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Theorem List for Metamath Proof Explorer - 42201-42300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremupgrwlkupwlkb 42201 In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.)
(𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃𝐹(UPWalks‘𝐺)𝑃))
 
TheoremupgrisupwlkALT 42202* Alternate proof of upgriswlk 26718 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹𝑈𝑃𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
 
20.35.10  Set of unordered pairs
 
Theoremsprid 42203 Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
{𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
 
Theoremelsprel 42204* An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4434, which is not an element of all unordered pairs, see spr0nelg 42205. (Contributed by AV, 21-Nov-2021.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
 
Theoremspr0nelg 42205* The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
 
Syntaxcspr 42206 Extend class notation with set of pairs.
class Pairs
 
Definitiondf-spr 42207* Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.)
Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}})
 
Theoremsprval 42208* The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 
Theoremsprvalpw 42209* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 
Theoremsprssspr 42210* The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
(Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
 
Theoremspr0el 42211 The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
∅ ∉ (Pairs‘𝑉)
 
Theoremsprvalpwn0 42212* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 
Theoremsprel 42213* An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
 
Theoremprssspr 42214* An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})
 
Theoremprelspr 42215 An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))
 
Theoremprsprel 42216 The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
 
Theoremprsssprel 42217 The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
 
Theoremsprvalpwle2 42218* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑝) ≤ 2})
 
Theoremsprsymrelfvlem 42219* Lemma for sprsymrelf 42224 and sprsymrelfv 42223. (Contributed by AV, 19-Nov-2021.)
(𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
 
Theoremsprsymrelf1lem 42220* Lemma for sprsymrelf1 42225. (Contributed by AV, 22-Nov-2021.)
((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))
 
Theoremsprsymrelfolem1 42221* Lemma 1 for sprsymrelfo 42226. (Contributed by AV, 22-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       𝑄 ∈ 𝒫 (Pairs‘𝑉)
 
Theoremsprsymrelfolem2 42222* Lemma 2 for sprsymrelfo 42226. (Contributed by AV, 23-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))
 
Theoremsprsymrelfv 42223* The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑋𝑃 → (𝐹𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})
 
Theoremsprsymrelf 42224* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃𝑅
 
Theoremsprsymrelf1 42225* The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃1-1𝑅
 
Theoremsprsymrelfo 42226* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃onto𝑅)
 
Theoremsprsymrelf1o 42227* The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃1-1-onto𝑅)
 
Theoremsprbisymrel 42228* There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)
 
Theoremsprsymrelen 42229* The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝑃𝑅)
 
Theoremupgredgssspr 42230 The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.)
(𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺)))
 
Theoremuspgropssxp 42231* The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of an Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 42241. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}       (𝑉𝑊𝐺 ⊆ (𝑊 × 𝑃))
 
Theoremuspgrsprfv 42232* The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 42238. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       (𝑋𝐺 → (𝐹𝑋) = (2nd𝑋))
 
Theoremuspgrsprf 42233* The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       𝐹:𝐺𝑃
 
Theoremuspgrsprf1 42234* The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       𝐹:𝐺1-1𝑃
 
Theoremuspgrsprfo 42235* The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       (𝑉𝑊𝐹:𝐺onto𝑃)
 
Theoremuspgrsprf1o 42236* The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 42241. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       (𝑉𝑊𝐹:𝐺1-1-onto𝑃)
 
Theoremuspgrex 42237* The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}       (𝑉𝑊𝐺 ∈ V)
 
Theoremuspgrbispr 42238* There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}       (𝑉𝑊 → ∃𝑓 𝑓:𝐺1-1-onto𝑃)
 
Theoremuspgrspren 42239* The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}       (𝑉𝑊𝐺𝑃)
 
Theoremuspgrymrelen 42240* The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 42241. (Contributed by AV, 27-Nov-2021.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝐺𝑅)
 
Theoremuspgrbisymrel 42241* There is a bijection between the "simple pseudographs" for a fixed set 𝑉 of vertices and the class 𝑅 of the symmetric relations on the fixed set 𝑉. The simple pseudographs, which are graphs without hyper- or multiedges, but which may contain loops, are expressed as ordered pairs of the vertices and the edges (as proper or improper unordered pairs of vertices, not as indexed edges!) in this theorem. That class 𝐺 of such simple pseudographs is a set (if 𝑉 is a set, see uspgrex 42237) of equivalence classes of graphs abstracting from the index sets of their edge functions.

Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ⟨𝑣, 𝑒⟩ ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.)

𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝐺1-1-onto𝑅)
 
TheoremuspgrbisymrelALT 42242* Alternate proof of uspgrbisymrel 42241 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝐺1-1-onto𝑅)
 
20.35.11  Monoids (extension)
 
20.35.11.1  Auxiliary theorems
 
Theoremovn0dmfun 42243 If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6375. (Contributed by AV, 27-Jan-2020.)
((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))
 
Theoremxpsnopab 42244* A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)
({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}
 
Theoremxpiun 42245* A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
(𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}
 
Theoremovn0ssdmfun 42246* If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6375. (Contributed by AV, 27-Jan-2020.)
(∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))
 
Theoremfnxpdmdm 42247 The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.)
(𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴)
 
Theoremcnfldsrngbas 42248 The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅))
 
Theoremcnfldsrngadd 42249 The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆𝑉 → + = (+g𝑅))
 
Theoremcnfldsrngmul 42250 The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆𝑉 → · = (.r𝑅))
 
20.35.11.2  Magmas and Semigroups (extension)
 
Theoremplusfreseq 42251 If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (+𝑓𝑀)       (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
 
Theoremmgmplusfreseq 42252 If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (+𝑓𝑀)       ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )
 
Theorem0mgm 42253 A set with an empty base set is always a magma". (Contributed by AV, 25-Feb-2020.)
(Base‘𝑀) = ∅       (𝑀𝑉𝑀 ∈ Mgm)
 
Theoremmgmpropd 42254* If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
 
Theoremismgmd 42255* Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐺𝑉)    &   (𝜑+ = (+g𝐺))    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)       (𝜑𝐺 ∈ Mgm)
 
20.35.11.3  Magma homomorphisms and submagmas
 
Syntaxcmgmhm 42256 Hom-set generator class for magmas.
class MgmHom
 
Syntaxcsubmgm 42257 Class function taking a magma to its lattice of submagmas.
class SubMgm
 
Definitiondf-mgmhm 42258* A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.)
MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦))})
 
Definitiondf-submgm 42259* A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.)
SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})
 
Theoremmgmhmrcl 42260 Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
(𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
 
Theoremsubmgmrcl 42261 Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
(𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
 
Theoremismgmhm 42262* Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)       (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
 
Theoremmgmhmf 42263 A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵𝐶)
 
Theoremmgmhmpropd 42264* Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   (𝜑𝐵 ≠ ∅)    &   (𝜑𝐶 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))       (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))
 
Theoremmgmhmlin 42265 A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 
Theoremmgmhmf1o 42266 A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))
 
Theoremidmgmhm 42267 The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀))
 
Theoremissubmgm 42268* Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
 
Theoremissubmgm2 42269 Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑀)    &   𝐻 = (𝑀s 𝑆)       (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵𝐻 ∈ Mgm)))
 
Theoremrabsubmgmd 42270* Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ Mgm)    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)    &   (𝑧 = 𝑥 → (𝜓𝜃))    &   (𝑧 = 𝑦 → (𝜓𝜏))    &   (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))       (𝜑 → {𝑧𝐵𝜓} ∈ (SubMgm‘𝑀))
 
Theoremsubmgmss 42271 Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑆 ∈ (SubMgm‘𝑀) → 𝑆𝐵)
 
Theoremsubmgmid 42272 Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀))
 
