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Theorem List for Metamath Proof Explorer - 36501-36600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrpnnen3 36501 Dedekind cut injection of into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
ℝ ≼ 𝒫 ℚ

20.24.36  More equivalents of the Axiom of Choice

Theoremaxac10 36502 Characterization of choice similar to dffin1-5 8973. (Contributed by Stefan O'Rear, 6-Jan-2015.)
( ≈ “ On) = V

Theoremharinf 36503 The Hartogs number of an infinite set is at least ω. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
((𝑆𝑉 ∧ ¬ 𝑆 ∈ Fin) → ω ⊆ (har‘𝑆))

Theoremwdom2d2 36504* Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝐴) → ∃𝑦𝐵𝑧𝐶 𝑥 = 𝑋)       (𝜑𝐴* (𝐵 × 𝐶))

Theoremttac 36505 Tarski's theorem about choice: infxpidm 9143 is equivalent to ax-ac 9044. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
(CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))

Theorempw2f1ocnv 36506* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7832, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       (𝐴𝑉 → (𝐹:(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑦, 1𝑜, ∅)))))

Theorempw2f1o2 36507* Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7832, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       (𝐴𝑉𝐹:(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)

Theorempw2f1o2val 36508* Function value of the pw2f1o2 36507 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       (𝑋 ∈ (2𝑜𝑚 𝐴) → (𝐹𝑋) = (𝑋 “ {1𝑜}))

Theorempw2f1o2val2 36509* Membership in a mapped set under the pw2f1o2 36507 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐹 = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))       ((𝑋 ∈ (2𝑜𝑚 𝐴) ∧ 𝑌𝐴) → (𝑌 ∈ (𝐹𝑋) ↔ (𝑋𝑌) = 1𝑜))

Theoremsoeq12d 36510 Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 Or 𝐴𝑆 Or 𝐵))

Theoremfreq12d 36511 Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐵))

Theoremweeq12d 36512 Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝜑𝑅 = 𝑆)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))

Theoremlimsuc2 36513 Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))

Theoremwepwsolem 36514* Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}    &   𝑈 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧) E (𝑦𝑧) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑥𝑤) = (𝑦𝑤)))}    &   𝐹 = (𝑎 ∈ (2𝑜𝑚 𝐴) ↦ (𝑎 “ {1𝑜}))       (𝐴 ∈ V → 𝐹 Isom 𝑈, 𝑇 ((2𝑜𝑚 𝐴), 𝒫 𝐴))

Theoremwepwso 36515* A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑧𝑦 ∧ ¬ 𝑧𝑥) ∧ ∀𝑤𝐴 (𝑤𝑅𝑧 → (𝑤𝑥𝑤𝑦)))}       ((𝐴𝑉𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)

Theoremdnnumch1 36516* Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 8616. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)

Theoremdnnumch2 36517* Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       (𝜑𝐴 ⊆ ran 𝐹)

Theoremdnnumch3lem 36518* Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       ((𝜑𝑤𝐴) → ((𝑥𝐴 (𝐹 “ {𝑥}))‘𝑤) = (𝐹 “ {𝑤}))

Theoremdnnumch3 36519* Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))       (𝜑 → (𝑥𝐴 (𝐹 “ {𝑥})):𝐴1-1→On)

Theoremdnwech 36520* Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))    &   𝐻 = {⟨𝑣, 𝑤⟩ ∣ (𝐹 “ {𝑣}) ∈ (𝐹 “ {𝑤})}       (𝜑𝐻 We 𝐴)

Theoremfnwe2val 36521* Lemma for fnwe2 36525. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}       (𝑎𝑇𝑏 ↔ ((𝐹𝑎)𝑅(𝐹𝑏) ∨ ((𝐹𝑎) = (𝐹𝑏) ∧ 𝑎(𝐹𝑎) / 𝑧𝑆𝑏)))

