval 16073 for an example of the use of +).
Class variables are selected in alphabetical order starting
from 𝐴 if there is no reason to do otherwise, but many
assertions select different class variables or a different order
to make their intended meaning clearer.
 Turnstile.
"⊢ ", meaning "It is provable that," is the first token
of all assertions
and hypotheses that aren't syntax constructions. This is a standard
convention in logic. For us, it also prevents any ambiguity with
statements that are syntax constructions, such as "wff ¬ 𝜑".
 Biconditional (↔).
There are basically two ways to maximize the effectiveness of
biconditionals (↔):
you can either have onedirectional simplifications of all theorems
that produce biconditionals, or you can have onedirectional
simplifications of theorems that consume biconditionals.
Some tools (like Lean) follow the first approach, but set.mm follows
the second approach. Practically, this means that in set.mm, for
every theorem that uses an implication in the hypothesis, like
axmp 5, there is a corresponding version with a biconditional or a
reversed biconditional, like mpbi 220 or mpbir 221. We prefer this
second approach because the number of duplications in the second
approach is bounded by the size of the propositional calculus section,
which is much smaller than the number of possible theorems in all later
sections that produce biconditionals. So although theorems like
biimpi 206 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 954, sylbir 225,
or 3imtr4i 281.
 Substitution.
"[𝑦 / 𝑥]𝜑" should be read "the wff that results from the
proper substitution of 𝑦 for 𝑥 in wff 𝜑." See dfsb 1878
and the related dfsbc 3423 and dfcsb 3520.
 Isaset.
"𝐴 ∈ V" should be read "Class 𝐴 is a set (i.e. exists)."
This is a convention based on Definition 2.9 of [Quine] p. 19.
See dfv 3192 and isset 3197.
However, instead of using 𝐼 ∈ V in the antecedent of a theorem for
some variable 𝐼, we now prefer to use 𝐼 ∈ 𝑉 (or another
variable if 𝑉 is not available) to make it more general. That way we
can often avoid needing extra uses of elex 3202 and syl 17 in the common
case where 𝐼 is already a member of something.
For hypotheses ($e statement) of theorems (mostly in inference form),
however, ⊢ 𝐴 ∈ V is used rather than ⊢ 𝐴 ∈ 𝑉 (e.g.
difexi 4779). This is because 𝐴 ∈ V is almost always satisfied using
an existence theorem stating "... ∈ V", and a hardcoded V in
the $e statement saves a couple of syntax building steps that substitute
V into 𝑉. Notice that this does not hold for hypotheses of
theorems in deduction form: Here still ⊢ (𝜑 → 𝐴 ∈ 𝑉) should be
used rather than ⊢ (𝜑 → 𝐴 ∈ V).
 Converse.
"^{◡}𝑅" should be read "converse of (relation) 𝑅"
and is the same as the more standard notation R^{1}
(the standard notation is ambiguous). See dfcnv 5092.
This can be used to define a subset, e.g., dftan 14746 notates
"the set of values whose cosine is a nonzero complex number" as
(^{◡}cos “ (ℂ ∖ {0})).
 Function application.
"(𝐹‘𝑥)" should be read "the value
of function 𝐹 at 𝑥" and has the same meaning as the more
familiar but ambiguous notation F(x). For example,
(cos‘0) = 1 (see cos0 14824). The left apostrophe notation
originated with Peano and was adopted in Definition *30.01 of
[WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
Definition 6.11 of [TakeutiZaring] p. 26. See dffv 5865.
In the ASCII (input) representation there are spaces around the grave
accent; there is a single accent when it is used directly,
and it is doubled within comments.
 Infix and parentheses.
When a function that takes two classes and produces a class
is applied as part of an infix expression, the expression is always
surrounded by parentheses (see dfov 6618).
For example, the + in (2 + 2); see 2p2e4 11104.
Function application is itself an example of this.
Similarly, predicate expressions
in infix form that take two or three wffs and produce a wff
are also always surrounded by parentheses, such as
(𝜑 → 𝜓), (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓), and
(𝜑 ↔ 𝜓)
(see wi 4, dfor 385, dfan 386, and dfbi 197 respectively).
In contrast, a binary relation (which compares two _classes_ and
produces a _wff_) applied in an infix expression is _not_
surrounded by parentheses.
This includes set membership 𝐴 ∈ 𝐵 (see wel 1988),
equality 𝐴 = 𝐵 (see dfcleq 2614),
subset 𝐴 ⊆ 𝐵 (see dfss 3574), and
lessthan 𝐴 < 𝐵 (see dflt 9909). For the general definition
of a binary relation in the form 𝐴𝑅𝐵, see dfbr 4624.
For example, 0 < 1 (see 0lt1 10510) does not use parentheses.
 Unary minus.
The symbol  is used to indicate a unary minus, e.g., 1.
It is specially defined because it is so commonly used.
See cneg 10227.
 Function definition.
Functions are typically defined by first defining the constant symbol
(using $c) and declaring that its symbol is a class with the
label cNAME (e.g., ccos 14739).
The function is then defined labeled dfNAME; definitions
are typically given using the mapsto notation (e.g., dfcos 14745).
Typically, there are other proofs such as its
closure labeled NAMEcl (e.g., coscl 14801), its
function application form labeled NAMEval (e.g., cosval 14797),
and at least one simple value (e.g., cos0 14824).
 Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
(!‘4) = ;24 (dffac 13017 and fac4 13024).
 Unambiguous symbols.
A given symbol has a single unambiguous meaning in general.
Thus, where the literature might use the same symbol with different
meanings, here we use different (variant) symbols for different
meanings. These variant symbols often have suffixes, subscripts,
or underlines to distinguish them. For example, here
"0" always means the value zero (df0 9903), while
"0_{g}" is the group identity element (df0g 16042),
"0." is the poset zero (dfp0 16979),
"0_{𝑝}" is the zero polynomial (df0p 23377),
"0_{vec}" is the zero vector in a normed subcomplex vector space
(df0v 27341), and
"0" is a class variable for use as a connective symbol
(this is used, for example, in p0val 16981).
There are other class variables used as connective symbols
where traditional notation would use ambiguous symbols, including
"1", "+", "∗", and "∥".
These symbols are very similar to traditional notation, but because
they are different symbols they eliminate ambiguity.
 ASCII representation of symbols.
We must have an ASCII representation for each symbol.
We generally choose short sequences, ideally digraphs, and generally
choose sequences that vaguely resemble the mathematical symbol.
Here are some of the conventions we use when selecting an
ASCII representation.
We generally do not include parentheses inside a symbol because
that confuses text editors (such as emacs).
Greek letters for wff variables always use the first two letters
of their English names, making them easy to type and easy to remember.
Symbols that almost look like letters, such as ∀,
are often represented by that letter followed by a period.
For example, "A." is used to represent ∀,
"e." is used to represent ∈, and
"E." is used to represent ∃.
Single letters are now always variable names, so constants that are
often shown as single letters are now typically preceded with "_"
in their ASCII representation, for example,
"_i" is the ASCII representation for the imaginary unit i.
A script font constant is often the letter
preceded by "~" meaning "curly", such as "~P" to represent
the power class 𝒫.
Originally, all setvar and class variables used only single letters
az and AZ, respectively. A big change in recent years was to
allow the use of certain symbols as variable names to make formulas
more readable, such as a variable representing an additive group
operation. The convention is to take the original constant token
(in this case "+" which means complex number addition) and put
a period in front of it to result in the ASCII representation of the
variable ".+", shown as +, that can
be used instead of say the letter "P" that had to be used before.
Choosing tokens for more advanced concepts that have no standard
symbols but are represented by words in books, is hard. A few are
reasonably obvious, like "Grp" for group and "Top" for topology,
but often they seem to end up being either too long or too
cryptic. It would be nice if the math community came up with
standardized short abbreviations for English math terminology,
like they have more or less done with symbols, but that probably
won't happen any time soon.
Another informal convention that we've somewhat followed, that is also
not uncommon in the literature, is to start tokens with a
capital letter for collectionlike objects and lower case for
functionlike objects. For example, we have the collections On
(ordinal numbers), Fin, Prime, Grp, and we have the functions sin,
tan, log, sup. Predicates like Ord and Lim also tend to start
with upper case, but in a sense they are really collectionlike,
e.g. Lim indirectly represents the collection of limit ordinals,
but it can't be an actual class since not all limit ordinals
are sets.
This initial capital vs. lower case letter convention is sometimes
ambiguous. In the past there's been a debate about whether
domain and range are collectionlike or functionlike, thus whether
we should use Dom, Ran or dom, ran. Both are used in the literature.
In the end dom, ran won out for aesthetic reasons
(Norm Megill simply just felt they looked nicer).
 Typography conventions.
Class symbols for functions (e.g., abs, sin)
should usually not have leading or trailing blanks in their
HTML/Latex representation.
This is in contrast to class symbols for operations
(e.g., gcd, sadd, eval), which usually do
include leading and trailing blanks in their representation.
If a class symbol is used for a function as well as an operation
(according to the definition dfov 6618, each operation value can be
written as function value of an ordered pair), the convention for its
primary usage should be used, e.g. (iEdg‘𝐺) versus
(𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸⟩.
 Number construction independence.
There are many ways to model complex numbers.
After deriving the complex number postulates we
reintroduce them as new axioms on top of set theory.
This lets us easily identify which axioms are needed
for a particular complex number proof, without the obfuscation
of the set theory used to derive them.
This also lets us be independent of the specific construction,
which we believe is valuable.
See mmcomplex.html for details.
Thus, for example, we don't allow the use of ∅ ∉ ℂ,
as handy as that would be, because that would be
constructionspecific. We want proofs about ℂ to be independent
of whether or not ∅ ∈ ℂ.
 Minimize hypotheses
(except for construction independence and number theorem domains).
In most cases we try to minimize hypotheses, that is,
we eliminate or reduce what must be true to prove something, so that
the proof is more general and easier to use.
There are exceptions. For example, we intentionally add hypotheses
if they help make proofs independent of a particular construction
(e.g., the contruction of complex numbers ℂ).
We also intentionally add hypotheses for many real and complex
number theorems to expressly state their domains even when they
aren't strictly needed. For example, we could show that
(𝐴 < 𝐵 → 𝐵 ≠ 𝐴) without any other hypotheses, but in
practice we also require proving at least some domains
(e.g., see ltnei 10121). Here are the reasons as discussed in
https://groups.google.com/g/metamath/c/2AW7T3d2YiQ/m/iSN7g87t3ikJ:
 Having the hypotheses immediately shows the intended domain of
applicability (is it ℝ, ℝ^{*}, ω, or something else?),
without having to trace back to definitions.
 Having the hypotheses forces its use in the intended
domain, which generally is desirable.
 The behavior is dependent on accidental behavior of definitions
outside of their domains, so the theorems are nonportable and
"brittle".
 Only a few theorems can have their hypotheses removed
in this fashion due to happy coincidences for our particular
settheoretical definitions. The poor user (especially a
novice learning real number arithmetic) is going to be
confused not knowing when hypotheses are needed and when
they are not. For someone who hasn't traced back the
settheoretical foundations of the definitions, it is
seemingly random and isn't intuitive at all.
 The consensus of opinion of people on this group seemed to be
against doing this.
 Natural numbers.
There are different definitions of "natural" numbers in the literature.
We use ℕ (dfnn 10981) for the set of positive integers starting
from 1, and ℕ_{0} (dfn0 11253) for the set of nonnegative integers
starting at zero.
 Decimal numbers.
Numbers larger than nine are often expressed in base 10 using the
decimal constructor dfdec 11454, e.g., ;;;4001 (see 4001prm 15795
for a proof that 4001 is prime).
 Theorem forms.
We will use the following descriptive terms to categorize theorems:
 A theorem is in "closed form" if it has no $e hypotheses
(e.g., unss 3771). The term "tautology" is also used, especially in
propositional calculus. This form was formerly called "theorem form"
or "closed theorem form".
 A theorem is in "deduction form" (or is a "deduction") if it
has zero or more $e hypotheses, and the hypotheses and the conclusion
are implications that share the same antecedent. More precisely, the
conclusion is an implication with a wff variable as the antecedent
(usually 𝜑), and every hypothesis ($e statement) is either:
 an implication with the same antecedent as the conclusion, or
 a definition. A definition can be for a class variable (this is a
class variable followed by =, e.g. the definition of 𝐷 in
lhop 23717) or a wff variable (this is a wff variable followed by
↔); class variable definitions are more common.
In practice, a proof of a theorem in deduction form will also contain
many steps that are implications where the antecedent is either that
wff variable (usually 𝜑) or is a conjunction (𝜑 ∩ ...)
including that wff variable (𝜑). E.g. a1d 25, unssd 3773.
Although they are no real deductions, theorems without $e hypotheses,
but in the form (𝜑 → ...), are also said to be in "deduction
form". Such theorems usually have a two step proof, applying a1i 11 to a
given theorem, and are used as convenience theorems to shorten many
proofs. E.g. eqidd 2622, which is used more than 1500 times.
 A theorem is in "inference form" (or is an "inference") if
it has one or more $e hypotheses, but is not in deduction form,
i.e. there is no common antecedent (e.g., unssi 3772).
Any theorem whose conclusion is an implication has an associated
inference, whose hypotheses are the hypotheses of that theorem
together with the antecedent of its conclusion, and whose conclusion is
the consequent of that conclusion. When both theorems are in set.mm,
then the associated inference is often labeled by adding the suffix "i"
to the label of the original theorem (for instance, con3i 150 is the
inference associated with con3 149). The inference associated with a
theorem is easily derivable from that theorem by a simple use of
axmp 5. The other direction is the subject of the Deduction Theorem
discussed below. We may also use the term "associated inference" when
the above process is iterated. For instance, syl 17 is an
inference associated with imim1 83 because it is the inference
associated with imim1i 63 which is itself the inference
associated with imim1 83.
"Deduction form" is the preferred form for theorems because this form
allows us to easily use the theorem in places where (in traditional
