P. 431 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wopprc 43001	Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})
Theorem	rpnnen3lem 43002*	Lemma for rpnnen3 43003. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})
Theorem	rpnnen3 43003	Dedekind cut injection of ℝ into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ ℝ ≼ 𝒫 ℚ
21.33.36  More equivalents of the Axiom of Choice
Theorem	axac10 43004	Characterization of choice similar to dffin1-5 10400. (Contributed by Stefan O'Rear, 6-Jan-2015.)
⊢ ( ≈ “ On) = V
Theorem	harinf 43005	The Hartogs number of an infinite set is at least ω. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
⊢ ((𝑆 ∈ 𝑉 ∧ ¬ 𝑆 ∈ Fin) → ω ⊆ (har‘𝑆))
Theorem	wdom2d2 43006*	Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)    ⇒   ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶))
Theorem	ttac 43007	Tarski's theorem about choice: infxpidm 10574 is equivalent to ax-ac 10471. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
⊢ (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
Theorem	pw2f1ocnv 43008*	Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9091, 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 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))))
Theorem	pw2f1o2 43009*	Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9091, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴)
Theorem	pw2f1o2val 43010*	Function value of the pw2f1o2 43009 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))
Theorem	pw2f1o2val2 43011*	Membership in a mapped set under the pw2f1o2 43009 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o))
Theorem	limsuc2 43012	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 = ∪ 𝐴) → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
Theorem	wepwsolem 43013*	Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))}    &   ⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝐹 = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o}))    ⇒   ⊢ (𝐴 ∈ V → 𝐹 Isom 𝑈, 𝑇 ((2o ↑m 𝐴), 𝒫 𝐴))
Theorem	wepwso 43014*	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 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
Theorem	dnnumch1 43015*	Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 10042. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
Theorem	dnnumch2 43016*	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))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
Theorem	dnnumch3lem 43017*	Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
Theorem	dnnumch3 43018*	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))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)
Theorem	dnwech 43019*	Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    &   ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}    ⇒   ⊢ (𝜑 → 𝐻 We 𝐴)
Theorem	fnwe2val 43020*	Lemma for fnwe2 43024. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    ⇒   ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
Theorem	fnwe2lem1 43021*	Lemma for fnwe2 43024. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
Theorem	fnwe2lem2 43022*	Lemma for fnwe2 43024. 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 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑎 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑎 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑇𝑏)
Theorem	fnwe2lem3 43023*	Lemma for fnwe2 43024. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑎 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑏 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
Theorem	fnwe2 43024*	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 8129 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 We 𝐴)
Theorem	aomclem1 43025*	Lemma for dfac11 43033. 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 (𝑅1‘𝐴). 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)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    ⇒   ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))
Theorem	aomclem2 43026*	Lemma for dfac11 43033. 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)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎))
Theorem	aomclem3 43027*	Lemma for dfac11 43033. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (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(𝑧‘𝑎) We (𝑅1‘𝑎))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧))
Theorem	aomclem4 43028*	Lemma for dfac11 43033. Limit case. Patch together well-orderings constructed so far using fnwe2 43024 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(𝑧‘𝑎) We (𝑅1‘𝑎))    ⇒   ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧))
Theorem	aomclem5 43029*	Lemma for dfac11 43033. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (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(𝑧‘𝑎) We (𝑅1‘𝑎))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))
Theorem	aomclem6 43030*	Lemma for dfac11 43033. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (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))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → (𝐻‘𝐴) We (𝑅1‘𝐴))
Theorem	aomclem7 43031*	Lemma for dfac11 43033. (𝑅1‘𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (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))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))
Theorem	aomclem8 43032*	Lemma for dfac11 43033. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))
Theorem	dfac11 43033*	The right-hand side of this theorem (compare with ac4 10487), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9604, 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 ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Theorem	kelac1 43034*	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.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ∈ (Clsd‘𝐽))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵:𝑆–1-1-onto→𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑈 ∈ ∪ 𝐽)    &   ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
Theorem	kelac2lem 43035	Lemma for kelac2 43036 and dfac21 43037: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
Theorem	kelac2 43036*	Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)    &   ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))) ∈ Comp)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
Theorem	dfac21 43037	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 → (∏t‘𝑓) ∈ Comp))
21.33.37  Finitely generated left modules
Syntax	clfig 43038	Extend class notation with the class of finitely generated left modules.
