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Theorem List for Metamath Proof Explorer - 37901-38000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjm3.1lem1 37901 Lemma for jm3.1 37904. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)       (𝜑 → (𝐾𝑁) < 𝐴)
 
Theoremjm3.1lem2 37902 Lemma for jm3.1 37904. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)       (𝜑 → (𝐾𝑁) < ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1))
 
Theoremjm3.1lem3 37903 Lemma for jm3.1 37904. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(𝜑𝐴 ∈ (ℤ‘2))    &   (𝜑𝐾 ∈ (ℤ‘2))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)       (𝜑 → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∈ ℕ)
 
Theoremjm3.1 37904 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(((𝐴 ∈ (ℤ‘2) ∧ 𝐾 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) → (𝐾𝑁) = (((𝐴 Xrm 𝑁) − ((𝐴𝐾) · (𝐴 Yrm 𝑁))) mod ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1)))
 
Theoremexpdiophlem1 37905* Lemma for expdioph 37907. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(𝐶 ∈ ℕ0 → (((𝐴 ∈ (ℤ‘2) ∧ 𝐵 ∈ ℕ) ∧ 𝐶 = (𝐴𝐵)) ↔ ∃𝑑 ∈ ℕ0𝑒 ∈ ℕ0𝑓 ∈ ℕ0 ((𝐴 ∈ (ℤ‘2) ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 ∈ (ℤ‘2) ∧ 𝑑 = (𝐴 Yrm (𝐵 + 1))) ∧ ((𝑑 ∈ (ℤ‘2) ∧ 𝑒 = (𝑑 Yrm 𝐵)) ∧ ((𝑑 ∈ (ℤ‘2) ∧ 𝑓 = (𝑑 Xrm 𝐵)) ∧ (𝐶 < ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∧ ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∥ ((𝑓 − ((𝑑𝐴) · 𝑒)) − 𝐶))))))))
 
Theoremexpdiophlem2 37906 Lemma for expdioph 37907. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{𝑎 ∈ (ℕ0𝑚 (1...3)) ∣ (((𝑎‘1) ∈ (ℤ‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3)
 
Theoremexpdioph 37907 The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{𝑎 ∈ (ℕ0𝑚 (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3)
 
20.25.35  Uncategorized stuff not associated with a major project
 
Theoremsetindtr 37908* Epsilon induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 8648; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∀𝑥(𝑥𝐴𝑥𝐴) → (∃𝑦(Tr 𝑦𝐵𝑦) → 𝐵𝐴))
 
Theoremsetindtrs 37909* Epsilon induction scheme without Infinity. See comments at setindtr 37908. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∀𝑦𝑥 𝜓𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
 
Theoremdford3lem1 37910* Lemma for dford3 37912. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))
 
Theoremdford3lem2 37911* Lemma for dford3 37912. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)
 
Theoremdford3 37912* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))
 
Theoremdford4 37913* dford3 37912 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ ∀𝑎𝑏𝑐((𝑎𝑁𝑏𝑎) → (𝑏𝑁 ∧ (𝑐𝑏𝑐𝑎))))
 
Theoremwopprc 37914 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1𝑜 ∈ {{{𝐴}, ∅}, {{𝐵}}})
 
Theoremrpnnen3lem 37915* Lemma for rpnnen3 37916. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})
 
Theoremrpnnen3 37916 Dedekind cut injection of into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
ℝ ≼ 𝒫 ℚ
 
20.25.36  More equivalents of the Axiom of Choice
 
Theoremaxac10 37917 Characterization of choice similar to dffin1-5 9248. (Contributed by Stefan O'Rear, 6-Jan-2015.)
( ≈ “ On) = V
 
Theoremharinf 37918 The Hartogs number of an infinite set is at least ω. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
((𝑆𝑉 ∧ ¬ 𝑆 ∈ Fin) → ω ⊆ (har‘𝑆))
 
Theoremwdom2d2 37919* Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)       (𝜑𝐴* (𝐵 × 𝐶))
 
Theoremttac 37920 Tarski's theorem about choice: infxpidm 9422 is equivalent to ax-ac 9319. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
(CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
 