textbook formalizations) the standard Deduction Theorem (see below)
would be used. We call this approach "deduction style".
In contrast, we usually avoid theorems in "inference form" when that
would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other
forms of the same theorem (e.g., pm2.43d 53). The labels for inferences
usually have the suffix "i" (e.g., pm2.43i 52). The labels of theorems
in "closed form" would have no special suffix (e.g., pm2.43 56). When
an inference is converted to a theorem by eliminating an "is a set"
hypothesis, we sometimes suffix the closed form with "g" (for "more
general") as in uniex 6918 vs. uniexg 6920.
 Deduction theorem.
The Deduction Theorem is a metalogical theorem that provides an
algorithm for constructing a proof of a theorem from the proof of its
corresponding deduction (its associated inference). See for instance
Theorem 3 in [Margaris] p. 56. In ordinary mathematics, no one actually
carries out the algorithm, because (in its most basic form) it involves
an exponential explosion of the number of proof steps as more hypotheses
are eliminated. Instead, in ordinary mathematics the Deduction Theorem
is invoked simply to claim that something can be done in principle,
without actually doing it. For more details, see mmdeduction.html.
The Deduction Theorem is a metalogical theorem that cannot be applied
directly in metamath, and the explosion of steps would be a problem
anyway, so alternatives are used. One alternative we use sometimes is
the "weak deduction theorem" dedth 4117, which works in certain cases in
set theory. We also sometimes use dedhb 3363. However, the primary
mechanism we use today for emulating the deduction theorem is to write
proofs in deduction form (aka "deduction style") as described earlier;
the prefixed 𝜑 → mimics the context in a deduction proof system.
In practice this mechanism works very well. This approach is described
in the deduction form and natural deduction page mmnatded.html; a
list of translations for common natural deduction rules is given in
natded 27148.
 Recursion.
We define recursive functions using various "recursion constructors".
These allow us to define, with compact direct definitions, functions
that are usually defined in textbooks with indirect selfreferencing
recursive definitions. This produces compact definition and much
simpler proofs, and greatly reduces the risk of creating unsound
definitions. Examples of recursion constructors include
recs(𝐹) in dfrecs 7428, rec(𝐹, 𝐼) in dfrdg 7466,
seq_{𝜔}(𝐹, 𝐼) in dfseqom 7503, and seq𝑀( + , 𝐹) in
dfseq 12758. These have characteristic function 𝐹 and initial value
𝐼. (Σ_{g} in dfgsum 16043 isn't really designed for arbitrary
recursion, but you could do it with the right magma.) The logically
primary one is dfrecs 7428, but for the "average user" the most useful
one is probably dfseq 12758 provided that a countable sequence is
sufficient for the recursion.
 Extensible structures.
Mathematics includes many structures such as ring, group, poset, etc.
We define an "extensible structure" which is then used to define group,
ring, poset, etc. This allows theorems from more general structures
(groups) to be reused for more specialized structures (rings) without
having to reprove them. See dfstruct 15802.
 Undefined results and "junk theorems".
Some expressions are only expected to be meaningful in certain contexts.
For example, consider Russell's definition description binder iota,
where (℩𝑥𝜑) is meant to be "the 𝑥 such that 𝜑"
(where 𝜑 typically depends on x).
What should that expression produce when there is no such 𝑥?
In set.mm we primarily use one of two approaches.
One approach is to make the expression evaluate to the empty set
whenever the expression is being used outside of its expected context.
While not perfect, it makes it a bit more clear when something
is undefined, and it has the advantage that it makes more
things equal outside their domain which can remove hypotheses when
you feel like exploiting these socalled junk theorems.
Note that Quine does this with iota (his definition of iota
evaluates to the empty set when there is no unique value of 𝑥).
Quine has no problem with that and we don't see why we should,
so we define iota exactly the same way that Quine does.
The main place where you see this being systematically exploited is in
"reverse closure" theorems like 𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹,
which is useful when 𝐹 is a family of sets. (by this we
mean it's a set set even in a type theoretic interpretation.)
The second approach uses "(New usage is discouraged.)" to prevent
unintentional uses of certain properties.
For example, you could define some construct dfNAME whose
usage is discouraged, and prove only the specific properties
you wish to use (and add those proofs to the list of permitted uses
of "discouraged" information). From then on, you can only use
those specific properties without a warning.
Other approaches often have hidden problems.
For example, you could try to "not define undefined terms"
by creating definitions like ${ $d 𝑦𝑥 $. $d 𝑦𝜑 $.
dfiota $a ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) $. $}.
This will be rejected by the definition checker, but the bigger
theoretical reason to reject this axiom is that it breaks equality 
the metatheorem (𝑥 = 𝑦 → P(x) = P(y) ) fails
to hold if definitions don't unfold without some assumptions.
(That is, iotabidv 5841 is no longer provable and must be added
as an axiom.) It is important for every syntax constructor to
satisfy equality theorems *unconditionally*, e.g., expressions
like (1 / 0) = (1 / 0) should not be rejected.
This is forced on us by the context free term
language, and anything else requires a lot more infrastructure
(e.g., a type checker) to support without making everything else
more painful to use.
Another approach would be to try to make nonsensical
statements syntactically invalid, but that can create its own
complexities; in some cases that would make parsing itself undecidable.
In practice this does not seem to be a serious issue.
No one does these things deliberately in "real" situations,
and some knowledgeable people (such as Mario Carneiro)
have never seen this happen accidentally.
Norman Megill doesn't agree that these "junk" consequences are
necessarily bad anyway, and they can significantly shorten proofs
in some cases. This database would be much larger if, for example,
we had to condition fvex 6168 on the argument being in the domain
of the function. It is impossible to derive a contradiction
from sound definitions (i.e. that pass the definition check),
assuming ZFC is consistent, and he doesn't see the point of all the
extra busy work and huge increase in set.mm size that would result
from restricting *all* definitions.
So instead of implementing a complex system to counter a
problem that does not appear to occur in practice, we use
a significantly simpler set of approaches.
 Organizing proofs.
Humans have trouble understanding long proofs.
It is often preferable to break longer proofs into
smaller parts (just as with traditional proofs). In Metamath
this is done by creating separate proofs of the separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 8030 is the first lemma for sbth 8040. Also, consider proving
reusable results separately, so that others will be able to easily
reuse that part of your work.
 Limit proof size.
It is often preferable to break longer proofs into
smaller parts, just as you would do with traditional proofs.
One reason is that humans have trouble understanding long proofs.
Another reason is that it's generally best to prove
reusable results separately,
so that others will be able to easily reuse them.
Finally, the "minimize" routine can take much longer with
very long proofs.
We encourage proofs to be no more than 200 essential steps, and
generally no more than 500 essential steps,
though these are simply guidelines and not hardandfast rules.
Much smaller proofs are fine!
We also acknowledge that some proofs, especially autogenerated ones,
should sometimes not be broken up (e.g., because
breaking them up might be useless and inefficient due to many
interconnections and reused terms within the proof).
In Metamath, breaking up longer proofs is done by creating multiple
separate proofs of separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 8030 is the first lemma for sbth 8040.
 Hypertext links.
We strongly encourage comments to have many links to related material,
with accompanying text that explains the relationship. These can help
readers understand the context. Links to other statements, or to
HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a
spaceseparated tilde (e.g., " ~ dfprm " results in " dfprm 15329").
When metamath.exe is used to generate HTML it automatically inserts
hypertext links for syntax used (e.g., every symbol used), every axiom
and definition depended on, the justification for each step in a proof,
and to both the next and previous assertion.
 Hypertext links to section headers.
Some section headers have text under them that describes or explains the
section. However, they are not part of the description of axioms or
theorems, and there is no way to link to them directly. To provide for
this, section headers with accompanying text (indicated with "*"
prefixed to mmtheorems.html#mmdtoc entries) have an anchor in
mmtheorems.html whose name is the first $a or $p statement that
follows the header. For example there is a glossary under the section
heading called GRAPH THEORY. The first $a or $p statement that follows
is cedgf 25801. To reference it we link to the anchor using a
spaceseparated tilde followed by the spaceseparated link
mmtheorems.html#cedgf, which will become the hyperlink
mmtheorems.html#cedgf. Note that no theorem in set.mm is allowed to
begin with "mm" (enforced by "verify markup" in the metamath program).
Whenever the software sees a tilde reference beginning with "http:",
"https:", or "mm", the reference is assumed to be a link to something
other than a statement label, and the tilde reference is used as is.
This can also be useful for relative links to other pages such as
mmcomplex.html.
 Bibliography references.
Please include a bibliographic reference to any external material used.
A name in square brackets in a comment indicates a
bibliographic reference. The full reference must be of the form
KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER 
note that this is a very specific form that requires a page number.
There should be no comma between the author reference and the
"p." (a constant indicator).
Whitespace, comma, period, or semicolon should follow NUMBER.
An example is Theorem 3.1 of [Monk1] p. 22,
The KEYWORD, which is not casesensitive,
must be one of the following: Axiom, Chapter, Compare, Condition,
Corollary, Definition, Equation, Example, Exercise, Figure, Item,
Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem,
Property, Proposition, Remark, Rule, Scheme, Section, or Theorem.
The IDENTIFIER is optional, as in for example
"Remark in [Monk1] p. 22".
The NOISEWORDS(S) are zero or more from the list: from, in, of, on.
The AUTHOR(S) must be present in the file identified with the
htmlbibliography assignment (e.g., mmset.html) as a named anchor
(NAME=). If there is more than one document by the same author(s),
add a numeric suffix (as shown here).
The NUMBER is a page number, and may be any alphanumeric string such as
an integer or Roman numeral.
Note that we _require_ page numbers in comments for individual
$a or $p statements. We allow names in square brackets without
page numbers (a reference to an entire document) in
heading comments.
If this is a new reference, please also add it to the
"Bibliography" section of mmset.html.
(The file mmbiblio.html is automatically rebuilt, e.g.,
using the metamath.exe "write bibliography" command.)
 Acceptable shorter proofs
Shorter proofs are welcome, and any shorter proof we accept
will be acknowledged in the theorem's description. However,
in some cases a proof may be "shorter" or not depending on
how it is formatted. This section provides general guidelines.
Usually we automatically accept shorter proofs that (1)
shorten the set.mm file (with compressed proofs), (2) reduce
the size of the HTML file generated with SHOW STATEMENT xx
/ HTML, (3) use only existing, unmodified theorems in the
database (the order of theorems may be changed, though), and
(4) use no additional axioms.
Usually we will also automatically accept a _new_ theorem
that is used to shorten multiple proofs, if the total size
of set.mm (including the comment of the new theorem, not
including the acknowledgment) decreases as a result.
In borderline cases, we typically place more importance on
the number of compressed proof steps and less on the length
of the label section (since the names are in principle
arbitrary). If two proofs have the same number of compressed
proof steps, we will typically give preference to the one
with the smaller number of different labels, or if these
numbers are the same, the proof with the fewest number of
characters that the proofs happen to have by chance when
label lengths are included.
A few theorems have a longer proof than necessary in order
to avoid the use of certain axioms, for pedagogical purposes,
and for other reasons. These theorems will (or should) have
a "(Proof modification is discouraged.)" tag in their
description. For example, idALT 23 shows a proof directly from
axioms. Shorter proofs for such cases won't be accepted,
of course, unless the criteria described continues to be
satisfied.
 Input format.
The input is in ASCII with twospace indents. Tab characters are not
allowed. Use embedded math comments or HTML entities for nonASCII
characters (e.g., "é" for "é").
 Information on syntax, axioms, and definitions.
For a hyperlinked list of syntax, axioms, and definitions, see
mmdefinitions.html.
If you have questions about a specific symbol or axiom, it is best
to go directly to its definition to learn more about it.
The generated HTML for each theorem and axiom includes hypertext
links to each symbol's definition.
 Reserved symbols: 'LETTER.
Some symbols are reserved for potential future use.
Symbols with the pattern 'LETTER are reserved for possibly
representing characters (this is somewhat similar to Lisp).
We would expect '\n to represent newline, 'sp for space, and perhaps
'\x24 for the dollar character.
 Language and spelling.
It is preferred to use American English for comments and symbols, e.g.
we use "neighborhood" instead of the British English "neighbourhood".
An exception is the word "analog", which can be either a noun or an
adjective. Furthermore, "analog" has the confounding meaning "not
digital", whereas "analogue" is often used in the sense something that
bears analogy to something else also in American English. Therefore,
"analogue" is used for the noun and "analogous" for the adjective in
set.mm.
 Comments and layout.
As for formatting of the file set.mm, and in particular formatting and
layout of the comments, the foremost rule is consistency. The first
sections of set.mm, in particular Part 1 "Classical firstorder logic
with equality" can serve as a model for contributors. Some formatting
rules are enforced when using the Metamath program's "WRITE SOURCE"
command with the "REWRAP" option. Here are a few other rules, which are
not enforced, but that we try follow:

