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Theorem List for Metamath Proof Explorer - 41201-41300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjm2.27dlem4 41201 Lemma for rmydioph 41203. Infer β„•-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
𝐴 ∈ β„•    &   π΅ = (𝐴 + 1)    β‡’   π΅ ∈ β„•
 
Theoremjm2.27dlem5 41202 Lemma for rmydioph 41203. Used with sselii 3939 to infer membership of midpoints of range; jm2.27dlem2 41199 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
𝐡 = (𝐴 + 1)    &   (1...𝐡) βŠ† (1...𝐢)    β‡’   (1...𝐴) βŠ† (1...𝐢)
 
Theoremrmydioph 41203 jm2.27 41197 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
{π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ ((π‘Žβ€˜1) ∈ (β„€β‰₯β€˜2) ∧ (π‘Žβ€˜3) = ((π‘Žβ€˜1) Yrm (π‘Žβ€˜2)))} ∈ (Diophβ€˜3)
 
21.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C
 
Theoremrmxdiophlem 41204* X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
((𝐴 ∈ (β„€β‰₯β€˜2) ∧ 𝑁 ∈ β„•0 ∧ 𝑋 ∈ β„•0) β†’ (𝑋 = (𝐴 Xrm 𝑁) ↔ βˆƒπ‘¦ ∈ β„•0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) βˆ’ (((𝐴↑2) βˆ’ 1) Β· (𝑦↑2))) = 1)))
 
Theoremrmxdioph 41205 X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ ((π‘Žβ€˜1) ∈ (β„€β‰₯β€˜2) ∧ (π‘Žβ€˜3) = ((π‘Žβ€˜1) Xrm (π‘Žβ€˜2)))} ∈ (Diophβ€˜3)
 
Theoremjm3.1lem1 41206 Lemma for jm3.1 41209. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(πœ‘ β†’ 𝐴 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ (𝐾 Yrm (𝑁 + 1)) ≀ 𝐴)    β‡’   (πœ‘ β†’ (𝐾↑𝑁) < 𝐴)
 
Theoremjm3.1lem2 41207 Lemma for jm3.1 41209. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(πœ‘ β†’ 𝐴 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ (𝐾 Yrm (𝑁 + 1)) ≀ 𝐴)    β‡’   (πœ‘ β†’ (𝐾↑𝑁) < ((((2 Β· 𝐴) Β· 𝐾) βˆ’ (𝐾↑2)) βˆ’ 1))
 
Theoremjm3.1lem3 41208 Lemma for jm3.1 41209. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(πœ‘ β†’ 𝐴 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ (𝐾 Yrm (𝑁 + 1)) ≀ 𝐴)    β‡’   (πœ‘ β†’ ((((2 Β· 𝐴) Β· 𝐾) βˆ’ (𝐾↑2)) βˆ’ 1) ∈ β„•)
 
Theoremjm3.1 41209 Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(((𝐴 ∈ (β„€β‰₯β€˜2) ∧ 𝐾 ∈ (β„€β‰₯β€˜2) ∧ 𝑁 ∈ β„•) ∧ (𝐾 Yrm (𝑁 + 1)) ≀ 𝐴) β†’ (𝐾↑𝑁) = (((𝐴 Xrm 𝑁) βˆ’ ((𝐴 βˆ’ 𝐾) Β· (𝐴 Yrm 𝑁))) mod ((((2 Β· 𝐴) Β· 𝐾) βˆ’ (𝐾↑2)) βˆ’ 1)))
 
Theoremexpdiophlem1 41210* Lemma for expdioph 41212. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
(𝐢 ∈ β„•0 β†’ (((𝐴 ∈ (β„€β‰₯β€˜2) ∧ 𝐡 ∈ β„•) ∧ 𝐢 = (𝐴↑𝐡)) ↔ βˆƒπ‘‘ ∈ β„•0 βˆƒπ‘’ ∈ β„•0 βˆƒπ‘“ ∈ β„•0 ((𝐴 ∈ (β„€β‰₯β€˜2) ∧ 𝐡 ∈ β„•) ∧ ((𝐴 ∈ (β„€β‰₯β€˜2) ∧ 𝑑 = (𝐴 Yrm (𝐡 + 1))) ∧ ((𝑑 ∈ (β„€β‰₯β€˜2) ∧ 𝑒 = (𝑑 Yrm 𝐡)) ∧ ((𝑑 ∈ (β„€β‰₯β€˜2) ∧ 𝑓 = (𝑑 Xrm 𝐡)) ∧ (𝐢 < ((((2 Β· 𝑑) Β· 𝐴) βˆ’ (𝐴↑2)) βˆ’ 1) ∧ ((((2 Β· 𝑑) Β· 𝐴) βˆ’ (𝐴↑2)) βˆ’ 1) βˆ₯ ((𝑓 βˆ’ ((𝑑 βˆ’ 𝐴) Β· 𝑒)) βˆ’ 𝐢))))))))
 