Theorempw2f1ocnv 37921* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8108, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       (𝐴𝑉 → (𝐹:(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))
 
Theorempw2f1o2 37922* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8108, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       (𝐴𝑉𝐹:(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)
 
Theorempw2f1o2val 37923* Function value of the pw2f1o2 37922 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1𝑜}))
 
Theorempw2f1o2val2 37924* Membership in a mapped set under the pw2f1o2 37922 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1𝑜))
 
Theoremsoeq12d 37925 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))
 
Theoremfreq12d 37926 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐵))
 
Theoremweeq12d 37927 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))
 
Theoremlimsuc2 37928 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
 
Theoremwepwsolem 37929* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}    &   𝑈 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝐹 = (𝑎 ∈ (2𝑜𝑚 𝐴) ↦ (𝑎 “ {1𝑜}))       (𝐴 ∈ V → 𝐹 Isom 𝑈, 𝑇 ((2𝑜𝑚 𝐴), 𝒫 𝐴))
 
Theoremwepwso 37930* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}       ((𝐴𝑉𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
 
Theoremdnnumch1 37931* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8891. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
 
Theoremdnnumch2 37932* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       (𝜑𝐴 ⊆ ran 𝐹)
 
Theoremdnnumch3lem 37933* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))
 
Theoremdnnumch3 37934* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)
 
Theoremdnwech 37935* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))    &   𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}       (𝜑𝐻 We 𝐴)
 
Theoremfnwe2val 37936* Lemma for fnwe2 37940. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}       (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))
 
Theoremfnwe2lem1 37937* Lemma for fnwe2 37940. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})       ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})
 
Theoremfnwe2lem2 37938* Lemma for fnwe2 37940. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus 𝑇 is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})    &   (𝜑 → (𝐹𝐴):𝐴𝐵)    &   (𝜑𝑅 We 𝐵)    &   (𝜑𝑎𝐴)    &   (𝜑𝑎 ≠ ∅)       (𝜑 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)
 
Theoremfnwe2lem3 37939* Lemma for fnwe2 37940. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})    &   (𝜑 → (𝐹𝐴):𝐴𝐵)    &   (𝜑𝑅 We 𝐵)    &   (𝜑𝑎𝐴)    &   (𝜑𝑏𝐴)       (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))
 
Theoremfnwe2 37940* A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 7338 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})    &   (𝜑 → (𝐹𝐴):𝐴𝐵)    &   (𝜑𝑅 We 𝐵)       (𝜑𝑇 We 𝐴)
 
Theoremaomclem1 37941* Lemma for dfac11 37949. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of (𝑅1𝐴). In what follows, 𝐴 is the index of the rank we wish to well-order, 𝑧 is the collection of well-orderings constructed so far, dom 𝑧 is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and 𝑦 is a postulated multiple-choice function.

Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)

𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = suc dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))       (𝜑𝐵 Or (𝑅1‘dom 𝑧))
 
Theoremaomclem2 37942* Lemma for dfac11 37949. Successor case 2, a choice function for subsets of (𝑅1‘dom 𝑧). (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = suc dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → dom 𝑧𝐴)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎))
 
Theoremaomclem3 37943* Lemma for dfac11 37949. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = suc dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → dom 𝑧𝐴)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑𝐸 We (𝑅1‘dom 𝑧))
 
Theoremaomclem4 37944* Lemma for dfac11 37949. Limit case. Patch together well-orderings constructed so far using fnwe2 37940 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))       (𝜑𝐹 We (𝑅1‘dom 𝑧))
 
Theoremaomclem5 37945* Lemma for dfac11 37949. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → dom 𝑧𝐴)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑𝐺 We (𝑅1‘dom 𝑧))
 
Theoremaomclem6 37946* Lemma for dfac11 37949. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   𝐻 = recs((𝑧 ∈ V ↦ 𝐺))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → (𝐻𝐴) We (𝑅1𝐴))
 
Theoremaomclem7 37947* Lemma for dfac11 37949. (𝑅1𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   𝐻 = recs((𝑧 ∈ V ↦ 𝐺))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
 
Theoremaomclem8 37948* Lemma for dfac11 37949. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))
 
Theoremdfac11 37949* The right-hand side of this theorem (compare with ac4 9335), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8538, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it.

