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Theorem List for Metamath Proof Explorer - 41201-41300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremweeq12d 41201 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 41202 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 41203* 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 41204* 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 41205* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9900. (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 41206* 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 41207* 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 41208* 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 41209* 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 41210* Lemma for fnwe2 41214. 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 41211* Lemma for fnwe2 41214. 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 41212* Lemma for fnwe2 41214. 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 41213* Lemma for fnwe2 41214. 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 41214* 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 8053 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 41215* Lemma for dfac11 41223. 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 41216* Lemma for dfac11 41223. 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 41217* Lemma for dfac11 41223. 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 41218* Lemma for dfac11 41223. Limit case. Patch together well-orderings constructed so far using fnwe2 41214 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 41219* Lemma for dfac11 41223. 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 41220* Lemma for dfac11 41223. 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 41221* Lemma for dfac11 41223. (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 41222* Lemma for dfac11 41223. 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 41223* The right-hand side of this theorem (compare with ac4 10345), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9462, 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 41224* 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 41225 Lemma for kelac2 41226 and dfac21 41227: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝑆 ∈ 𝑉 β†’ (topGenβ€˜{𝑆, {𝒫 βˆͺ 𝑆}}) ∈ Comp)
 
Theoremkelac2 41226* 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 41227 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 41228 Extend class notation with the class of finitely generated left modules.
class LFinGen
 
Definitiondf-lfig 41229 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 41230* 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 41231* 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 41232* 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 41233 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 41234 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝑀 ∈ LFinGen β†’ 𝑀 ∈ LMod)
 
Theoremlsmfgcl 41235 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 41236 Extend class notation with the class of Noetherian left modules.
class LNoeM
 
Definitiondf-lnm 41237* 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 41238* 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 41239* 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 41240 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM β†’ 𝑀 ∈ LMod)
 
Theoremlnmlssfg 41241 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 41242 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 41243 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM β†’ 𝑀 ∈ LFinGen)
 
Theoremkercvrlsm 41244 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 41245 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 41246 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 41247 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 41248 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 41249 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 41250* 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 41251 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 41252 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 41253* 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 41254 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 41255 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 41256* Weaker version of unxpwdom 9459 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 41257* The pw2f1o 8955 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 41258* 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 41259 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 41260 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
(𝐺 ≃𝑔 𝐻 β†’ (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
 
Theoremimasgim 41261 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 41262 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 41263 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 41264 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 41265 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 41266 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 41267 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 41268 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 41269 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…}))
 
21.29.41  Noetherian rings and left modules II
 
Syntaxclnr 41270 Extend class notation with the class of left Noetherian rings.
class LNoeR
 
Definitiondf-lnr 41271 A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR = {π‘Ž ∈ Ring ∣ (ringLModβ€˜π‘Ž) ∈ LNoeM}
 
Theoremislnr 41272 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLModβ€˜π΄) ∈ LNoeM))
 
Theoremlnrring 41273 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR β†’ 𝐴 ∈ Ring)
 
Theoremlnrlnm 41274 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR β†’ (ringLModβ€˜π΄) ∈ LNoeM)
 
Theoremislnr2 41275* Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    &   π‘ = (RSpanβ€˜π‘…)    β‡’   (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ βˆ€π‘– ∈ π‘ˆ βˆƒπ‘” ∈ (𝒫 𝐡 ∩ Fin)𝑖 = (π‘β€˜π‘”)))
 
Theoremislnr3 41276 Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ π‘ˆ ∈ (NoeACSβ€˜π΅)))
 
Theoremlnr2i 41277* Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
π‘ˆ = (LIdealβ€˜π‘…)    &   π‘ = (RSpanβ€˜π‘…)    β‡’   ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ π‘ˆ) β†’ βˆƒπ‘” ∈ (𝒫 𝐼 ∩ Fin)𝐼 = (π‘β€˜π‘”))
 
Theoremlpirlnr 41278 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR β†’ 𝑅 ∈ LNoeR)
 
Theoremlnrfrlm 41279 Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
π‘Œ = (𝑅 freeLMod 𝐼)    β‡’   ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ Fin) β†’ π‘Œ ∈ LNoeM)
 
Theoremlnrfg 41280 Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
𝑆 = (Scalarβ€˜π‘€)    β‡’   ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) β†’ 𝑀 ∈ LNoeM)
 
Theoremlnrfgtr 41281 A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
𝑆 = (Scalarβ€˜π‘€)    &   π‘ˆ = (LSubSpβ€˜π‘€)    &   π‘ = (𝑀 β†Ύs 𝑃)    β‡’   ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃 ∈ π‘ˆ) β†’ 𝑁 ∈ LFinGen)
 