Theoremexpdiophlem2 41211 Lemma for expdioph 41212. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
{π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ (((π‘Žβ€˜1) ∈ (β„€β‰₯β€˜2) ∧ (π‘Žβ€˜2) ∈ β„•) ∧ (π‘Žβ€˜3) = ((π‘Žβ€˜1)↑(π‘Žβ€˜2)))} ∈ (Diophβ€˜3)
 
Theoremexpdioph 41212 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.)
{π‘Ž ∈ (β„•0 ↑m (1...3)) ∣ (π‘Žβ€˜3) = ((π‘Žβ€˜1)↑(π‘Žβ€˜2))} ∈ (Diophβ€˜3)
 
21.29.35  Uncategorized stuff not associated with a major project
 
Theoremsetindtr 41213* 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 9603; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(βˆ€π‘₯(π‘₯ βŠ† 𝐴 β†’ π‘₯ ∈ 𝐴) β†’ (βˆƒπ‘¦(Tr 𝑦 ∧ 𝐡 ∈ 𝑦) β†’ 𝐡 ∈ 𝐴))
 
Theoremsetindtrs 41214* Set induction scheme without Infinity. See comments at setindtr 41213. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(βˆ€π‘¦ ∈ π‘₯ πœ“ β†’ πœ‘)    &   (π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    &   (π‘₯ = 𝐡 β†’ (πœ‘ ↔ πœ’))    β‡’   (βˆƒπ‘§(Tr 𝑧 ∧ 𝐡 ∈ 𝑧) β†’ πœ’)
 
Theoremdford3lem1 41215* Lemma for dford3 41217. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr 𝑁 ∧ βˆ€π‘¦ ∈ 𝑁 Tr 𝑦) β†’ βˆ€π‘ ∈ 𝑁 (Tr 𝑏 ∧ βˆ€π‘¦ ∈ 𝑏 Tr 𝑦))
 
Theoremdford3lem2 41216* Lemma for dford3 41217. (Contributed by Stefan O'Rear, 28-Oct-2014.)
((Tr π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ Tr 𝑦) β†’ π‘₯ ∈ On)
 
Theoremdford3 41217* Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ (Tr 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 Tr π‘₯))
 
Theoremdford4 41218* dford3 41217 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(Ord 𝑁 ↔ βˆ€π‘Žβˆ€π‘βˆ€π‘((π‘Ž ∈ 𝑁 ∧ 𝑏 ∈ π‘Ž) β†’ (𝑏 ∈ 𝑁 ∧ (𝑐 ∈ 𝑏 β†’ 𝑐 ∈ π‘Ž))))
 
Theoremwopprc 41219 Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ V ∧ 𝐡 ∈ V) ↔ Β¬ 1o ∈ {{{𝐴}, βˆ…}, {{𝐡}}})
 
Theoremrpnnen3lem 41220* Lemma for rpnnen3 41221. (Contributed by Stefan O'Rear, 18-Jan-2015.)
(((π‘Ž ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ π‘Ž < 𝑏) β†’ {𝑐 ∈ β„š ∣ 𝑐 < π‘Ž} β‰  {𝑐 ∈ β„š ∣ 𝑐 < 𝑏})
 
Theoremrpnnen3 41221 Dedekind cut injection of ℝ into 𝒫 β„š. (Contributed by Stefan O'Rear, 18-Jan-2015.)
ℝ β‰Ό 𝒫 β„š
 
21.29.36  More equivalents of the Axiom of Choice
 
Theoremaxac10 41222 Characterization of choice similar to dffin1-5 10257. (Contributed by Stefan O'Rear, 6-Jan-2015.)
( β‰ˆ β€œ On) = V
 
Theoremharinf 41223 The Hartogs number of an infinite set is at least Ο‰. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
((𝑆 ∈ 𝑉 ∧ Β¬ 𝑆 ∈ Fin) β†’ Ο‰ βŠ† (harβ€˜π‘†))
 
Theoremwdom2d2 41224* Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝐢 ∈ 𝑋)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ βˆƒπ‘¦ ∈ 𝐡 βˆƒπ‘§ ∈ 𝐢 π‘₯ = 𝑋)    β‡’   (πœ‘ β†’ 𝐴 β‰Ό* (𝐡 Γ— 𝐢))
 