A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)

(CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
 
Theoremkelac1 37950* Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝜑𝑥𝐼) → 𝑆 ≠ ∅)    &   ((𝜑𝑥𝐼) → 𝐽 ∈ Top)    &   ((𝜑𝑥𝐼) → 𝐶 ∈ (Clsd‘𝐽))    &   ((𝜑𝑥𝐼) → 𝐵:𝑆1-1-onto𝐶)    &   ((𝜑𝑥𝐼) → 𝑈 𝐽)    &   (𝜑 → (∏t‘(𝑥𝐼𝐽)) ∈ Comp)       (𝜑X𝑥𝐼 𝑆 ≠ ∅)
 
Theoremkelac2lem 37951 Lemma for kelac2 37952 and dfac21 37953: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)
 
Theoremkelac2 37952* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝜑𝑥𝐼) → 𝑆𝑉)    &   ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)    &   (𝜑 → (∏t‘(𝑥𝐼 ↦ (topGen‘{𝑆, {𝒫 𝑆}}))) ∈ Comp)       (𝜑X𝑥𝐼 𝑆 ≠ ∅)
 
Theoremdfac21 37953 Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
(CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
 
20.25.37  Finitely generated left modules
 
Syntaxclfig 37954 Extend class notation with the class of finitely generated left modules.
class LFinGen
 
Definitiondf-lfig 37955 Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
 
Theoremislmodfg 37956* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐵 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁𝑏) = 𝐵)))
 
Theoremislssfg 37957* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝑈(𝑏 ∈ Fin ∧ (𝑁𝑏) = 𝑈)))
 
Theoremislssfg2 37958* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐵 = (Base‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑁𝑏) = 𝑈))
 
Theoremislssfgi 37959 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑁 = (LSpan‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑋 = (𝑊s (𝑁𝐵))       ((𝑊 ∈ LMod ∧ 𝐵𝑉𝐵 ∈ Fin) → 𝑋 ∈ LFinGen)
 
Theoremfglmod 37960 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
 
Theoremlsmfgcl 37961 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑈 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐷 = (𝑊s 𝐴)    &   𝐸 = (𝑊s 𝐵)    &   𝐹 = (𝑊s (𝐴 𝐵))    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷 ∈ LFinGen)    &   (𝜑𝐸 ∈ LFinGen)       (𝜑𝐹 ∈ LFinGen)
 
20.25.38  Noetherian left modules I
 
Syntaxclnm 37962 Extend class notation with the class of Noetherian left modules.
class LNoeM
 
Definitiondf-lnm 37963* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen}
 
Theoremislnm 37964* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑆 = (LSubSp‘𝑀)       (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))
 
Theoremislnm2 37965* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑀)    &   𝑆 = (LSubSp‘𝑀)    &   𝑁 = (LSpan‘𝑀)       (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁𝑔)))
 
Theoremlnmlmod 37966 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
 
Theoremlnmlssfg 37967 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑆 = (LSubSp‘𝑀)    &   𝑅 = (𝑀s 𝑈)       ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)
 
Theoremlnmlsslnm 37968 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑆 = (LSubSp‘𝑀)    &   𝑅 = (𝑀s 𝑈)       ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LNoeM)
 
Theoremlnmfg 37969 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)
 
Theoremkercvrlsm 37970 The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑈 = (LSubSp‘𝑆)    &    = (LSSum‘𝑆)    &    0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐷𝑈)    &   (𝜑 → (𝐹𝐷) = ran 𝐹)       (𝜑 → (𝐾 𝐷) = 𝐵)
 
Theoremlmhmfgima 37971 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑇s (𝐹𝐴))    &   𝑋 = (𝑆s 𝐴)    &   𝑈 = (LSubSp‘𝑆)    &   (𝜑𝑋 ∈ LFinGen)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))       (𝜑𝑌 ∈ LFinGen)
 
Theoremlnmepi 37972 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵) → 𝑇 ∈ LNoeM)
 