The file set.mm should have a double blank line before each section
header, and at no other places. In particular, there are no triple
blank lines. If there is a "@( Begin $[ ... $] @)" comment (where "@"
is actually "$") before the section header, then the double blank line
should go before that comment.

The header comments should be spaced as those of Part 1, namely, with
a blank line before and after the comment, and an indentation of two
spaces.

Header comments are not rewrapped by the Metamath program [as of
24Oct2021], but similar spacing and wrapping should be used as for
other comments: double spaces after a period ending a sentence, line
wrapping with line width of 79, and no trailing spaces at the end of
lines.
The challenge of varying mathematical conventions
We try to follow mathematical conventions, but in many cases
different texts use different conventions.
In those cases we pick some reasonably common convention and stick to
it.
We have already mentioned that the term "natural number" has
varying definitions (some start from 0, others start from 1), but
that is not the only such case.
A useful example is the set of metavariables used to represent
arbitrary wellformed formulas (wffs).
We use an open phi, φ, to represent the first arbitrary wff in an
assertion with one or more wffs; this is a common convention and
this symbol is easily distinguished from the empty set symbol.
That said, it is impossible to please everyone or simply "follow
the literature" because there are many different conventions for
a variable that represents any arbitrary wff.
To demonstrate the point,
here are some conventions for variables that represent an arbitrary
wff and some texts that use each convention:
 open phi φ (and so on): Tarski's papers,
Rasiowa & Sikorski's
The Mathematics of Metamathematics (1963),
Monk's Introduction to Set Theory (1969),
Enderton's Elements of Set Theory (1977),
Bell & Machover's A Course in Mathematical Logic (1977),
Jech's Set Theory (1978),
Takeuti & Zaring's
Introduction to Axiomatic Set Theory (1982).
 closed phi ϕ (and so on):
Levy's Basic Set Theory (1979),
Kunen's Set Theory (1980),
Paulson's Isabelle: A Generic Theorem Prover (1994),
Huth and Ryan's Logic in Computer Science (2004/2006).
 Greek α, β, γ:
Duffy's Principles of Automated Theorem Proving (1991).
 Roman A, B, C:
Kleene's Introduction to Metamathematics (1974),
Smullyan's FirstOrder Logic (1968/1995).
 script A, B, C:
Hamilton's Logic for Mathematicians (1988).
 italic A, B, C:
Mendelson's Introduction to Mathematical Logic (1997).
 italic P, Q, R:
Suppes's Axiomatic Set Theory (1972),
Gries and Schneider's A Logical Approach to Discrete Math
(1993/1994),
Rosser's Logic for Mathematicians (2008).
 italic p, q, r:
Quine's Set Theory and Its Logic (1969),
Kuratowski & Mostowski's Set Theory (1976).
 italic X, Y, Z:
Dijkstra and Scholten's
Predicate Calculus and Program Semantics (1990).
 Fraktur letters:
Fraenkel et. al's Foundations of Set Theory (1973).
Distinctness or freeness
Here are some conventions that address distinctness or freeness of a
variable:
 Ⅎ𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑";
see dfnf 1707 (whose description has some important technical
details). Similarly, Ⅎ𝑥𝐴 is read 𝑥 is not free in (class)
𝐴, see dfnfc 2750.
 "$d x y $." should be read "Assume x and y are distinct
variables."
 "$d x 𝜑 $." should be read "Assume x does not occur in phi $."
Sometimes a theorem is proved using
Ⅎ𝑥𝜑 (dfnf 1707) in place of
"$d 𝑥𝜑 $." when a more general result is desired;
ax5 1836 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2477 from dfeu 2473.
 "$d x A $." should be read "Assume x is not a variable occurring in
class A."
 "$d x A $. $d x ps $. $e  (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) $."
is an idiom
often used instead of explicit substitution, meaning "Assume psi results
from the proper substitution of A for x in phi."
 " ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ..." occurs early in some cases, and
should be read "If x and y are distinct
variables, then..." This antecedent provides us with a technical
device (called a "distinctor" in Section 7 of [Megill] p. 444)
to avoid the need for the
$d statement early in our development of predicate calculus, permitting
unrestricted substitutions as conceptually simple as those in
propositional calculus. However, the $d eventually becomes a
requirement, and after that this device is rarely used.
There is a general technique to replace a $d x A or
$d x ph condition in a theorem with the corresponding
Ⅎ𝑥𝐴 or Ⅎ𝑥𝜑; here it is.
⊢ T[x, A] where ,
and you wish to prove ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ T[x, A].
You apply the theorem substituting 𝑦 for 𝑥 and 𝐴 for 𝐴,
where 𝑦 is a new dummy variable, so that
$d y A is satisfied.
You obtain ⊢ T[y, A], and apply chvar to obtain ⊢
T[x, A] (or just use mpbir 221 if T[x, A] binds 𝑥).
The side goal is ⊢ (𝑥 = 𝑦 → ( T[y, A] ↔ T[x, A] )),
where you can use equality theorems, except
that when you get to a bound variable you use a nondv bound variable
renamer theorem like cbval 2270. The section
mmtheorems32.html#mm3146s also describes the
metatheorem that underlies this.
Standard Metamath verifiers do not distinguish between axioms and
definitions (both are $a statements).
In practice, we require that definitions (1) be conservative
(a definition should not allow an expression
that previously qualified as a wff but was not provable
to become provable) and be eliminable
(there should exist an algorithmic method for converting any
expression using the definition into
a logically equivalent expression that previously qualified as a wff).
To ensure this, we have additional rules on almost all definitions
($a statements with a label that does not begin with ax).
These additional rules are not applied in a few cases where they
are too strict (dfbi 197, dfclab 2608, dfcleq 2614, and dfclel 2617);
see those definitions for more information.
These additional rules for definitions are checked by at least
mmj2's definition check (see
mmj2 master file mmj2jar/macros/definitionCheck.js).
This definition check relies on the database being very much like
set.mm, down to the names of certain constants and types, so it
cannot apply to all Metamath databases... but it is useful in set.mm.
In this definition check, a $astatement with a given label and
typecode ⊢ passes the test if and only if it
respects the following rules (these rules require that we have
an unambiguous tree parse, which is checked separately):
 The expression must be a biconditional or an equality (i.e. its
rootsymbol must be ↔ or =).
If the proposed definition passes this first rule, we then
define its definiendum as its left hand side (LHS) and
its definiens as its right hand side (RHS).
We define the *defined symbol* as the rootsymbol of the LHS.
We define a *dummy variable* as a variable occurring
in the RHS but not in the LHS.
Note that the "rootsymbol" is the root of the considered tree;
it need not correspond to a single token in the database
(e.g., see w3o 1035 or wsb 1877).
 The defined expression must not appear in any statement
between its syntax axiom () and its definition,
and the defined expression must not be used in its definiens.
See df3an 1038 for an example where the same symbol is used in
different ways (this is allowed).
 No two variables occurring in the LHS may share a
disjoint variable (DV) condition.
 All dummy variables are required to be disjoint from any
other (dummy or not) variable occurring in this labeled expression.
 Either
(a) there must be no nonsetvar dummy variables, or
(b) there must be a justification theorem.
The justification theorem must be of form
⊢ ( definiens rootsymbol definiens' )
where definiens' is definiens but the dummy variables are all
replaced with other unused dummy variables of the same type.
Note that rootsymbol is ↔ or =, and that setvar
variables are simply variables with the setvar typecode.
 One of the following must be true:
(a) there must be no setvar dummy variables,
(b) there must be a justification theorem as described in rule 5, or
(c) if there are setvar dummy variables, every one must not be free.
That is, it must be true that
(𝜑 → ∀𝑥𝜑) for each setvar dummy variable 𝑥
where 𝜑 is the definiens.
We use two different tests for nonfreeness; one must succeed
for each setvar dummy variable 𝑥.
The first test requires that the setvar dummy variable 𝑥
be syntactically bound
(this is sometimes called the "fast" test, and this implies
that we must track binding operators).
The second test requires a successful
search for the directlystated proof of (𝜑 → ∀𝑥𝜑)
Part c of this rule is how most setvar dummy variables
are handled.
Rule 3 may seem unnecessary, but it is needed.
Without this rule, you can define something like
cbar $a wff Foo x y $.
${ $d x y $. dffoo $a  ( Foo x y <> x = y ) $. $}
and now "Foo x x" is not eliminable;
there is no way to prove that it means anything in particular,
because the definitional theorem that is supposed to be
responsible for connecting it to the original language wants
nothing to do with this expression, even though it is well formed.
A justification theorem for a definition (if used this way)
must be proven before the definition that depends on it.
One example of a justification theorem is vjust 3191.
The definition dfv 3192 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} is justified
by the justification theorem vjust 3191
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}.
Another example of a justification theorem is trujust 1482;
the definition dftru 1483 ⊢ (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥))
is justified by trujust 1482 ⊢ ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).
Here is more information about our processes for checking and
contributing to this work:
 Multiple verifiers.
This entire file is verified by multiple independentlyimplemented
verifiers when it is checked in, giving us extremely high
confidence that all proofs follow from the assumptions.
The checkers also check for various other problems such as
overly long lines.
 Maximum text line length is 79 characters.
You can fix comment line length by running the commands scripts/rewrap
or metamath 'read set.mm' 'save proof */c/f'
'write source set.mm/rewrap' quit .
As a general rule, a math string in a comment should be surrounded
by backquotes on the same line, and if it is too long it should
be broken into multiple adjacent mathstrings on multiple lines.
Those commands don't modify the math content of statements.
In statements we try to break before the outermost important connective
(not including the typecode and perhaps not the antecedent).
For examples, see sqrtmulii 14076 and absmax 14019.
 Discouraged information.
A separate file named "discouraged" lists all
discouraged statements and uses of them, and this file is checked.
If you change the use of discouraged things, you will need to change
this file.
This makes it obvious when there is a change to anything discouraged
(triggering further review).
 LRParser check.
Metamath verifiers ensure that $p statements follow from previous
$a and $p statements.
However, by itself the Metamath language permits certain kinds of
syntactic ambiguity that we choose to avoid in this database.
Thus, we require that this database unambiguously parse
using the "LRParser" check (implemented by at least mmj2).
(For details, see mmj2 master file src/mmj/verify/LRParser.java).
This check
counters, for example, a devious ambiguous construct
developed by saueran at oregonstate dot edu
posted on Mon, 11 Feb 2019 17:32:32 0800 (PST)
based on creating definitions with mismatched parentheses.
 Proposing specific changes.
Please propose specific changes as pull requests (PRs) against the
"develop" branch of set.mm, at:
https://github.com/metamath/set.mm/tree/develop
 Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/forum/#!forum/metamath.
(Contributed by DAW, 27Dec2016.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