Theoremttac 41225 Tarski's theorem about choice: infxpidm 10431 is equivalent to ax-ac 10328. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
(CHOICE ↔ βˆ€π‘(Ο‰ β‰Ό 𝑐 β†’ (𝑐 Γ— 𝑐) β‰ˆ 𝑐))
 
Theorempw2f1ocnv 41226* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8956, 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 = (π‘₯ ∈ (2o ↑m 𝐴) ↦ (β—‘π‘₯ β€œ {1o}))    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◑𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, βˆ…)))))
 
Theorempw2f1o2 41227* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8956, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = (π‘₯ ∈ (2o ↑m 𝐴) ↦ (β—‘π‘₯ β€œ {1o}))    β‡’   (𝐴 ∈ 𝑉 β†’ 𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴)
 
Theorempw2f1o2val 41228* Function value of the pw2f1o2 41227 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐹 = (π‘₯ ∈ (2o ↑m 𝐴) ↦ (β—‘π‘₯ β€œ {1o}))    β‡’   (𝑋 ∈ (2o ↑m 𝐴) β†’ (πΉβ€˜π‘‹) = (◑𝑋 β€œ {1o}))
 
Theorempw2f1o2val2 41229* Membership in a mapped set under the pw2f1o2 41227 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐹 = (π‘₯ ∈ (2o ↑m 𝐴) ↦ (β—‘π‘₯ β€œ {1o}))    β‡’   ((𝑋 ∈ (2o ↑m 𝐴) ∧ π‘Œ ∈ 𝐴) β†’ (π‘Œ ∈ (πΉβ€˜π‘‹) ↔ (π‘‹β€˜π‘Œ) = 1o))
 
Theoremsoeq12d 41230 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(πœ‘ β†’ 𝑅 = 𝑆)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐡))
 
Theoremfreq12d 41231 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(πœ‘ β†’ 𝑅 = 𝑆)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐡))
 
Theoremweeq12d 41232 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(πœ‘ β†’ 𝑅 = 𝑆)    &   (πœ‘ β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ (𝑅 We 𝐴 ↔ 𝑆 We 𝐡))
 
Theoremlimsuc2 41233 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
((Ord 𝐴 ∧ 𝐴 = βˆͺ 𝐴) β†’ (𝐡 ∈ 𝐴 ↔ suc 𝐡 ∈ 𝐴))
 
Theoremwepwsolem 41234* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘§ ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ Β¬ 𝑧 ∈ π‘₯) ∧ βˆ€π‘€ ∈ 𝐴 (𝑀𝑅𝑧 β†’ (𝑀 ∈ π‘₯ ↔ 𝑀 ∈ 𝑦)))}    &   π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘§ ∈ 𝐴 ((π‘₯β€˜π‘§) E (π‘¦β€˜π‘§) ∧ βˆ€π‘€ ∈ 𝐴 (𝑀𝑅𝑧 β†’ (π‘₯β€˜π‘€) = (π‘¦β€˜π‘€)))}    &   πΉ = (π‘Ž ∈ (2o ↑m 𝐴) ↦ (β—‘π‘Ž β€œ {1o}))    β‡’   (𝐴 ∈ V β†’ 𝐹 Isom π‘ˆ, 𝑇 ((2o ↑m 𝐴), 𝒫 𝐴))
 
Theoremwepwso 41235* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴 ∈ 𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘§ ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ Β¬ 𝑧 ∈ π‘₯) ∧ βˆ€π‘€ ∈ 𝐴 (𝑀𝑅𝑧 β†’ (𝑀 ∈ π‘₯ ↔ 𝑀 ∈ 𝑦)))}    β‡’   ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) β†’ 𝑇 Or 𝒫 𝐴)
 
Theoremdnnumch1 41236* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9899. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (πΊβ€˜(𝐴 βˆ– ran 𝑧))))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 β‰  βˆ… β†’ (πΊβ€˜π‘¦) ∈ 𝑦))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ On (𝐹 β†Ύ π‘₯):π‘₯–1-1-onto→𝐴)
 
Theoremdnnumch2 41237* 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 ↦ (πΊβ€˜(𝐴 βˆ– ran 𝑧))))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 β‰  βˆ… β†’ (πΊβ€˜π‘¦) ∈ 𝑦))    β‡’   (πœ‘ β†’ 𝐴 βŠ† ran 𝐹)
 
Theoremdnnumch3lem 41238* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (πΊβ€˜(𝐴 βˆ– ran 𝑧))))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 β‰  βˆ… β†’ (πΊβ€˜π‘¦) ∈ 𝑦))    β‡’   ((πœ‘ ∧ 𝑀 ∈ 𝐴) β†’ ((π‘₯ ∈ 𝐴 ↦ ∩ (◑𝐹 β€œ {π‘₯}))β€˜π‘€) = ∩ (◑𝐹 β€œ {𝑀}))
 