21.29.42  Hilbert's Basis Theorem
 
Syntaxcldgis 41282 The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq
 
Definitiondf-ldgis 41283* Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- π‘₯ elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 41291. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq = (π‘Ÿ ∈ V ↦ (𝑖 ∈ (LIdealβ€˜(Poly1β€˜π‘Ÿ)) ↦ (π‘₯ ∈ β„•0 ↦ {𝑗 ∣ βˆƒπ‘˜ ∈ 𝑖 ((( deg1 β€˜π‘Ÿ)β€˜π‘˜) ≀ π‘₯ ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘₯))})))
 
Theoremhbtlem1 41284* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   π· = ( deg1 β€˜π‘…)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = {𝑗 ∣ βˆƒπ‘˜ ∈ 𝐼 ((π·β€˜π‘˜) ≀ 𝑋 ∧ 𝑗 = ((coe1β€˜π‘˜)β€˜π‘‹))})
 
Theoremhbtlem2 41285 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   π‘‡ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜πΌ)β€˜π‘‹) ∈ 𝑇)
 
Theoremhbtlem7 41286 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   π‘‡ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) β†’ (π‘†β€˜πΌ):β„•0βŸΆπ‘‡)
 
Theoremhbtlem4 41287 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ β„•0)    &   (πœ‘ β†’ π‘Œ ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    β‡’   (πœ‘ β†’ ((π‘†β€˜πΌ)β€˜π‘‹) βŠ† ((π‘†β€˜πΌ)β€˜π‘Œ))
 
Theoremhbtlem3 41288 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐽 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐼 βŠ† 𝐽)    &   (πœ‘ β†’ 𝑋 ∈ β„•0)    β‡’   (πœ‘ β†’ ((π‘†β€˜πΌ)β€˜π‘‹) βŠ† ((π‘†β€˜π½)β€˜π‘‹))
 
Theoremhbtlem5 41289* The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐽 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐼 βŠ† 𝐽)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„•0 ((π‘†β€˜π½)β€˜π‘₯) βŠ† ((π‘†β€˜πΌ)β€˜π‘₯))    β‡’   (πœ‘ β†’ 𝐼 = 𝐽)
 
Theoremhbtlem6 41290* There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘ƒ)    &   π‘† = (ldgIdlSeqβ€˜π‘…)    &   π‘ = (RSpanβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝑅 ∈ LNoeR)    &   (πœ‘ β†’ 𝐼 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑋 ∈ β„•0)    β‡’   (πœ‘ β†’ βˆƒπ‘˜ ∈ (𝒫 𝐼 ∩ Fin)((π‘†β€˜πΌ)β€˜π‘‹) βŠ† ((π‘†β€˜(π‘β€˜π‘˜))β€˜π‘‹))
 
Theoremhbt 41291 The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑃 = (Poly1β€˜π‘…)    β‡’   (𝑅 ∈ LNoeR β†’ 𝑃 ∈ LNoeR)
 
21.29.43  Additional material on polynomials [DEPRECATED]
 
Syntaxcmnc 41292 Extend class notation with the class of monic polynomials.
class Monic
 
Syntaxcplylt 41293 Extend class notatin with the class of limited-degree polynomials.
class Poly<
 
Definitiondf-mnc 41294* Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Monic = (𝑠 ∈ 𝒫 β„‚ ↦ {𝑝 ∈ (Polyβ€˜π‘ ) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1})
 
Definitiondf-plylt 41295* Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.)
Poly< = (𝑠 ∈ 𝒫 β„‚, π‘₯ ∈ β„•0 ↦ {𝑝 ∈ (Polyβ€˜π‘ ) ∣ (𝑝 = 0𝑝 ∨ (degβ€˜π‘) < π‘₯)})
 
Theoremdgrsub2 41296 Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
𝑁 = (degβ€˜πΉ)    β‡’   (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘‡)) ∧ ((degβ€˜πΊ) = 𝑁 ∧ 𝑁 ∈ β„• ∧ ((coeffβ€˜πΉ)β€˜π‘) = ((coeffβ€˜πΊ)β€˜π‘))) β†’ (degβ€˜(𝐹 ∘f βˆ’ 𝐺)) < 𝑁)
 
Theoremelmnc 41297 Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic β€˜π‘†) ↔ (𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1))
 
Theoremmncply 41298 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic β€˜π‘†) β†’ 𝑃 ∈ (Polyβ€˜π‘†))
 
Theoremmnccoe 41299 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic β€˜π‘†) β†’ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1)
 
Theoremmncn0 41300 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)
(𝑃 ∈ ( Monic β€˜π‘†) β†’ 𝑃 β‰  0𝑝)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 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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 46901-46966
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