Theorem  conventionslabel 27147 
The following explains some of the label conventions in use
in the Metamath Proof Explorer ("set.mm").
For the general conventions, see conventions 27146.
Every statement has a unique identifying label, which serves the
same purpose as an equation number in a book.
We use various label naming conventions to provide
easytoremember hints about their contents.
Labels are not a 1to1 mapping, because that would create
long names that would be difficult to remember and tedious to type.
Instead, label names are relatively short while
suggesting their purpose.
Names are occasionally changed to make them more consistent or
as we find better ways to name them.
Here are a few of the label naming conventions:
 Axioms, definitions, and wff syntax.
As noted earlier, axioms are named "axNAME",
proofs of proven axioms are named "axNAME", and
definitions are named "dfNAME".
Wff syntax declarations have labels beginning with "w"
followed by short fragment suggesting its purpose.
 Hypotheses.
Hypotheses have the name of the final axiom or theorem, followed by
".", followed by a unique id (these ids are usually consecutive integers
starting with 1, e.g. for rgen 2918"rgen.1 $e  ( x e. A > ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 20352: "mdet0.d $e  D = ( N maDet R ) $.").
 Common names.
If a theorem has a wellknown name, that name (or a short version of it)
is sometimes used directly. Examples include
barbara 2562 and stirling 39643.
 Principia Mathematica.
Proofs of theorems from Principia Mathematica often use a special
naming convention: "pm" followed by its identifier.
For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named
pm2.27 42.
 19.x series of theorems.
Similar to the conventions for the theorems from Principia Mathematica,
theorems from Section 19 of [Margaris] p. 90 often use a special naming
convention: "19." resp. "r19." (for corresponding restricted quantifier
versions) followed by its identifier.
For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled
19.38 1763, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 2952.
 Characters to be used for labels
Although the specification of Metamath allows for dots/periods "." in
any label, it is usually used only in labels for hypotheses (see above).
Exceptions are the labels of theorems from Principia Mathematica and the
19.x series of theorems from Section 19 of [Margaris] p. 90 (see above)
and 0.999... 14556. Furthermore, the underscore "_" should not be used.
 Syntax label fragments.
Most theorems are named using a concatenation of syntax label fragments
(omitting variables) that represent the important part of the theorem's
main conclusion. Almost every syntactic construct has a definition
labeled "dfNAME", and normally NAME is the syntax label fragment. For
example, the class difference construct (𝐴 ∖ 𝐵) is defined in
dfdif 3563, and thus its syntax label fragment is "dif". Similarly, the
subclass relation 𝐴 ⊆ 𝐵 has syntax label fragment "ss"
because it is defined in dfss 3574. Most theorem names follow from
these fragments, for example, the theorem proving (𝐴 ∖ 𝐵) ⊆ 𝐴
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 3721. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in dfsn 4156), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from dfpr 4158). Digits are used to represent
themselves. Suffixes (e.g., with numbers) are sometimes used to
distinguish multiple theorems that would otherwise produce the same
label.
 Phantom definitions.
In some cases there are common label fragments for something that could
be in a definition, but for technical reasons is not. The iselementof
(is member of) construct 𝐴 ∈ 𝐵 does not have a dfNAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses iselementof
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4294. An "n" is often used for negation (¬), e.g.,
nan 603.
 Exceptions.
Sometimes there is a definition dfNAME but the label fragment is not
the NAME part. The definition should note this exception as part of its
definition. In addition, the table below attempts to list all such
cases and marks them in bold. For example, the label fragment "cn"
represents complex numbers ℂ (even though its definition is in
dfc 9902) and "re" represents real numbers ℝ ( definition dfr 9906).
The empty set ∅ often uses fragment 0, even though it is defined
in dfnul 3898. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with dfadd 9907), but "p" is used as
the fragment for constant theorems. Equality (𝐴 = 𝐵) often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 11104.
 Other markings.
In labels we sometimes use "com" for "commutative", "ass" for
"associative", "rot" for "rotation", and "di" for "distributive".
 Focus on the important part of the conclusion.
Typically the conclusion is the part the user is most interested in.
So, a rough guideline is that a label typically provides a hint
about only the conclusion; a label rarely says anything about the
hypotheses or antecedents.
If there are multiple theorems with the same conclusion
but different hypotheses/antecedents, then the labels will need
to differ; those label differences should emphasize what is different.
There is no need to always fully describe the conclusion; just
identify the important part. For example,
cos0 14824 is the theorem that provides the value for the cosine of 0;
we would need to look at the theorem itself to see what that value is.
The label "cos0" is concise and we use it instead of "cos0eq1".
There is no need to add the "eq1", because there will never be a case
where we have to disambiguate between different values produced by
the cosine of zero, and we generally prefer shorter labels if
they are unambiguous.
 Closures and values.
As noted above, if a function dfNAME is defined, there is typically a
proof of its value labeled "NAMEval" and of its closure labeld "NAMEcl".
E.g., for cosine (dfcos 14745) we have value cosval 14797 and closure
coscl 14801.
 Special cases.
Sometimes, syntax and related markings are insufficient to distinguish
different theorems. For example, there are over a hundred different
implicationonly theorems. They are grouped in a more adhoc way that
attempts to make their distinctions clearer. These often use
abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and
"id" for "identity". It is especially hard to give good names in the
propositional calculus section because there are so few primitives.
However, in most cases this is not a serious problem. There are a few
very common theorems like axmp 5 and syl 17 that you will have no
trouble remembering, a few theorem series like syl*anc and simp* that
you can use parametrically, and a few other useful glue things for
destructuring 'and's and 'or's (see natded 27148 for a list), and that is
about all you need for most things. As for the rest, you can just
assume that if it involves at most three connectives, then it is
probably already proved in set.mm, and searching for it will give you
the label.
 Suffixes.
Suffixes are used to indicate the form of a theorem (see above).
Additionally, we sometimes suffix with "v" the label of a theorem
eliminating a hypothesis such as Ⅎ𝑥𝜑 in 19.21 2073 via the use of
disjoint variable conditions combined with nfv 1840. If two (or three)
such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used,
e.g. exlimivv 1857.
Conversely, we sometimes suffix with "f" the label of a theorem
introducing such a hypothesis to eliminate the need for the disjoint
variable condition; e.g. euf 2477 derived from dfeu 2473. The "f" stands
for "not free in" which is less restrictive than "does not occur in."
The suffix "b" often means "biconditional" (↔, "iff" , "if and
only if"), e.g. sspwb 4888.
We sometimes suffix with "s" the label of an inference that manipulates
an antecedent, leaving the consequent unchanged. The "s" means that the
inference eliminates the need for a syllogism (syl 17) type inference
in a proof. A theorem label is suffixed with "ALT" if it provides an
alternate lesspreferred proof of a theorem (e.g., the proof is
clearer but uses more axioms than the preferred version).
The "ALT" may be further suffixed with a number if there is more
than one alternate theorem.
Furthermore, a theorem label is suffixed with "OLD" if there is a new
version of it and the OLD version is obsolete (and will be removed
within one year).
Finally, it should be mentioned that suffixes can be combined, for
example in cbvaldva 2280 (cbval 2270 in deduction form "d" with a not free
variable replaced by a disjoint variable condition "v" with a
conjunction as antecedent "a").
Here is a nonexhaustive list of common suffixes:
 a : theorem having a conjunction as antecedent
 b : theorem expressing a logical equivalence
 c : contraction (e.g., sylc 65, syl2anc 692), commutes
(e.g., biimpac 503)
 d : theorem in deduction form
 f : theorem with a hypothesis such as Ⅎ𝑥𝜑
 g : theorem in closed form having an "is a set" antecedent
 i : theorem in inference form
 l : theorem concerning something at the left
 r : theorem concerning something at the right
 r : theorem with something reversed (e.g., a biconditional)
 s : inference that manipulates an antecedent ("s" refers to an
application of syl 17 that is eliminated)
 v : theorem with one (main) disjoint variable condition
 vv : theorem with two (main) disjoint variable conditions
 w : weak(er) form of a theorem
 ALT : alternate proof of a theorem
 ALTV : alternate version of a theorem or definition
 OLD : old/obsolete version of a theorem/definition/proof
 Reuse.
When creating a new theorem or axiom, try to reuse abbreviations used
elsewhere. A comment should explain the first use of an abbreviation.
The following table shows some commonly used abbreviations in labels, in
alphabetical order. For each abbreviation we provide a mnenomic, the
source theorem or the assumption defining it, an expression showing what
it looks like, whether or not it is a "syntax fragment" (an abbreviation
that indicates a particular kind of syntax), and hyperlinks to label
examples that use the abbreviation. The abbreviation is bolded if there
is a dfNAME definition but the label fragment is not NAME. This is
not a complete list of abbreviations, though we do want this to
eventually be a complete list of exceptions.
Abbreviation  Mnenomic  Source 
Expression  Syntax?  Example(s) 
a  and (suffix)  
 No  biimpa 501, rexlimiva 3023 
abl  Abelian group  dfabl 18136 
Abel  Yes  ablgrp 18138, zringabl 19762 
abs  absorption    No 
ressabs 15879 
abs  absolute value (of a complex number) 
dfabs 13926  (abs‘𝐴)  Yes 
absval 13928, absneg 13967, abs1 13987 
ad  adding  
 No  adantr 481, ad2antlr 762 
add  add (see "p")  dfadd 9907 
(𝐴 + 𝐵)  Yes 
addcl 9978, addcom 10182, addass 9983 
al  "for all"  
∀𝑥𝜑  No  alim 1735, alex 1750 
ALT  alternative/less preferred (suffix)  
 No  idALT 23 
an  and  dfan 386 
(𝜑 ∧ 𝜓)  Yes 
anor 510, iman 440, imnan 438 
ant  antecedent  
 No  adantr 481 
ass  associative  
 No  biass 374, orass 546, mulass 9984 
asym  asymmetric, antisymmetric  
 No  intasym 5480, asymref 5481, posasymb 16892 
ax  axiom  
 No  ax6dgen 2002, ax1cn 9930 
bas, base 
base (set of an extensible structure)  dfbase 15805 
(Base‘𝑆)  Yes 
baseval 15858, ressbas 15870, cnfldbas 19690 
b, bi  biconditional ("iff", "if and only if")
 dfbi 197  (𝜑 ↔ 𝜓)  Yes 
impbid 202, sspwb 4888 
br  binary relation  dfbr 4624 
𝐴𝑅𝐵  Yes  brab1 4670, brun 4673 
cbv  change bound variable   
No  cbvalivw 1931, cbvrex 3160 
cl  closure    No 
ifclda 4098, ovrcl 6651, zaddcl 11377 
cn  complex numbers  dfc 9902 
ℂ  Yes  nnsscn 10985, nncn 10988 
cnfld  field of complex numbers  dfcnfld 19687 
ℂ_{fld}  Yes  cnfldbas 19690, cnfldinv 19717 
cntz  centralizer  dfcntz 17690 
(Cntz‘𝑀)  Yes 
cntzfval 17693, dprdfcntz 18354 
cnv  converse  dfcnv 5092 
^{◡}𝐴  Yes  opelcnvg 5272, f1ocnv 6116 
co  composition  dfco 5093 
(𝐴 ∘ 𝐵)  Yes  cnvco 5278, fmptco 6362 
com  commutative  
 No  orcom 402, bicomi 214, eqcomi 2630 
con  contradiction, contraposition  
 No  condan 834, con2d 129 
csb  class substitution  dfcsb 3520 
⦋𝐴 / 𝑥⦌𝐵  Yes 
csbid 3527, csbie2g 3550 
cyg  cyclic group  dfcyg 18220 
CycGrp  Yes 
iscyg 18221, zringcyg 19779 
d  deduction form (suffix)  
 No  idd 24, impbid 202 
df  (alternate) definition (prefix)  
 No  dfrel2 5552, dffn2 6014 
di, distr  distributive  
 No 
andi 910, imdi 378, ordi 907, difindi 3863, ndmovdistr 6788 
dif  class difference  dfdif 3563 
(𝐴 ∖ 𝐵)  Yes 
difss 3721, difindi 3863 
div  division  dfdiv 10645 
(𝐴 / 𝐵)  Yes 
divcl 10651, divval 10647, divmul 10648 
dm  domain  dfdm 5094 
dom 𝐴  Yes  dmmpt 5599, iswrddm0 13284 
e, eq, equ  equals  dfcleq 2614 
𝐴 = 𝐵  Yes 
2p2e4 11104, uneqri 3739, equtr 1945 
edg  edge  dfedg 25874 
(Edg‘𝐺)  Yes 
edgopval 25876, usgredgppr 26015 
el  element of  
𝐴 ∈ 𝐵  Yes 
eldif 3570, eldifsn 4294, elssuni 4440 
eu  "there exists exactly one"  dfeu 2473 
∃!𝑥𝜑  Yes  euex 2493, euabsn 4238 
ex  exists (i.e. is a set)  
 No  brrelex 5126, 0ex 4760 
ex  "there exists (at least one)"  dfex 1702 
∃𝑥𝜑  Yes  exim 1758, alex 1750 
exp  export  
 No  expt 168, expcom 451 
f  "not free in" (suffix)  
 No  equs45f 2349, sbf 2379 
f  function  dff 5861 
𝐹:𝐴⟶𝐵  Yes  fssxp 6027, opelf 6032 
fal  false  dffal 1486 
⊥  Yes  bifal 1494, falantru 1505 
fi  finite intersection  dffi 8277 
(fi‘𝐵)  Yes  fival 8278, inelfi 8284 
fi, fin  finite  dffin 7919 
Fin  Yes 
isfi 7939, snfi 7998, onfin 8111 
fld  field (Note: there is an alternative
definition Fld of a field, see dffld 33462)  dffield 18690 
Field  Yes  isfld 18696, fldidom 19245 
fn  function with domain  dffn 5860 
𝐴 Fn 𝐵  Yes  ffn 6012, fndm 5958 
frgp  free group  dffrgp 18063 
(freeGrp‘𝐼)  Yes 
frgpval 18111, frgpadd 18116 
fsupp  finitely supported function 
dffsupp 8236  𝑅 finSupp 𝑍  Yes 
isfsupp 8239, fdmfisuppfi 8244, fsuppco 8267 
fun  function  dffun 5859 
Fun 𝐹  Yes  funrel 5874, ffun 6015 
fv  function value  dffv 5865 
(𝐹‘𝐴)  Yes  fvres 6174, swrdfv 13378 
fz  finite set of sequential integers 
dffz 12285 
(𝑀...𝑁)  Yes  fzval 12286, eluzfz 12295 
fz0  finite set of sequential nonnegative integers 