Theoremdnnumch3 41239* 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 ↦ (πΊβ€˜(𝐴 βˆ– ran 𝑧))))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 β‰  βˆ… β†’ (πΊβ€˜π‘¦) ∈ 𝑦))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ ∩ (◑𝐹 β€œ {π‘₯})):𝐴–1-1β†’On)
 
Theoremdnwech 41240* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (πΊβ€˜(𝐴 βˆ– ran 𝑧))))    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ βˆ€π‘¦ ∈ 𝒫 𝐴(𝑦 β‰  βˆ… β†’ (πΊβ€˜π‘¦) ∈ 𝑦))    &   π» = {βŸ¨π‘£, π‘€βŸ© ∣ ∩ (◑𝐹 β€œ {𝑣}) ∈ ∩ (◑𝐹 β€œ {𝑀})}    β‡’   (πœ‘ β†’ 𝐻 We 𝐴)
 
Theoremfnwe2val 41241* Lemma for fnwe2 41245. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (πΉβ€˜π‘₯) β†’ 𝑆 = π‘ˆ)    &   π‘‡ = {⟨π‘₯, π‘¦βŸ© ∣ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯π‘ˆπ‘¦))}    β‡’   (π‘Žπ‘‡π‘ ↔ ((πΉβ€˜π‘Ž)𝑅(πΉβ€˜π‘) ∨ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ∧ π‘Žβ¦‹(πΉβ€˜π‘Ž) / π‘§β¦Œπ‘†π‘)))
 
Theoremfnwe2lem1 41242* Lemma for fnwe2 41245. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (πΉβ€˜π‘₯) β†’ 𝑆 = π‘ˆ)    &   π‘‡ = {⟨π‘₯, π‘¦βŸ© ∣ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯π‘ˆπ‘¦))}    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘ˆ We {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯)})    β‡’   ((πœ‘ ∧ π‘Ž ∈ 𝐴) β†’ ⦋(πΉβ€˜π‘Ž) / π‘§β¦Œπ‘† We {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Ž)})
 
Theoremfnwe2lem2 41243* Lemma for fnwe2 41245. 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 = (πΉβ€˜π‘₯) β†’ 𝑆 = π‘ˆ)    &   π‘‡ = {⟨π‘₯, π‘¦βŸ© ∣ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯π‘ˆπ‘¦))}    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘ˆ We {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯)})    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐴):𝐴⟢𝐡)    &   (πœ‘ β†’ 𝑅 We 𝐡)    &   (πœ‘ β†’ π‘Ž βŠ† 𝐴)    &   (πœ‘ β†’ π‘Ž β‰  βˆ…)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ π‘Ž βˆ€π‘ ∈ π‘Ž Β¬ 𝑐𝑇𝑏)
 
Theoremfnwe2lem3 41244* Lemma for fnwe2 41245. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (πΉβ€˜π‘₯) β†’ 𝑆 = π‘ˆ)    &   π‘‡ = {⟨π‘₯, π‘¦βŸ© ∣ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯π‘ˆπ‘¦))}    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘ˆ We {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯)})    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐴):𝐴⟢𝐡)    &   (πœ‘ β†’ 𝑅 We 𝐡)    &   (πœ‘ β†’ π‘Ž ∈ 𝐴)    &   (πœ‘ β†’ 𝑏 ∈ 𝐴)    β‡’   (πœ‘ β†’ (π‘Žπ‘‡π‘ ∨ π‘Ž = 𝑏 ∨ π‘π‘‡π‘Ž))
 
Theoremfnwe2 41245* 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 8052 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (πΉβ€˜π‘₯) β†’ 𝑆 = π‘ˆ)    &   π‘‡ = {⟨π‘₯, π‘¦βŸ© ∣ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯π‘ˆπ‘¦))}    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘ˆ We {𝑦 ∈ 𝐴 ∣ (πΉβ€˜π‘¦) = (πΉβ€˜π‘₯)})    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐴):𝐴⟢𝐡)    &   (πœ‘ β†’ 𝑅 We 𝐡)    β‡’   (πœ‘ β†’ 𝑇 We 𝐴)
 
Theoremaomclem1 41246* Lemma for dfac11 41254. 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 (𝑅1β€˜π΄). 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.)

𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}    &   (πœ‘ β†’ dom 𝑧 ∈ On)    &   (πœ‘ β†’ dom 𝑧 = suc βˆͺ dom 𝑧)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))    β‡’   (πœ‘ β†’ 𝐡 Or (𝑅1β€˜dom 𝑧))
 
Theoremaomclem2 41247* Lemma for dfac11 41254. Successor case 2, a choice function for subsets of (𝑅1β€˜dom 𝑧). (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}    &   πΆ = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))    &   (πœ‘ β†’ dom 𝑧 ∈ On)    &   (πœ‘ β†’ dom 𝑧 = suc βˆͺ dom 𝑧)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))    &   (πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ dom 𝑧 βŠ† 𝐴)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))    β‡’   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜dom 𝑧)(π‘Ž β‰  βˆ… β†’ (πΆβ€˜π‘Ž) ∈ π‘Ž))
 
Theoremaomclem3 41248* Lemma for dfac11 41254. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}    &   πΆ = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))    &   π· = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))    &   πΈ = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}    &   (πœ‘ β†’ dom 𝑧 ∈ On)    &   (πœ‘ β†’ dom 𝑧 = suc βˆͺ dom 𝑧)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))    &   (πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ dom 𝑧 βŠ† 𝐴)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))    β‡’   (πœ‘ β†’ 𝐸 We (𝑅1β€˜dom 𝑧))
 
Theoremaomclem4 41249* Lemma for dfac11 41254. Limit case. Patch together well-orderings constructed so far using fnwe2 41245 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐹 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}    &   (πœ‘ β†’ dom 𝑧 ∈ On)    &   (πœ‘ β†’ dom 𝑧 = βˆͺ dom 𝑧)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))    β‡’   (πœ‘ β†’ 𝐹 We (𝑅1β€˜dom 𝑧))
 
Theoremaomclem5 41250* Lemma for dfac11 41254. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}    &   πΆ = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))    &   π· = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))    &   πΈ = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}    &   πΉ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}    &   πΊ = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))    &   (πœ‘ β†’ dom 𝑧 ∈ On)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ dom 𝑧(π‘§β€˜π‘Ž) We (𝑅1β€˜π‘Ž))    &   (πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ dom 𝑧 βŠ† 𝐴)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))    β‡’   (πœ‘ β†’ 𝐺 We (𝑅1β€˜dom 𝑧))
 
Theoremaomclem6 41251* Lemma for dfac11 41254. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}    &   πΆ = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))    &   π· = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))    &   πΈ = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}    &   πΉ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}    &   πΊ = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))    &   π» = recs((𝑧 ∈ V ↦ 𝐺))    &   (πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))    β‡’   (πœ‘ β†’ (π»β€˜π΄) We (𝑅1β€˜π΄))
 
Theoremaomclem7 41252* Lemma for dfac11 41254. (𝑅1β€˜π΄) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐡 = {βŸ¨π‘Ž, π‘βŸ© ∣ βˆƒπ‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)((𝑐 ∈ 𝑏 ∧ Β¬ 𝑐 ∈ π‘Ž) ∧ βˆ€π‘‘ ∈ (𝑅1β€˜βˆͺ dom 𝑧)(𝑑(π‘§β€˜βˆͺ dom 𝑧)𝑐 β†’ (𝑑 ∈ π‘Ž ↔ 𝑑 ∈ 𝑏)))}    &   πΆ = (π‘Ž ∈ V ↦ sup((π‘¦β€˜π‘Ž), (𝑅1β€˜dom 𝑧), 𝐡))    &   π· = recs((π‘Ž ∈ V ↦ (πΆβ€˜((𝑅1β€˜dom 𝑧) βˆ– ran π‘Ž))))    &   πΈ = {βŸ¨π‘Ž, π‘βŸ© ∣ ∩ (◑𝐷 β€œ {π‘Ž}) ∈ ∩ (◑𝐷 β€œ {𝑏})}    &   πΉ = {βŸ¨π‘Ž, π‘βŸ© ∣ ((rankβ€˜π‘Ž) E (rankβ€˜π‘) ∨ ((rankβ€˜π‘Ž) = (rankβ€˜π‘) ∧ π‘Ž(π‘§β€˜suc (rankβ€˜π‘Ž))𝑏))}    &   πΊ = (if(dom 𝑧 = βˆͺ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1β€˜dom 𝑧) Γ— (𝑅1β€˜dom 𝑧)))    &   π» = recs((𝑧 ∈ V ↦ 𝐺))    &   (πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))    β‡’   (πœ‘ β†’ βˆƒπ‘ 𝑏 We (𝑅1β€˜π΄))
 
Theoremaomclem8 41253* Lemma for dfac11 41254. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(πœ‘ β†’ 𝐴 ∈ On)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝒫 (𝑅1β€˜π΄)(π‘Ž β‰  βˆ… β†’ (π‘¦β€˜π‘Ž) ∈ ((𝒫 π‘Ž ∩ Fin) βˆ– {βˆ…})))    β‡’   (πœ‘ β†’ βˆƒπ‘ 𝑏 We (𝑅1β€˜π΄))
 