class LFinGen
Definition	df-lfig 43039	Define the class of finitely generated left modules. Finite generation of subspaces can be interpreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
Theorem	islmodfg 43040*	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 ∧ (𝑁‘𝑏) = 𝐵)))
Theorem	islssfg 43041*	Property of a finitely generated left (sub)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝑈(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝑈)))
Theorem	islssfg2 43042*	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 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑁‘𝑏) = 𝑈))
Theorem	islssfgi 43043	Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s (𝑁‘𝐵))    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑋 ∈ LFinGen)
Theorem	fglmod 43044	Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
Theorem	lsmfgcl 43045	The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐷 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐸 = (𝑊 ↾s 𝐵)    &   ⊢ 𝐹 = (𝑊 ↾s (𝐴 ⊕ 𝐵))    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ LFinGen)    &   ⊢ (𝜑 → 𝐸 ∈ LFinGen)    ⇒   ⊢ (𝜑 → 𝐹 ∈ LFinGen)
21.33.38  Noetherian left modules I
Syntax	clnm 43046	Extend class notation with the class of Noetherian left modules.
class LNoeM
Definition	df-lnm 43047*	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)(𝑤 ↾s 𝑖) ∈ LFinGen}
Theorem	islnm 43048*	Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ 𝑆 = (LSubSp‘𝑀)    ⇒   ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))
Theorem	islnm2 43049*	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 ∧ ∀𝑖 ∈ 𝑆 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔)))
Theorem	lnmlmod 43050	A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
Theorem	lnmlssfg 43051	A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (𝑀 ↾s 𝑈)    ⇒   ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)
Theorem	lnmlsslnm 43052	All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (𝑀 ↾s 𝑈)    ⇒   ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LNoeM)
Theorem	lnmfg 43053	A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)
Theorem	kercvrlsm 43054	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 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐹 “ 𝐷) = ran 𝐹)    ⇒   ⊢ (𝜑 → (𝐾 ⊕ 𝐷) = 𝐵)
Theorem	lmhmfgima 43055	A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑇 ↾s (𝐹 “ 𝐴))    &   ⊢ 𝑋 = (𝑆 ↾s 𝐴)    &   ⊢ 𝑈 = (LSubSp‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ LFinGen)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    ⇒   ⊢ (𝜑 → 𝑌 ∈ LFinGen)
Theorem	lnmepi 43056	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)
Theorem	lmhmfgsplit 43057	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 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑈 = (𝑆 ↾s 𝐾)    &   ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)
Theorem	lmhmlnmsplit 43058	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 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑈 = (𝑆 ↾s 𝐾)    &   ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
Theorem	lnmlmic 43059	Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
21.33.39  Addenda for structure powers
Theorem	pwssplit4 43060*	Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐸 = (𝑅 ↑s (𝐴 ∪ 𝐵))    &   ⊢ 𝐺 = (Base‘𝐸)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵))    &   ⊢ 𝐶 = (𝑅 ↑s 𝐴)    &   ⊢ 𝐷 = (𝑅 ↑s 𝐵)    &   ⊢ 𝐿 = (𝐸 ↾s 𝐾)    ⇒   ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Theorem	filnm 43061	Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)
Theorem	pwslnmlem0 43062	Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s ∅)    ⇒   ⊢ (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem1 43063*	First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s {𝑖})    ⇒   ⊢ (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem2 43064	A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑋 = (𝑊 ↑s 𝐴)    &   ⊢ 𝑌 = (𝑊 ↑s 𝐵)    &   ⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ LNoeM)    &   ⊢ (𝜑 → 𝑌 ∈ LNoeM)    ⇒   ⊢ (𝜑 → 𝑍 ∈ LNoeM)
Theorem	pwslnm 43065	Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s 𝐼)    ⇒   ⊢ ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM)
21.33.40  Every set admits a group structure iff choice
Theorem	unxpwdom3 43066*	Weaker version of unxpwdom 9601 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
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐷) → (𝑎 + 𝑏) ∈ (𝐴 ∪ 𝐵))    &   ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐶) ∧ (𝑏 ∈ 𝐷 ∧ 𝑐 ∈ 𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))    &   ⊢ (((𝜑 ∧ 𝑑 ∈ 𝐷) ∧ (𝑎 ∈ 𝐶 ∧ 𝑐 ∈ 𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))    &   ⊢ (𝜑 → ¬ 𝐷 ≼ 𝐴)    ⇒   ⊢ (𝜑 → 𝐶 ≼* (𝐷 × 𝐵))