(0...𝑁)  Yes  nn0fz0 12394, fz0tp 12397 
fzo  halfopen integer range  dffzo 12423 
(𝑀..^𝑁)  Yes 
elfzo 12429, elfzofz 12442 
g  more general (suffix); eliminates "is a set"
hypothsis  
 No  uniexg 6920 
gr  graph  
 No  uhgrf 25887, isumgr 25919, usgrres1 26129 
grp  group  dfgrp 17365 
Grp  Yes  isgrp 17368, tgpgrp 21822 
gsum  group sum  dfgsum 16043 
(𝐺 Σ_{g} 𝐹)  Yes 
gsumval 17211, gsumwrev 17736 
hash  size (of a set)  dfhash 13074 
(#‘𝐴)  Yes 
hashgval 13076, hashfz1 13090, hashcl 13103 
hb  hypothesis builder (prefix)  
 No  hbxfrbi 1749, hbald 2038, hbequid 33713 
hm  (monoid, group, ring) homomorphism  
 No  ismhm 17277, isghm 17600, isrhm 18661 
i  inference (suffix)  
 No  eleq1i 2689, tcsni 8579 
i  implication (suffix)  
 No  brwdomi 8433, infeq5i 8493 
id  identity  
 No  biid 251 
iedg  indexed edge  dfiedg 25811 
(iEdg‘𝐺)  Yes 
iedgval0 25866, edgiedgb 25879 
idm  idempotent  
 No  anidm 675, tpidm13 4268 
im, imp  implication (label often omitted) 
dfim 13791  (𝐴 → 𝐵)  Yes 
iman 440, imnan 438, impbidd 200 
ima  image  dfima 5097 
(𝐴 “ 𝐵)  Yes  resima 5400, imaundi 5514 
imp  import  
 No  biimpa 501, impcom 446 
in  intersection  dfin 3567 
(𝐴 ∩ 𝐵)  Yes  elin 3780, incom 3789 
inf  infimum  dfinf 8309 
inf(ℝ^{+}, ℝ^{*}, < )  Yes 
fiinfcl 8367, infiso 8373 
is...  is (something a) ...?  
 No  isring 18491 
j  joining, disjoining  
 No  jc 159, jaoi 394 
l  left  
 No  olcd 408, simpl 473 
map  mapping operation or set exponentiation 
dfmap 7819  (𝐴 ↑_{𝑚} 𝐵)  Yes 
mapvalg 7827, elmapex 7838 
mat  matrix  dfmat 20154 
(𝑁 Mat 𝑅)  Yes 
matval 20157, matring 20189 
mdet  determinant (of a square matrix) 
dfmdet 20331  (𝑁 maDet 𝑅)  Yes 
mdetleib 20333, mdetrlin 20348 
mgm  magma  dfmgm 17182 
Magma  Yes 
mgmidmo 17199, mgmlrid 17206, ismgm 17183 
mgp  multiplicative group  dfmgp 18430 
(mulGrp‘𝑅)  Yes 
mgpress 18440, ringmgp 18493 
mnd  monoid  dfmnd 17235 
Mnd  Yes  mndass 17242, mndodcong 17901 
mo  "there exists at most one"  dfmo 2474 
∃*𝑥𝜑  Yes  eumo 2498, moim 2518 
mp  modus ponens  axmp 5 
 No  mpd 15, mpi 20 
mpt  modus ponendo tollens  
 No  mptnan 1690, mptxor 1691 
mpt  mapsto notation for a function 
dfmpt 4685  (𝑥 ∈ 𝐴 ↦ 𝐵)  Yes 
fconstmpt 5133, resmpt 5418 
mpt2  mapsto notation for an operation 
dfmpt2 6620  (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)  Yes 
mpt2mpt 6717, resmpt2 6723 
mul  multiplication (see "t")  dfmul 9908 
(𝐴 · 𝐵)  Yes 
mulcl 9980, divmul 10648, mulcom 9982, mulass 9984 
n, not  not  
¬ 𝜑  Yes 
nan 603, notnotr 125 
ne  not equal  dfne  𝐴 ≠ 𝐵 
Yes  exmidne 2800, neeqtrd 2859 
nel  not element of  dfnel  𝐴 ∉ 𝐵