Theoremdfac11 41254* The right-hand side of this theorem (compare with ac4 10344), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9461, 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 ↔ βˆ€π‘₯βˆƒπ‘“βˆ€π‘§ ∈ π‘₯ (𝑧 β‰  βˆ… β†’ (π‘“β€˜π‘§) ∈ ((𝒫 𝑧 ∩ Fin) βˆ– {βˆ…})))
 
Theoremkelac1 41255* 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.)
((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝑆 β‰  βˆ…)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐽 ∈ Top)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐢 ∈ (Clsdβ€˜π½))    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐡:𝑆–1-1-onto→𝐢)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ π‘ˆ ∈ βˆͺ 𝐽)    &   (πœ‘ β†’ (∏tβ€˜(π‘₯ ∈ 𝐼 ↦ 𝐽)) ∈ Comp)    β‡’   (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 𝑆 β‰  βˆ…)
 
Theoremkelac2lem 41256 Lemma for kelac2 41257 and dfac21 41258: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝑆 ∈ 𝑉 β†’ (topGenβ€˜{𝑆, {𝒫 βˆͺ 𝑆}}) ∈ Comp)
 
Theoremkelac2 41257* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝑆 ∈ 𝑉)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝑆 β‰  βˆ…)    &   (πœ‘ β†’ (∏tβ€˜(π‘₯ ∈ 𝐼 ↦ (topGenβ€˜{𝑆, {𝒫 βˆͺ 𝑆}}))) ∈ Comp)    β‡’   (πœ‘ β†’ Xπ‘₯ ∈ 𝐼 𝑆 β‰  βˆ…)
 
Theoremdfac21 41258 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 ↔ βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
 
21.29.37  Finitely generated left modules
 
Syntaxclfig 41259 Extend class notation with the class of finitely generated left modules.
class LFinGen
 
Definitiondf-lfig 41260 Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using β†Ύs. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen = {𝑀 ∈ LMod ∣ (Baseβ€˜π‘€) ∈ ((LSpanβ€˜π‘€) β€œ (𝒫 (Baseβ€˜π‘€) ∩ Fin))}
 
Theoremislmodfg 41261* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘Š ∈ LFinGen ↔ βˆƒπ‘ ∈ 𝒫 𝐡(𝑏 ∈ Fin ∧ (π‘β€˜π‘) = 𝐡)))
 
Theoremislssfg 41262* Property of a finitely generated left (sub)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (𝑋 ∈ LFinGen ↔ βˆƒπ‘ ∈ 𝒫 π‘ˆ(𝑏 ∈ Fin ∧ (π‘β€˜π‘) = π‘ˆ)))
 
Theoremislssfg2 41263* 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (𝑋 ∈ LFinGen ↔ βˆƒπ‘ ∈ (𝒫 𝐡 ∩ Fin)(π‘β€˜π‘) = π‘ˆ))
 
Theoremislssfgi 41264 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑁 = (LSpanβ€˜π‘Š)    &   π‘‰ = (Baseβ€˜π‘Š)    &   π‘‹ = (π‘Š β†Ύs (π‘β€˜π΅))    β‡’   ((π‘Š ∈ LMod ∧ 𝐡 βŠ† 𝑉 ∧ 𝐡 ∈ Fin) β†’ 𝑋 ∈ LFinGen)
 
Theoremfglmod 41265 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝑀 ∈ LFinGen β†’ 𝑀 ∈ LMod)
 
Theoremlsmfgcl 41266 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
π‘ˆ = (LSubSpβ€˜π‘Š)    &    βŠ• = (LSSumβ€˜π‘Š)    &   π· = (π‘Š β†Ύs 𝐴)    &   πΈ = (π‘Š β†Ύs 𝐡)    &   πΉ = (π‘Š β†Ύs (𝐴 βŠ• 𝐡))    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐷 ∈ LFinGen)    &   (πœ‘ β†’ 𝐸 ∈ LFinGen)    β‡’   (πœ‘ β†’ 𝐹 ∈ LFinGen)
 
21.29.38  Noetherian left modules I
 
Syntaxclnm 41267 Extend class notation with the class of Noetherian left modules.
class LNoeM
 
Definitiondf-lnm 41268* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM = {𝑀 ∈ LMod ∣ βˆ€π‘– ∈ (LSubSpβ€˜π‘€)(𝑀 β†Ύs 𝑖) ∈ LFinGen}
 
Theoremislnm 41269* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑆 = (LSubSpβ€˜π‘€)    β‡’   (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ βˆ€π‘– ∈ 𝑆 (𝑀 β†Ύs 𝑖) ∈ LFinGen))
 