Theoremlmhmfgsplit 37973 If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &   𝑈 = (𝑆s 𝐾)    &   𝑉 = (𝑇s ran 𝐹)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)
 
Theoremlmhmlnmsplit 37974 If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &   𝑈 = (𝑆s 𝐾)    &   𝑉 = (𝑇s ran 𝐹)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
 
Theoremlnmlmic 37975 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
 
20.25.39  Addenda for structure powers
 
Theorempwssplit4 37976* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐸 = (𝑅s (𝐴𝐵))    &   𝐺 = (Base‘𝐸)    &    0 = (0g𝑅)    &   𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })}    &   𝐹 = (𝑥𝐾 ↦ (𝑥𝐵))    &   𝐶 = (𝑅s 𝐴)    &   𝐷 = (𝑅s 𝐵)    &   𝐿 = (𝐸s 𝐾)       ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
 
Theoremfilnm 37977 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)
 
Theorempwslnmlem0 37978 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s ∅)       (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)
 
Theorempwslnmlem1 37979* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s {𝑖})       (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)
 
Theorempwslnmlem2 37980 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑋 = (𝑊s 𝐴)    &   𝑌 = (𝑊s 𝐵)    &   𝑍 = (𝑊s (𝐴𝐵))    &   (𝜑𝑊 ∈ LMod)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑋 ∈ LNoeM)    &   (𝜑𝑌 ∈ LNoeM)       (𝜑𝑍 ∈ LNoeM)
 
Theorempwslnm 37981 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝐼)       ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM)
 
20.25.40  Every set admits a group structure iff choice
 
Theoremunxpwdom3 37982* Weaker version of unxpwdom 8535 where a function is required only to be cancellative, not an injection. 𝐷 and 𝐵 are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into 𝐴, each row must hit an element of 𝐵; by column injectivity, each row can be identified in at least one way by the 𝐵 element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷𝑋)    &   ((𝜑𝑎𝐶𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))    &   (((𝜑𝑎𝐶) ∧ (𝑏𝐷𝑐𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))    &   (((𝜑𝑑𝐷) ∧ (𝑎𝐶𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))    &   (𝜑 → ¬ 𝐷𝐴)       (𝜑𝐶* (𝐷 × 𝐵))
 
Theorempwfi2f1o 37983* The pw2f1o 8106 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
𝑆 = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅}    &   𝐹 = (𝑥𝑆 ↦ (𝑥 “ {1𝑜}))       (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
 
Theorempwfi2en 37984* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
𝑆 = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅}       (𝐴𝑉𝑆 ≈ (𝒫 𝐴 ∩ Fin))
 
Theoremfrlmpwfi 37985 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.)
𝑅 = (ℤ/nℤ‘2)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)       (𝐼𝑉𝐵 ≈ (𝒫 𝐼 ∩ Fin))
 
Theoremgicabl 37986 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
(𝐺𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
 
Theoremimasgim 37987 A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅 ∈ Grp)       (𝜑𝐹 ∈ (𝑅 GrpIso 𝑈))
 
Theoremisnumbasgrplem1 37988 A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Abel ∧ 𝐶𝐵) → 𝐶 ∈ (Base “ Abel))
 
Theoremharn0 37989 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆𝑉 → (har‘𝑆) ≠ ∅)
 
Theoremnuminfctb 37990 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
 
Theoremisnumbasgrplem2 37991 If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)
 
Theoremisnumbasgrplem3 37992 Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.)
((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel))
 
Theoremisnumbasabl 37993 A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel))
 
Theoremisnumbasgrp 37994 A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp))
 
Theoremdfacbasgrp 37995 A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.)
(CHOICE ↔ (Base “ Grp) = (V ∖ {∅}))
 
20.25.41  Noetherian rings and left modules II
 
Syntaxclnr 37996 Extend class notation with the class of left Noetherian rings.
class LNoeR
 
Definitiondf-lnr 37997 A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
 
Theoremislnr 37998 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
 
Theoremlnrring 37999 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR → 𝐴 ∈ Ring)
 
Theoremlnrlnm 38000 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)
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