Yes  neli 2895, nnel 2902 
ne0  not equal to zero (see n0)  
≠ 0  No 
negne0d 10350, ine0 10425, gt0ne0 10453 
nf  "not free in" (prefix)  
 No  nfnd 1782 
ngp  normed group  dfngp 22328 
NrmGrp  Yes  isngp 22340, ngptps 22346 
nm  norm (on a group or ring)  dfnm 22327 
(norm‘𝑊)  Yes 
nmval 22334, subgnm 22377 
nn  positive integers  dfnn 10981 
ℕ  Yes  nnsscn 10985, nncn 10988 
nn0  nonnegative integers  dfn0 11253 
ℕ_{0}  Yes  nnnn0 11259, nn0cn 11262 
n0  not the empty set (see ne0)  
≠ ∅  No  n0i 3902, vn0 3906, ssn0 3954 
OLD  old, obsolete (to be removed soon)  
 No  19.43OLD 1808 
op  ordered pair  dfop 4162 
⟨𝐴, 𝐵⟩  Yes  dfopif 4374, opth 4915 
or  or  dfor 385 
(𝜑 ∨ 𝜓)  Yes 
orcom 402, anor 510 
ot  ordered triple  dfot 4164 
⟨𝐴, 𝐵, 𝐶⟩  Yes 
euotd 4945, fnotovb 6659 
ov  operation value  dfov 6618 
(𝐴𝐹𝐵)  Yes
 fnotovb 6659, fnovrn 6774 
p  plus (see "add"), for allconstant
theorems  dfadd 9907 
(3 + 2) = 5  Yes 
3p2e5 11120 
pfx  prefix  dfpfx 40711 
(𝑊 prefix 𝐿)  Yes 
pfxlen 40720, ccatpfx 40738 
pm  Principia Mathematica  
 No  pm2.27 42 
pm  partial mapping (operation)  dfpm 7820 
(𝐴 ↑_{pm} 𝐵)  Yes  elpmi 7836, pmsspw 7852 
pr  pair  dfpr 4158 
{𝐴, 𝐵}  Yes 
elpr 4176, prcom 4244, prid1g 4272, prnz 4287 
prm, prime  prime (number)  dfprm 15329 
ℙ  Yes  1nprm 15335, dvdsprime 15343 
pss  proper subset  dfpss 3576 
𝐴 ⊊ 𝐵  Yes  pssss 3686, sspsstri 3693 
q  rational numbers ("quotients")  dfq 11749 
ℚ  Yes  elq 11750 
r  right  
 No  orcd 407, simprl 793 
rab  restricted class abstraction 
dfrab 2917  {𝑥 ∈ 𝐴 ∣ 𝜑}  Yes 
rabswap 3114, dfoprab 6619 
ral  restricted universal quantification 
dfral 2913  ∀𝑥 ∈ 𝐴𝜑  Yes 
ralnex 2988, ralrnmpt2 6740 
rcl  reverse closure  
 No  ndmfvrcl 6186, nnarcl 7656 
re  real numbers  dfr 9906 
ℝ  Yes  recn 9986, 0re 10000 
rel  relation  dfrel 5091  Rel 𝐴 
Yes  brrelex 5126, relmpt2opab 7219 
res  restriction  dfres 5096 
(𝐴 ↾ 𝐵)  Yes 
opelres 5371, f1ores 6118 
reu  restricted existential uniqueness 
dfreu 2915  ∃!𝑥 ∈ 𝐴𝜑  Yes 
nfreud 3106, reurex 3153 
rex  restricted existential quantification 
dfrex 2914  ∃𝑥 ∈ 𝐴𝜑  Yes 
rexnal 2991, rexrnmpt2 6741 
rmo  restricted "at most one" 
dfrmo 2916  ∃*𝑥 ∈ 𝐴𝜑  Yes 
nfrmod 3107, nrexrmo 3156 
rn  range  dfrn 5095  ran 𝐴 
Yes  elrng 5284, rncnvcnv 5319 
rng  (unital) ring  dfring 18489 
Ring  Yes 
ringidval 18443, isring 18491, ringgrp 18492 
rot  rotation  
 No  3anrot 1041, 3orrot 1042 
s  eliminates need for syllogism (suffix) 
  No  ancoms 469 
sb  (proper) substitution (of a set) 
dfsb 1878  [𝑦 / 𝑥]𝜑  Yes 
spsbe 1881, sbimi 1883 
sbc  (proper) substitution of a class 
dfsbc 3423  [𝐴 / 𝑥]𝜑  Yes 
sbc2or 3431, sbcth 3437 
sca  scalar  dfsca 15897 
(Scalar‘𝐻)  Yes 
resssca 15971, mgpsca 18436 
simp  simple, simplification  
 No  simpl 473, simp3r3 1169 
sn  singleton  dfsn 4156 
{𝐴}  Yes  eldifsn 4294 
sp  specialization  
 No  spsbe 1881, spei 2260 
ss  subset  dfss 3574 
𝐴 ⊆ 𝐵  Yes  difss 3721 
struct  structure  dfstruct 15802 
Struct  Yes  brstruct 15808, structfn 15816 
sub  subtract  dfsub 10228 
(𝐴 − 𝐵)  Yes 
subval 10232, subaddi 10328 
sup  supremum  dfsup 8308 
sup(𝐴, 𝐵, < )  Yes 
fisupcl 8335, supmo 8318 
supp  support (of a function)  dfsupp 7256 
(𝐹 supp 𝑍)  Yes 
ressuppfi 8261, mptsuppd 7278 
swap  swap (two parts within a theorem) 
  No  rabswap 3114, 2reuswap 3397 
syl  syllogism  syl 17 
 No  3syl 18 
sym  symmetric  
 No  dfsymdif 3828, cnvsym 5479 
symg  symmetric group  dfsymg 17738 
(SymGrp‘𝐴)  Yes 
symghash 17745, pgrpsubgsymg 17768 
t 
times (see "mul"), for allconstant theorems 
dfmul 9908 
(3 · 2) = 6  Yes 
3t2e6 11139 
th  theorem  
 No  nfth 1724, sbcth 3437, weth 9277 
tp  triple  dftp 4160 
{𝐴, 𝐵, 𝐶}  Yes 
eltpi 4207, tpeq1 4254 
tr  transitive  
 No  bitrd 268, biantr 971 
tru  true  dftru 1483 
⊤  Yes  bitru 1493, truanfal 1504 
un  union  dfun 3565 
(𝐴 ∪ 𝐵)  Yes 
uneqri 3739, uncom 3741 
unit  unit (in a ring) 
dfunit 18582  (Unit‘𝑅)  Yes 
isunit 18597, nzrunit 19207 
v  disjoint variable conditions used when
a notfree hypothesis (suffix) 
  No  spimv 2256 
vtx  vertex  dfvtx 25810 
(Vtx‘𝐺)  Yes 
vtxval0 25865, opvtxov 25819 
vv  2 disjoint variables (in a notfree hypothesis)
(suffix)    No  19.23vv 1900 
w  weak (version of a theorem) (suffix)  
 No  ax11w 2004, spnfw 1925 
wrd  word 
dfword 13254  Word 𝑆  Yes 
iswrdb 13266, wrdfn 13274, ffz0iswrd 13287 
xp  cross product (Cartesian product) 
dfxp 5090  (𝐴 × 𝐵)  Yes 
elxp 5101, opelxpi 5118, xpundi 5142 
xr  eXtended reals  dfxr 10038 
ℝ^{*}  Yes  ressxr 10043, rexr 10045, 0xr 10046 
z  integers (from German "Zahlen") 
dfz 11338  ℤ  Yes 
elz 11339, zcn 11342 
zn  ring of integers mod 𝑛  dfzn 19795 
(ℤ/nℤ‘𝑁)  Yes 
znval 19823, zncrng 19833, znhash 19847 
zring  ring of integers  dfzring 19759 
ℤ_{ring}  Yes  zringbas 19764, zringcrng 19760

0, z 
slashed zero (empty set) (see n0)  dfnul 3898 
∅  Yes 
n0i 3902, vn0 3906; snnz 4286, prnz 4287 
(Contributed by DAW, 27Dec2016.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

17.1.2 Natural deduction


Theorem  natded 27148 
Here are typical natural deduction (ND) rules in the style of Gentzen
and Jaśkowski, along with MPE translations of them. This also
shows the recommended theorems when you find yourself needing these
rules (the recommendations encourage a slightly different proof style
that works more naturally with metamath). A decent list of the standard
rules of natural deduction can be found beginning with definition /\I in
[Pfenning] p. 18. For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer. Many more citations could be added.
Name  Natural Deduction Rule  Translation 
Recommendation  Comments 
IT 
Γ⊢ 𝜓 => Γ⊢ 𝜓 
idi 2 
nothing  Reiteration is always redundant in Metamath.
Definition "new rule" in [Pfenning] p. 18,
definition IT in [Clemente] p. 10. 
∧I 
Γ⊢ 𝜓 & Γ⊢ 𝜒 => Γ⊢ 𝜓 ∧ 𝜒 
jca 554 
jca 554, pm3.2i 471 
Definition ∧I in [Pfenning] p. 18,
definition I∧m,n in [Clemente] p. 10, and
definition ∧I in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
∧E_{L} 
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜓 
simpld 475 
simpld 475, adantr 481 
Definition ∧E_{L} in [Pfenning] p. 18,
definition E∧(1) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
∧E_{R} 
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜒 
simprd 479 
simpr 477, adantl 482 
Definition ∧E_{R} in [Pfenning] p. 18,
definition E∧(2) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
→I 
Γ, 𝜓⊢ 𝜒 => Γ⊢ 𝜓 → 𝜒 
ex 450  ex 450 
Definition →I in [Pfenning] p. 18,
definition I=>m,n in [Clemente] p. 11, and
definition →I in [Indrzejczak] p. 33. 
→E 
Γ⊢ 𝜓 → 𝜒 & Γ⊢ 𝜓 => Γ⊢ 𝜒 
mpd 15  axmp 5, mpd 15, mpdan 701, imp 445 
Definition →E in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition →E in [Indrzejczak] p. 33. 
∨I_{L}  Γ⊢ 𝜓 =>
Γ⊢ 𝜓 ∨ 𝜒 
olcd 408 
olc 399, olci 406, olcd 408 
Definition ∨I in [Pfenning] p. 18,
definition I∨n(1) in [Clemente] p. 12 
∨I_{R}  Γ⊢ 𝜒 =>
Γ⊢ 𝜓 ∨ 𝜒 
orcd 407 
orc 400, orci 405, orcd 407 
Definition ∨I_{R} in [Pfenning] p. 18,
definition I∨n(2) in [Clemente] p. 12. 
∨E  Γ⊢ 𝜓 ∨ 𝜒 & Γ, 𝜓⊢ 𝜃 &
Γ, 𝜒⊢ 𝜃 => Γ⊢ 𝜃 
mpjaodan 826 
mpjaodan 826, jaodan 825, jaod 395 
Definition ∨E in [Pfenning] p. 18,
definition E∨m,n,p in [Clemente] p. 12. 
¬I  Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓 
inegd 1500  pm2.01d 181 