Theoremislnm2 41270* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐡 = (Baseβ€˜π‘€)    &   π‘† = (LSubSpβ€˜π‘€)    &   π‘ = (LSpanβ€˜π‘€)    β‡’   (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ βˆ€π‘– ∈ 𝑆 βˆƒπ‘” ∈ (𝒫 𝐡 ∩ Fin)𝑖 = (π‘β€˜π‘”)))
 
Theoremlnmlmod 41271 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM β†’ 𝑀 ∈ LMod)
 
Theoremlnmlssfg 41272 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑆 = (LSubSpβ€˜π‘€)    &   π‘… = (𝑀 β†Ύs π‘ˆ)    β‡’   ((𝑀 ∈ LNoeM ∧ π‘ˆ ∈ 𝑆) β†’ 𝑅 ∈ LFinGen)
 
Theoremlnmlsslnm 41273 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑆 = (LSubSpβ€˜π‘€)    &   π‘… = (𝑀 β†Ύs π‘ˆ)    β‡’   ((𝑀 ∈ LNoeM ∧ π‘ˆ ∈ 𝑆) β†’ 𝑅 ∈ LNoeM)
 
Theoremlnmfg 41274 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM β†’ 𝑀 ∈ LFinGen)
 
Theoremkercvrlsm 41275 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.)
π‘ˆ = (LSubSpβ€˜π‘†)    &    βŠ• = (LSSumβ€˜π‘†)    &    0 = (0gβ€˜π‘‡)    &   πΎ = (◑𝐹 β€œ { 0 })    &   π΅ = (Baseβ€˜π‘†)    &   (πœ‘ β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))    &   (πœ‘ β†’ 𝐷 ∈ π‘ˆ)    &   (πœ‘ β†’ (𝐹 β€œ 𝐷) = ran 𝐹)    β‡’   (πœ‘ β†’ (𝐾 βŠ• 𝐷) = 𝐡)
 
Theoremlmhmfgima 41276 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
π‘Œ = (𝑇 β†Ύs (𝐹 β€œ 𝐴))    &   π‘‹ = (𝑆 β†Ύs 𝐴)    &   π‘ˆ = (LSubSpβ€˜π‘†)    &   (πœ‘ β†’ 𝑋 ∈ LFinGen)    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))    β‡’   (πœ‘ β†’ π‘Œ ∈ LFinGen)
 
Theoremlnmepi 41277 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐡 = (Baseβ€˜π‘‡)    β‡’   ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐡) β†’ 𝑇 ∈ LNoeM)
 
Theoremlmhmfgsplit 41278 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 = (0gβ€˜π‘‡)    &   πΎ = (◑𝐹 β€œ { 0 })    &   π‘ˆ = (𝑆 β†Ύs 𝐾)    &   π‘‰ = (𝑇 β†Ύs ran 𝐹)    β‡’   ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑆 ∈ LFinGen)
 
Theoremlmhmlnmsplit 41279 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 = (0gβ€˜π‘‡)    &   πΎ = (◑𝐹 β€œ { 0 })    &   π‘ˆ = (𝑆 β†Ύs 𝐾)    &   π‘‰ = (𝑇 β†Ύs ran 𝐹)    β‡’   ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LNoeM ∧ 𝑉 ∈ LNoeM) β†’ 𝑆 ∈ LNoeM)
 
Theoremlnmlmic 41280 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅 β‰ƒπ‘š 𝑆 β†’ (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
 
21.29.39  Addenda for structure powers
 
Theorempwssplit4 41281* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐸 = (𝑅 ↑s (𝐴 βˆͺ 𝐡))    &   πΊ = (Baseβ€˜πΈ)    &    0 = (0gβ€˜π‘…)    &   πΎ = {𝑦 ∈ 𝐺 ∣ (𝑦 β†Ύ 𝐴) = (𝐴 Γ— { 0 })}    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ β†Ύ 𝐡))    &   πΆ = (𝑅 ↑s 𝐴)    &   π· = (𝑅 ↑s 𝐡)    &   πΏ = (𝐸 β†Ύs 𝐾)    β‡’   ((𝑅 ∈ LMod ∧ (𝐴 βˆͺ 𝐡) ∈ 𝑉 ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ 𝐹 ∈ (𝐿 LMIso 𝐷))
 
Theoremfilnm 41282 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐡 = (Baseβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝐡 ∈ Fin) β†’ π‘Š ∈ LNoeM)
 
Theorempwslnmlem0 41283 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
π‘Œ = (π‘Š ↑s βˆ…)    β‡’   (π‘Š ∈ LMod β†’ π‘Œ ∈ LNoeM)
 
Theorempwslnmlem1 41284* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
π‘Œ = (π‘Š ↑s {𝑖})    β‡’   (π‘Š ∈ LNoeM β†’ π‘Œ ∈ LNoeM)
 
