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Theorem List for Metamath Proof Explorer - 39601-39700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrmydioph 39601 jm2.27 39595 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
{𝑎 ∈ (ℕ0m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ‘2) ∧ (𝑎‘3) = ((𝑎‘1) Yrm (𝑎‘2)))} ∈ (Dioph‘3)

20.28.34  X and Y sequences 5: Diophantine representability of X, ^, _C

Theoremrmxdiophlem 39602* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℕ0𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1)))

Theoremrmxdioph 39603 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{𝑎 ∈ (ℕ0m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ‘2) ∧ (𝑎‘3) = ((𝑎‘1) Xrm (𝑎‘2)))} ∈ (Dioph‘3)

Theoremjm3.1lem1 39604 Lemma for jm3.1 39607. (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 39605 Lemma for jm3.1 39607. (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 39606 Lemma for jm3.1 39607. (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 39607 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 39608* Lemma for expdioph 39610. 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 39609 Lemma for expdioph 39610. 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 39610 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)

20.28.35  Uncategorized stuff not associated with a major project

Theoremsetindtr 39611* Set 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 9168; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∀𝑥(𝑥𝐴𝑥𝐴) → (∃𝑦(Tr 𝑦𝐵𝑦) → 𝐵𝐴))

Theoremsetindtrs 39612* Set induction scheme without Infinity. See comments at setindtr 39611. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∀𝑦𝑥 𝜓𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)

Theoremdford3lem1 39613* Lemma for dford3 39615. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑁 ∧ ∀𝑦𝑁 Tr 𝑦) → ∀𝑏𝑁 (Tr 𝑏 ∧ ∀𝑦𝑏 Tr 𝑦))

Theoremdford3lem2 39614* Lemma for dford3 39615. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑥 ∧ ∀𝑦𝑥 Tr 𝑦) → 𝑥 ∈ On)

Theoremdford3 39615* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥𝑁 Tr 𝑥))

Theoremdford4 39616* dford3 39615 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ ∀𝑎𝑏𝑐((𝑎𝑁𝑏𝑎) → (𝑏𝑁 ∧ (𝑐𝑏𝑐𝑎))))

Theoremwopprc 39617 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})

Theoremrpnnen3lem 39618* Lemma for rpnnen3 39619. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})

Theoremrpnnen3 39619 Dedekind cut injection of into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
ℝ ≼ 𝒫 ℚ

20.28.36  More equivalents of the Axiom of Choice

Theoremaxac10 39620 Characterization of choice similar to dffin1-5 9802. (Contributed by Stefan O'Rear, 6-Jan-2015.)
( ≈ “ On) = V

Theoremharinf 39621 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 39622* 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 39623 Tarski's theorem about choice: infxpidm 9976 is equivalent to ax-ac 9873. (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 39624* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8616, 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 39625* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8616, 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 39626* Function value of the pw2f1o2 39625 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 39627* Membership in a mapped set under the pw2f1o2 39625 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 39628 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))

Theoremfreq12d 39629 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐵))

Theoremweeq12d 39630 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))

Theoremlimsuc2 39631 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 39632* 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 39633* 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 39634* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9448. (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 39635* 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 39636* 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 39637* 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 39638* 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 39639* Lemma for fnwe2 39643. 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 39640* Lemma for fnwe2 39643. 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 39641* Lemma for fnwe2 39643. 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 39642* Lemma for fnwe2 39643. 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 39643* 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 7818 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 39644* Lemma for dfac11 39652. 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 indices 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 39645* Lemma for dfac11 39652. 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 39646* Lemma for dfac11 39652. 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 39647* Lemma for dfac11 39652. Limit case. Patch together well-orderings constructed so far using fnwe2 39643 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 39648* Lemma for dfac11 39652. 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 39649* Lemma for dfac11 39652. 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 39650* Lemma for dfac11 39652. (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 39651* Lemma for dfac11 39652. 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 39652* The right-hand side of this theorem (compare with ac4 9889), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9048, 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 39653* 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 39654 Lemma for kelac2 39655 and dfac21 39656: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)

Theoremkelac2 39655* 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 39656 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))

20.28.37  Finitely generated left modules

Syntaxclfig 39657 Extend class notation with the class of finitely generated left modules.
class LFinGen

Definitiondf-lfig 39658 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 39659* 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 39660* 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 39661* 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 39662 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 39663 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Theoremlsmfgcl 39664 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)

20.28.38  Noetherian left modules I

Syntaxclnm 39665 Extend class notation with the class of Noetherian left modules.
class LNoeM

Definitiondf-lnm 39666* 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 39667* 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 39668* 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 39669 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM → 𝑀 ∈ LMod)

Theoremlnmlssfg 39670 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 39671 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 39672 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)

Theoremkercvrlsm 39673 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 39674 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 39675 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 39676 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 39677 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 39678 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))

Theorempwssplit4 39679* 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 39680 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)

Theorempwslnmlem0 39681 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s ∅)       (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)

Theorempwslnmlem1 39682* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s {𝑖})       (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)

Theorempwslnmlem2 39683 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 39684 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝐼)       ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM)

20.28.40  Every set admits a group structure iff choice

Theoremunxpwdom3 39685* Weaker version of unxpwdom 9045 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 39686* The pw2f1o 8614 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 39687* 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 39688 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 39689 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
(𝐺𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))

Theoremimasgim 39690 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 39691 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 39692 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆𝑉 → (har‘𝑆) ≠ ∅)

Theoremnuminfctb 39693 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 39694 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 39695 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 39696 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 39697 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 39698 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 ∖ {∅}))

20.28.41  Noetherian rings and left modules II

Syntaxclnr 39699 Extend class notation with the class of left Noetherian rings.
class LNoeR

Definitiondf-lnr 39700 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}

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