¬I  Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 =>
Γ⊢ ¬ 𝜓 
mtand 690  mtand 690 
definition I¬m,n,p in [Clemente] p. 13. 
¬I  Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 =>
Γ⊢ ¬ 𝜓 
pm2.65da 599  pm2.65da 599 
Contradiction. 
¬I 
Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓 
pm2.01da 458  pm2.01d 181, pm2.65da 599, pm2.65d 187 
For an alternative falsumfree natural deduction ruleset 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥ 
pm2.21fal 1502 
pm2.21dd 186  
¬E 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 

pm2.21dd 186 
definition →E in [Indrzejczak] p. 33. 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃 
pm2.21dd 186  pm2.21dd 186, pm2.21d 118, pm2.21 120 
For an alternative falsumfree natural deduction ruleset.
Definition ¬E in [Pfenning] p. 18. 
⊤I  Γ⊢ ⊤ 
a1tru 1497  tru 1484, a1tru 1497, trud 1490 
Definition ⊤I in [Pfenning] p. 18. 
⊥E  Γ, ⊥⊢ 𝜃 
falimd 1496  falim 1495 
Definition ⊥E in [Pfenning] p. 18. 
∀I 
Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓 
alrimiv 1852  alrimiv 1852, ralrimiva 2962 
Definition ∀I^{a} in [Pfenning] p. 18,
definition I∀n in [Clemente] p. 32. 
∀E 
Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓 
spsbcd 3436  spcv 3289, rspcv 3295 
Definition ∀E in [Pfenning] p. 18,
definition E∀n,t in [Clemente] p. 32. 
∃I 
Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓 
spesbcd 3508  spcev 3290, rspcev 3299 
Definition ∃I in [Pfenning] p. 18,
definition I∃n,t in [Clemente] p. 32. 
∃E 
Γ⊢ ∃𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓⊢ 𝜃 =>
Γ⊢ 𝜃 
exlimddv 1860  exlimddv 1860, exlimdd 2086,
exlimdv 1858, rexlimdva 3026 
Definition ∃E^{a,u} in [Pfenning] p. 18,
definition E∃m,n,p,a in [Clemente] p. 32. 
⊥C 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 
efald 1501  efald 1501 
Proof by contradiction (classical logic),
definition ⊥C in [Pfenning] p. 17. 
⊥C 
Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓 
pm2.18da 459  pm2.18da 459, pm2.18d 124, pm2.18 122 
For an alternative falsumfree natural deduction ruleset 
¬ ¬C 
Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓 
notnotrd 128  notnotrd 128, notnotr 125 
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition E¬n in [Clemente] p. 14. 
EM  Γ⊢ 𝜓 ∨ ¬ 𝜓 
exmidd 432  exmid 431 
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. 
=I  Γ⊢ 𝐴 = 𝐴 
eqidd 2622  eqid 2621, eqidd 2622 
Introduce equality,
definition =I in [Pfenning] p. 127. 
=E  Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 =>
Γ⊢ [𝐵 / 𝑥]𝜓 
sbceq1dd 3428  sbceq1d 3427, equality theorems 
Eliminate equality,
definition =E in [Pfenning] p. 127. (Both E1 and E2.) 
Note that MPE uses classical logic, not intuitionist logic. As is
conventional, the "I" rules are introduction rules, "E" rules are
elimination rules, the "C" rules are conversion rules, and Γ
represents the set of (current) hypotheses. We use wff variable names
beginning with 𝜓 to provide a closer representation
of the Metamath
equivalents (which typically use the antedent 𝜑 to represent the
context Γ).
Most of this information was developed by Mario Carneiro and posted on
3Feb2017. For more information, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
For annotated examples where some traditional ND rules
are directly applied in MPE, see exnatded5.2 27149, exnatded5.3 27152,
exnatded5.5 27155, exnatded5.7 27156, exnatded5.8 27158, exnatded5.13 27160,
exnatded9.20 27162, and exnatded9.26 27164.
(Contributed by DAW, 4Feb2017.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

17.1.3 Natural deduction examples
These are examples of how natural deduction rules can be applied in Metamath
(both as lineforline translations of ND rules, and as a way to apply
deduction forms without being limited to applying ND rules). For more
information, see natded 27148 and mmnatded.html 27148. Since these examples should
not be used within proofs of other theorems, especially in Mathboxes, they
are marked with "(New usage is discouraged.)".


Theorem  exnatded5.2 27149 
Theorem 5.2 of [Clemente] p. 15, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  5  ((𝜓 ∧ 𝜒) → 𝜃) 
(𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) 
Given 
$e. 
2  2  (𝜒 → 𝜓) 
(𝜑 → (𝜒 → 𝜓)) 
Given 
$e. 
3  1  𝜒 
(𝜑 → 𝜒) 
Given 
$e. 
4  3  𝜓 
(𝜑 → 𝜓) 
→E 2,3 
mpd 15, the MPE equivalent of →E, 1,2 
5  4  (𝜓 ∧ 𝜒) 
(𝜑 → (𝜓 ∧ 𝜒)) 
∧I 4,3 
jca 554, the MPE equivalent of ∧I, 3,1 
6  6  𝜃 
(𝜑 → 𝜃) 
→E 1,5 
mpd 15, the MPE equivalent of →E, 4,5 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
Below is the final metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.22 27150.
A proof without context is shown in exnatded5.2i 27151.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) & ⊢ (𝜑 → (𝜒 → 𝜓)) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝜃) 

Theorem  exnatded5.22 27150 
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
exnatded5.2 27149 and exnatded5.2i 27151. (Contributed by Mario Carneiro,
9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) & ⊢ (𝜑 → (𝜒 → 𝜓)) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝜃) 

Theorem  exnatded5.2i 27151 
The same as exnatded5.2 27149 and exnatded5.22 27150 but with no context.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜒) → 𝜃)
& ⊢ (𝜒 → 𝜓)
& ⊢ 𝜒 ⇒ ⊢ 𝜃 

Theorem  exnatded5.3 27152 
Theorem 5.3 of [Clemente] p. 16, translated line by line using an
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.32 27153.
A proof without context is shown in exnatded5.3i 27154.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3  (𝜓 → 𝜒) 
(𝜑 → (𝜓 → 𝜒)) 
Given 
$e; adantr 481 to move it into the ND hypothesis 
2  5;6  (𝜒 → 𝜃) 
(𝜑 → (𝜒 → 𝜃)) 
Given 
$e; adantr 481 to move it into the ND hypothesis 
3  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 477, to access the new assumption 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15, the MPE equivalent of →E, 1.3.
adantr 481 was used to transform its dependency
(we could also use imp 445 to get this directly from 1)

5  7  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15, the MPE equivalent of →E, 4,6.
adantr 481 was used to transform its dependency 
6  8  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) 
∧I 4,5 
jca 554, the MPE equivalent of ∧I, 4,7 
7  9  (𝜓 → (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 
→I 3,6 
ex 450, the MPE equivalent of →I, 8 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 

Theorem  exnatded5.32 27153 
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
exnatded5.3 27152 and exnatded5.3i 27154. (Contributed by Mario Carneiro,
9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 

Theorem  exnatded5.3i 27154 
The same as exnatded5.3 27152 and exnatded5.32 27153 but with no context.
Identical to jccir 561, which should be used instead. (Contributed
by
Mario Carneiro, 9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜓 → 𝜒)
& ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜓 → (𝜒 ∧ 𝜃)) 

Theorem  exnatded5.5 27155 
Theorem 5.5 of [Clemente] p. 18, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3 
(𝜓 → 𝜒) 
(𝜑 → (𝜓 → 𝜒)) 
Given 
$e; adantr 481 to move it into the ND hypothesis 
2  5  ¬ 𝜒 
(𝜑 → ¬ 𝜒)  Given 
$e; we'll use adantr 481 to move it into the ND hypothesis 
3  1 
... 𝜓  ((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 477 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15 1,3 
5  6  ... ¬ 𝜒 
((𝜑 ∧ 𝜓) → ¬ 𝜒) 
IT 2 
adantr 481 5 
6  7  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
∧I 3,4,5 
pm2.65da 599 4,6 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 481; simpr 477 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof is mtod 189;
a proof without context is shown in mto 188.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.7 27156 
Theorem 5.7 of [Clemente] p. 19, translated line by line using the
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.72 27157.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  6 
(𝜓 ∨ (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) 
Given 
$e. No need for adantr 481 because we do not move this
into an ND hypothesis 
2  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption (new scope) 
simpr 477 
3  2  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒)) 
∨I_{L} 2 
orcd 407, the MPE equivalent of ∨I_{L}, 1 
4  3  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜒 ∧ 𝜃)) 
ND hypothesis assumption (new scope) 
simpr 477 
5  4  ... 𝜒 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜒) 
∧E_{L} 4 
simpld 475, the MPE equivalent of ∧E_{L}, 3 
6  6  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜓 ∨ 𝜒)) 
∨I_{R} 5 
olcd 408, the MPE equivalent of ∨I_{R}, 4 
7  7  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
∨E 1,3,6 
mpjaodan 826, the MPE equivalent of ∨E, 2,5,6 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) 

Theorem  exnatded5.72 27157 
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
exnatded5.7 27156. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) 

Theorem  exnatded5.8 27158 
Theorem 5.8 of [Clemente] p. 20, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  10;11 
((𝜓 ∧ 𝜒) → ¬ 𝜃) 
(𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) 
Given 
$e; adantr 481 to move it into the ND hypothesis 
2  3;4  (𝜏 → 𝜃) 
(𝜑 → (𝜏 → 𝜃))  Given 
$e; adantr 481 to move it into the ND hypothesis 
3  7;8 
𝜒  (𝜑 → 𝜒) 
Given 
$e; adantr 481 to move it into the ND hypothesis 
4  1;2  𝜏  (𝜑 → 𝜏) 
Given 
$e. adantr 481 to move it into the ND hypothesis 
5  6  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND Hypothesis/Assumption 
simpr 477. New ND hypothesis scope, each reference outside
the scope must change antecedent 𝜑 to (𝜑 ∧ 𝜓). 
6  9  ... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜒)) 
∧I 5,3 
jca 554 (∧I), 6,8 (adantr 481 to bring in scope) 
7  5  ... ¬ 𝜃 
((𝜑 ∧ 𝜓) → ¬ 𝜃) 
→E 1,6 
mpd 15 (→E), 2,4 
8  12  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 (→E), 9,11;
note the contradiction with ND line 7 (MPE line 5) 
9  13  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
¬I 5,7,8 
pm2.65da 599 (¬I), 5,12; proof by contradiction.
MPE step 6 (ND#5) does not need a reference here, because
the assumption is embedded in the antecedents 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 481; simpr 477 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.82 27159.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & ⊢ (𝜑 → (𝜏 → 𝜃)) & ⊢ (𝜑 → 𝜒)
& ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.82 27159 
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
linebyline translation, see exnatded5.8 27158. (Contributed by Mario
Carneiro, 9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & ⊢ (𝜑 → (𝜏 → 𝜃)) & ⊢ (𝜑 → 𝜒)
& ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.13 27160 
Theorem 5.13 of [Clemente] p. 20, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.132 27161.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  15  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
Given 
$e. 
2;3  2  (𝜓 → 𝜃) 
(𝜑 → (𝜓 → 𝜃))  Given 
$e. adantr 481 to move it into the ND hypothesis 
3  9  (¬ 𝜏 → ¬ 𝜒) 
(𝜑 → (¬ 𝜏 → ¬ 𝜒)) 
Given 
$e. ad2antrr 761 to move it into the ND subhypothesis 
4  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 477 
5  4  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 1,3 
6  5  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏)) 
∨I 5 
orcd 407 4 
7  6  ... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 477 
8  8  ... ... ¬ 𝜏 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) 
(sub) ND hypothesis assumption 
simpr 477 
9  11  ... ... ¬ 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) 
→E 3,8 
mpd 15 8,10 
10  7  ... ... 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒) 
IT 7 
adantr 481 6 
11  12  ... ¬ ¬ 𝜏 
((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏) 
¬I 8,9,10 
pm2.65da 599 7,11 
12  13  ... 𝜏 
((𝜑 ∧ 𝜒) → 𝜏) 
¬E 11 
notnotrd 128 12 
13  14  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏)) 
∨I 12 
olcd 408 13 
14  16  (𝜃 ∨ 𝜏) 
(𝜑 → (𝜃 ∨ 𝜏)) 
∨E 1,6,13 
mpjaodan 826 5,14,15 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 481; simpr 477 is useful when you want to
depend directly on the new assumption).
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → (𝜃 ∨ 𝜏)) 