Theorem	pwfi2f1o 43067*	The pw2f1o 9089 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 ∅}    &   ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))
Theorem	pwfi2en 43068*	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 ∅}    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝑆 ≈ (𝒫 𝐴 ∩ Fin))
Theorem	frlmpwfi 43069	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)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 ≈ (𝒫 𝐼 ∩ Fin))
Theorem	gicabl 43070	Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
⊢ (𝐺 ≃𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))
Theorem	imasgim 43071	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.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈))
Theorem	isnumbasgrplem1 43072	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 ∧ 𝐶 ≈ 𝐵) → 𝐶 ∈ (Base “ Abel))
Theorem	harn0 43073	The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
⊢ (𝑆 ∈ 𝑉 → (har‘𝑆) ≠ ∅)
Theorem	numinfctb 43074	A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)
Theorem	isnumbasgrplem2 43075	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)
Theorem	isnumbasgrplem3 43076	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))
Theorem	isnumbasabl 43077	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))
Theorem	isnumbasgrp 43078	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))
Theorem	dfacbasgrp 43079	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 ∖ {∅}))
21.33.41  Noetherian rings and left modules II
Syntax	clnr 43080	Extend class notation with the class of left Noetherian rings.
class LNoeR
Definition	df-lnr 43081	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}
Theorem	islnr 43082	Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
Theorem	lnrring 43083	Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR → 𝐴 ∈ Ring)
Theorem	lnrlnm 43084	Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)
Theorem	islnr2 43085*	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.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑁 = (RSpan‘𝑅)    ⇒   ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔)))
Theorem	islnr3 43086	Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ 𝑈 ∈ (NoeACS‘𝐵)))
Theorem	lnr2i 43087*	Given an ideal in a left-Noetherian ring, there is a finite subset which generates it. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑁 = (RSpan‘𝑅)    ⇒   ⊢ ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ 𝑈) → ∃𝑔 ∈ (𝒫 𝐼 ∩ Fin)𝐼 = (𝑁‘𝑔))
Theorem	lpirlnr 43088	Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ (𝑅 ∈ LPIR → 𝑅 ∈ LNoeR)
Theorem	lnrfrlm 43089	Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM)
Theorem	lnrfg 43090	Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
⊢ 𝑆 = (Scalar‘𝑀)    ⇒   ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM)
Theorem	lnrfgtr 43091	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 = (Scalar‘𝑀)    &   ⊢ 𝑈 = (LSubSp‘𝑀)    &   ⊢ 𝑁 = (𝑀 ↾s 𝑃)    ⇒   ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃 ∈ 𝑈) → 𝑁 ∈ LFinGen)
21.33.42  Hilbert's Basis Theorem
Syntax	cldgis 43092	The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq
Definition	df-ldgis 43093*	Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- 𝑥 elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 43101. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 (((deg1‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
Theorem	hbtlem1 43094*	Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})
Theorem	hbtlem2 43095	Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ 𝑇 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇)
Theorem	hbtlem7 43096	Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ 𝑇 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇)
Theorem	hbtlem4 43097	The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑌 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐼)‘𝑌))
Theorem	hbtlem3 43098	The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐼 ⊆ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋))
Theorem	hbtlem5 43099*	The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐼 ⊆ 𝐽)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥))    ⇒   ⊢ (𝜑 → 𝐼 = 𝐽)
Theorem	hbtlem6 43100*	There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑃)    &   ⊢ 𝑆 = (ldgIdlSeq‘𝑅)    &   ⊢ 𝑁 = (RSpan‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ LNoeR)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))