Theorempwslnmlem2 41285 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐴 ∈ V    &   π΅ ∈ V    &   π‘‹ = (π‘Š ↑s 𝐴)    &   π‘Œ = (π‘Š ↑s 𝐡)    &   π‘ = (π‘Š ↑s (𝐴 βˆͺ 𝐡))    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)    &   (πœ‘ β†’ 𝑋 ∈ LNoeM)    &   (πœ‘ β†’ π‘Œ ∈ LNoeM)    β‡’   (πœ‘ β†’ 𝑍 ∈ LNoeM)
 
Theorempwslnm 41286 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
π‘Œ = (π‘Š ↑s 𝐼)    β‡’   ((π‘Š ∈ LNoeM ∧ 𝐼 ∈ Fin) β†’ π‘Œ ∈ LNoeM)
 
21.29.40  Every set admits a group structure iff choice
 
Theoremunxpwdom3 41287* Weaker version of unxpwdom 9458 where a function is required only to be cancellative, not an injection. 𝐷 and 𝐡 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 𝐡; by column injectivity, each row can be identified in at least one way by the 𝐡 element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝐷 ∈ 𝑋)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐢 ∧ 𝑏 ∈ 𝐷) β†’ (π‘Ž + 𝑏) ∈ (𝐴 βˆͺ 𝐡))    &   (((πœ‘ ∧ π‘Ž ∈ 𝐢) ∧ (𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) β†’ ((π‘Ž + 𝑏) = (π‘Ž + 𝑐) ↔ 𝑏 = 𝑐))    &   (((πœ‘ ∧ 𝑑 ∈ 𝐷) ∧ (π‘Ž ∈ 𝐢 ∧ 𝑐 ∈ 𝐢)) β†’ ((𝑐 + 𝑑) = (π‘Ž + 𝑑) ↔ 𝑐 = π‘Ž))    &   (πœ‘ β†’ Β¬ 𝐷 β‰Ό 𝐴)    β‡’   (πœ‘ β†’ 𝐢 β‰Ό* (𝐷 Γ— 𝐡))
 
Theorempwfi2f1o 41288* The pw2f1o 8954 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 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp βˆ…}    &   πΉ = (π‘₯ ∈ 𝑆 ↦ (β—‘π‘₯ β€œ {1o}))    β‡’   (𝐴 ∈ 𝑉 β†’ 𝐹:𝑆–1-1-ontoβ†’(𝒫 𝐴 ∩ Fin))
 
Theorempwfi2en 41289* 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 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp βˆ…}    β‡’   (𝐴 ∈ 𝑉 β†’ 𝑆 β‰ˆ (𝒫 𝐴 ∩ Fin))
 
Theoremfrlmpwfi 41290 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 = (β„€/nβ„€β€˜2)    &   π‘Œ = (𝑅 freeLMod 𝐼)    &   π΅ = (Baseβ€˜π‘Œ)    β‡’   (𝐼 ∈ 𝑉 β†’ 𝐡 β‰ˆ (𝒫 𝐼 ∩ Fin))
 
Theoremgicabl 41291 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
(𝐺 ≃𝑔 𝐻 β†’ (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
 
Theoremimasgim 41292 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.)
(πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝐹:𝑉–1-1-onto→𝐡)    &   (πœ‘ β†’ 𝑅 ∈ Grp)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 GrpIso π‘ˆ))
 
Theoremisnumbasgrplem1 41293 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.)
𝐡 = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Abel ∧ 𝐢 β‰ˆ 𝐡) β†’ 𝐢 ∈ (Base β€œ Abel))
 
Theoremharn0 41294 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆 ∈ 𝑉 β†’ (harβ€˜π‘†) β‰  βˆ…)
 
Theoremnuminfctb 41295 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
((𝑆 ∈ dom card ∧ Β¬ 𝑆 ∈ Fin) β†’ Ο‰ β‰Ό 𝑆)
 
Theoremisnumbasgrplem2 41296 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 βˆͺ (harβ€˜π‘†)) ∈ (Base β€œ Grp) β†’ 𝑆 ∈ dom card)
 
Theoremisnumbasgrplem3 41297 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 β‰  βˆ…) β†’ 𝑆 ∈ (Base β€œ Abel))
 
Theoremisnumbasabl 41298 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 βˆͺ (harβ€˜π‘†)) ∈ (Base β€œ Abel))
 
Theoremisnumbasgrp 41299 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 βˆͺ (harβ€˜π‘†)) ∈ (Base β€œ Grp))
 
Theoremdfacbasgrp 41300 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 β€œ Grp) = (V βˆ– {βˆ…}))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46997
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