Theorem  exnatded5.132 27161 
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with exnatded5.13 27160. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → (𝜃 ∨ 𝜏)) 

Theorem  exnatded9.20 27162 
Theorem 9.20 of [Clemente] p. 43, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  1 
(𝜓 ∧ (𝜒 ∨ 𝜃)) 
(𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) 
Given 
$e 
2  2  𝜓 
(𝜑 → 𝜓) 
∧E_{L} 1 
simpld 475 1 
3  11 
(𝜒 ∨ 𝜃) 
(𝜑 → (𝜒 ∨ 𝜃)) 
∧E_{R} 1 
simprd 479 1 
4  4 
... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 477 
5  5 
... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) 
∧I 2,4 
jca 554 3,4 
6  6 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜒) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{R} 5 
orcd 407 5 
7  8 
... 𝜃 
((𝜑 ∧ 𝜃) → 𝜃) 
ND hypothesis assumption 
simpr 477 
8  9 
... (𝜓 ∧ 𝜃) 
((𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜃)) 
∧I 2,7 
jca 554 7,8 
9  10 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{L} 8 
olcd 408 9 
10  12 
((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
(𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨E 3,6,9 
mpjaodan 826 6,10,11 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 481; simpr 477 is useful when you want to
depend directly on the new assumption).
Below is the final metamath proof (which reorders some steps).
A much more efficient proof is exnatded9.202 27163.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 

Theorem  exnatded9.202 27163 
A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare
with exnatded9.20 27162. (Contributed by David A. Wheeler,
19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 

Theorem  exnatded9.26 27164* 
Theorem 9.26 of [Clemente] p. 45, translated line by line using an
interpretation of natural deduction in Metamath. This proof has some
additional complications due to the fact that Metamath's existential
elimination rule does not change bound variables, so we need to verify
that 𝑥 is bound in the conclusion.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  3  ∃𝑥∀𝑦𝜓(𝑥, 𝑦) 
(𝜑 → ∃𝑥∀𝑦𝜓) 
Given 
$e. 
2  6  ... ∀𝑦𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) 
ND hypothesis assumption 
simpr 477. Later statements will have this scope. 
3  7;5,4  ... 𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → 𝜓) 
∀E 2,y 
spsbcd 3436 (∀E), 5,6. To use it we need a1i 11 and vex 3193.
This could be immediately done with 19.21bi 2057, but we want to show
the general approach for substitution.

4  12;8,9,10,11  ... ∃𝑥𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) 
∃I 3,a 
spesbcd 3508 (∃I), 11.
To use it we need sylibr 224, which in turn requires sylib 208 and
two uses of sbcid 3439.
This could be more immediately done using 19.8a 2049, but we want to show
the general approach for substitution.

5  13;1,2  ∃𝑥𝜓(𝑥, 𝑦) 
(𝜑 → ∃𝑥𝜓)  ∃E 1,2,4,a 
exlimdd 2086 (∃E), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1840 and nfe1 2024 (MPE# 1,2) 
6  14  ∀𝑦∃𝑥𝜓(𝑥, 𝑦) 
(𝜑 → ∀𝑦∃𝑥𝜓) 
∀I 5 
alrimiv 1852 (∀I), 13 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
Below is the final metamath proof (which reorders some steps).
Note that in the original proof, 𝜓(𝑥, 𝑦) has explicit
parameters. In Metamath, these parameters are always implicit, and the
parameters upon which a wff variable can depend are recorded in the
"allowed substitution hints" below.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded9.262 27165.
(Contributed by Mario Carneiro, 9Feb2017.)
(Revised by David A. Wheeler, 18Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) 

Theorem  exnatded9.262 27165* 
A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare
with exnatded9.26 27164. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) 

17.1.4 Definitional examples


Theorem  exor 27166 
Example for dfor 385. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)

⊢ (2 = 3 ∨ 4 = 4) 

Theorem  exan 27167 
Example for dfan 386. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)

⊢ (2 = 2 ∧ 3 = 3) 

Theorem  exdif 27168 
Example for dfdif 3563. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ ({1, 3} ∖ {1, 8}) =
{3} 

Theorem  exun 27169 
Example for dfun 3565. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ ({1, 3} ∪ {1, 8}) = {1, 3,
8} 

Theorem  exin 27170 
Example for dfin 3567. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ ({1, 3} ∩ {1, 8}) = {1} 

Theorem  exuni 27171 
Example for dfuni 4410. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 2Jul2016.)

⊢ ∪ {{1, 3}, {1,
8}} = {1, 3, 8} 

Theorem  exss 27172 
Example for dfss 3574. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ {1, 2} ⊆ {1, 2, 3} 

Theorem  expss 27173 
Example for dfpss 3576. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ {1, 2} ⊊ {1, 2, 3} 

Theorem  expw 27174 
Example for dfpw 4138. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 2Jul2016.)

⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5},
{7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))) 

Theorem  expr 27175 
Example for dfpr 4158. (Contributed by Mario Carneiro,
7May2015.)

⊢ (𝐴 ∈ {1, 1} → (𝐴↑2) = 1) 

Theorem  exbr 27176 
Example for dfbr 4624. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩}
→ 3𝑅9) 

Theorem  exopab 27177* 
Example for dfopab 4684. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) 

Theorem  exeprel 27178 
Example for dfeprel 4995. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ 5 E {1, 5} 

Theorem  exid 27179 
Example for dfid 4999. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ (5 I 5 ∧ ¬ 4 I 5) 

Theorem  expo 27180 
Example for dfpo 5005. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ ( < Po ℝ ∧ ¬ ≤ Po
ℝ) 

Theorem  exxp 27181 
Example for dfxp 5090. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ ({1, 5} × {2, 7}) = ({⟨1,
2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5,
7⟩}) 

Theorem  excnv 27182 
Example for dfcnv 5092. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ ^{◡}{⟨2, 6⟩, ⟨3, 9⟩} =
{⟨6, 2⟩, ⟨9, 3⟩} 

Theorem  exco 27183 
Example for dfco 5093. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ ((exp ∘ cos)‘0) =
e 

Theorem  exdm 27184 
Example for dfdm 5094. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ dom 𝐹 = {2,
3}) 

Theorem  exrn 27185 
Example for dfrn 5095. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ ran 𝐹 = {6,
9}) 

Theorem  exres 27186 
Example for dfres 5096. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)

⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
∧ 𝐵 = {1, 2}) →
(𝐹 ↾ 𝐵) = {⟨2,
6⟩}) 

Theorem  exima 27187 
Example for dfima 5097. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)

⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
∧ 𝐵 = {1, 2}) →
(𝐹 “ 𝐵) = {6}) 

Theorem  exfv 27188 
Example for dffv 5865. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ (𝐹‘3) =
9) 

Theorem  ex1st 27189 
Example for df1st 7128. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (1^{st} ‘⟨3, 4⟩) =
3 

Theorem  ex2nd 27190 
Example for df2nd 7129. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (2^{nd} ‘⟨3, 4⟩) =
4 

Theorem  1kp2ke3k 27191 
Example for dfdec 11454, 1000 + 2000 = 3000.
This proof disproves (by counterexample) the assertion of Hao Wang, who
stated, "There is a theorem in the primitive notation of set theory
that
corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula
would be forbiddingly long... even if (one) knows the definitions and is
asked to simplify the long formula according to them, chances are he will
make errors and arrive at some incorrect result." (Hao Wang,
"Theory and
practice in mathematics" , In Thomas Tymoczko, editor, New
Directions in
the Philosophy of Mathematics, pp 129152, Birkauser Boston, Inc.,
Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)
This is noted in Metamath: A Computer Language for Pure
Mathematics by
Norman Megill (2007) section 1.1.3. Megill then states, "A number of
writers have conveyed the impression that the kind of absolute rigor
provided by Metamath is an impossible dream, suggesting that a complete,
formal verification of a typical theorem would take millions of steps in
untold volumes of books... These writers assume, however, that in order
to achieve the kind of complete formal verification they desire one must
break down a proof into individual primitive steps that make direct
reference to the axioms. This is not necessary. There is no reason not
to make use of previously proved theorems rather than proving them over
and over... A hierarchy of theorems and definitions permits an
exponential growth in the formula sizes and primitive proof steps to be
described with only a linear growth in the number of symbols used. Of
course, this is how ordinary informal mathematics is normally done anyway,
but with Metamath it can be done with absolute rigor and precision."
The proof here starts with (2 + 1) = 3, commutes
it, and repeatedly
multiplies both sides by ten. This is certainly longer than traditional
mathematical proofs, e.g., there are a number of steps explicitly shown
here to show that we're allowed to do operations such as multiplication.
However, while longer, the proof is clearly a manageable size  even
though every step is rigorously derived all the way back to the primitive
notions of set theory and logic. And while there's a risk of making
errors, the many independent verifiers make it much less likely that an
incorrect result will be accepted.
This proof heavily relies on the decimal constructor dfdec 11454 developed by
Mario Carneiro in 2015. The underlying Metamath language has an
intentionally very small set of primitives; it doesn't even have a
builtin construct for numbers. Instead, the digits are defined using
these primitives, and the decimal constructor is used to make it easy to
express larger numbers as combinations of digits.
(Contributed by David A. Wheeler, 29Jun2016.) (Shortened by Mario
Carneiro using the arithmetic algorithm in mmj2, 30Jun2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000 

Theorem  exfl 27192 
Example for dffl 12549. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧
(⌊‘(3 / 2)) = 2) 

Theorem  exceil 27193 
Example for dfceil 12550. (Contributed by AV, 4Sep2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧
(⌈‘(3 / 2)) = 1) 

Theorem  exmod 27194 
Example for dfmod 12625. (Contributed by AV, 3Sep2021.)

⊢ ((5 mod 3) = 2 ∧ (7 mod 2) =
1) 

Theorem  exexp 27195 
Example for dfexp 12817. (Contributed by AV, 4Sep2021.)

⊢ ((5↑2) = ;25 ∧ (3↑2) = (1 / 9)) 

Theorem  exfac 27196 
Example for dffac 13017. (Contributed by AV, 4Sep2021.)

⊢ (!‘5) = ;;120 

Theorem  exbc 27197 
Example for dfbc 13046. (Contributed by AV, 4Sep2021.)

⊢ (5C3) = ;10 

Theorem  exhash 27198 
Example for dfhash 13074. (Contributed by AV, 4Sep2021.)

⊢ (#‘{0, 1, 2}) = 3 

Theorem  exsqrt 27199 
Example for dfsqrt 13925. (Contributed by AV, 4Sep2021.)

⊢ (√‘;25) = 5 

Theorem  exabs 27200 
Example for dfabs 13926. (Contributed by AV, 4Sep2021.)

⊢ (abs‘2) = 2 