Theoremsubmgmcl 42273 Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
+ = (+g𝑀)       ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
 
Theoremsubmgmmgm 42274 Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMgm‘𝑀) → 𝐻 ∈ Mgm)
 
Theoremsubmgmbas 42275 The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻))
 
Theoremsubsubmgm 42276 A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)))
 
Theoremresmgmhm 42277 Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MgmHom 𝑇))
 
Theoremresmgmhm2 42278 One direction of resmgmhm2b 42279. (Contributed by AV, 26-Feb-2020.)
𝑈 = (𝑇s 𝑋)       ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
 
Theoremresmgmhm2b 42279 Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
𝑈 = (𝑇s 𝑋)       ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))
 
Theoremmgmhmco 42280 The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MgmHom 𝑈))
 
Theoremmgmhmima 42281 The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹𝑋) ∈ (SubMgm‘𝑁))
 
Theoremmgmhmeql 42282 The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMgm‘𝑆))
 
Theoremsubmgmacs 42283 Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵))
 
Theoremismhm0 42284 Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &    0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))
 
Theoremmhmismgmhm 42285 Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
(𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆))
 
20.35.11.4  Examples and counterexamples for magmas, semigroups and monoids (extension)
 
Theoremopmpt2ismgm 42286* A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)       (𝜑𝑀 ∈ Mgm)
 
Theoremcopissgrp 42287* A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)    &   (𝜑𝐵 ≠ ∅)    &   (𝜑𝐶𝐵)       (𝜑𝑀 ∈ SGrp)
 
Theoremcopisnmnd 42288* A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)    &   (𝜑𝐶𝐵)    &   (𝜑 → 1 < (♯‘𝐵))       (𝜑𝑀 ∉ Mnd)
 
Theorem0nodd 42289* 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}       0 ∉ 𝑂
 
Theorem1odd 42290* 1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}       1 ∈ 𝑂
 
Theorem2nodd 42291* 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}       2 ∉ 𝑂
 
Theoremoddibas 42292* Lemma 1 for oddinmgm 42294: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   𝑀 = (ℂflds 𝑂)       𝑂 = (Base‘𝑀)
 
Theoremoddiadd 42293* Lemma 2 for oddinmgm 42294: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   𝑀 = (ℂflds 𝑂)        + = (+g𝑀)
 
Theoremoddinmgm 42294* The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 42423, and even a non-unital ring, see 2zrng 42414. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   𝑀 = (ℂflds 𝑂)       𝑀 ∉ Mgm
 
Theoremnnsgrpmgm 42295 The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
𝑀 = (ℂflds ℕ)       𝑀 ∈ Mgm
 
Theoremnnsgrp 42296 The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
𝑀 = (ℂflds ℕ)       𝑀 ∈ SGrp
 
Theoremnnsgrpnmnd 42297 The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
𝑀 = (ℂflds ℕ)       𝑀 ∉ Mnd
 
20.35.12  Magmas and internal binary operations (alternate approach)

With df-mpt2 6806, binary operations are defined by a rule, and with df-ov 6804, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆𝑆. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, we call binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 internal binary operations, see df-intop 42314. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 42315. Therefore, a "binary operation on a set 𝑆" according to Wikipedia is a "closed internal binary operation" in our terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations ).

Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 42315 and df-assintop 42316), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obey these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves.

In the following, an alternate definition df-cllaw 42301 for an internal binary operation is provided, which does not require function-ness, but only closure. Therefore, this definition could be used as binary operation (Slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 42309, or for an alternate definition df-mgm2 42334 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups).

 
20.35.12.1  Laws for internal binary operations

In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi.

 
Syntaxccllaw 42298 Extend class notation for the closure law.
class clLaw
 
Syntaxcasslaw 42299 Extend class notation for the associative law.
class assLaw
 
Syntaxccomlaw 42300 Extend class notation for the commutative law.
class comLaw
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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