Theoremfnwe2lem1 36522* Lemma for fnwe2 36525. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})       ((𝜑𝑎𝐴) → (𝐹𝑎) / 𝑧𝑆 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑎)})

Theoremfnwe2lem2 36523* Lemma for fnwe2 36525. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus 𝑇 is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})    &   (𝜑 → (𝐹𝐴):𝐴𝐵)    &   (𝜑𝑅 We 𝐵)    &   (𝜑𝑎𝐴)    &   (𝜑𝑎 ≠ ∅)       (𝜑 → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑇𝑏)

Theoremfnwe2lem3 36524* Lemma for fnwe2 36525. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})    &   (𝜑 → (𝐹𝐴):𝐴𝐵)    &   (𝜑𝑅 We 𝐵)    &   (𝜑𝑎𝐴)    &   (𝜑𝑏𝐴)       (𝜑 → (𝑎𝑇𝑏𝑎 = 𝑏𝑏𝑇𝑎))

Theoremfnwe2 36525* 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 7060 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
(𝑧 = (𝐹𝑥) → 𝑆 = 𝑈)    &   𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹𝑥)𝑅(𝐹𝑦) ∨ ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑈𝑦))}    &   ((𝜑𝑥𝐴) → 𝑈 We {𝑦𝐴 ∣ (𝐹𝑦) = (𝐹𝑥)})    &   (𝜑 → (𝐹𝐴):𝐴𝐵)    &   (𝜑𝑅 We 𝐵)       (𝜑𝑇 We 𝐴)

Theoremaomclem1 36526* Lemma for dfac11 36534. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of (𝑅1𝐴). In what follows, 𝐴 is the index of the rank we wish to well-order, 𝑧 is the collection of well-orderings constructed so far, dom 𝑧 is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and 𝑦 is a postulated multiple-choice function.

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

𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = suc dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))       (𝜑𝐵 Or (𝑅1‘dom 𝑧))

Theoremaomclem2 36527* Lemma for dfac11 36534. Successor case 2, a choice function for subsets of (𝑅1‘dom 𝑧). (Contributed by Stefan O'Rear, 18-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = suc dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → dom 𝑧𝐴)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶𝑎) ∈ 𝑎))

Theoremaomclem3 36528* Lemma for dfac11 36534. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = suc dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → dom 𝑧𝐴)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑𝐸 We (𝑅1‘dom 𝑧))

Theoremaomclem4 36529* Lemma for dfac11 36534. Limit case. Patch together well-orderings constructed so far using fnwe2 36525 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → dom 𝑧 = dom 𝑧)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))       (𝜑𝐹 We (𝑅1‘dom 𝑧))

Theoremaomclem5 36530* Lemma for dfac11 36534. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   (𝜑 → dom 𝑧 ∈ On)    &   (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → dom 𝑧𝐴)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑𝐺 We (𝑅1‘dom 𝑧))

Theoremaomclem6 36531* Lemma for dfac11 36534. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   𝐻 = recs((𝑧 ∈ V ↦ 𝐺))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → (𝐻𝐴) We (𝑅1𝐴))

Theoremaomclem7 36532* Lemma for dfac11 36534. (𝑅1𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}    &   𝐶 = (𝑎 ∈ V ↦ sup((𝑦𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   𝐸 = {⟨𝑎, 𝑏⟩ ∣ (𝐷 “ {𝑎}) ∈ (𝐷 “ {𝑏})}    &   𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   𝐺 = (if(dom 𝑧 = dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   𝐻 = recs((𝑧 ∈ V ↦ 𝐺))    &   (𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))

Theoremaomclem8 36533* Lemma for dfac11 36534. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1𝐴)(𝑎 ≠ ∅ → (𝑦𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))       (𝜑 → ∃𝑏 𝑏 We (𝑅1𝐴))

Theoremdfac11 36534* The right-hand side of this theorem (compare with ac4 9060), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 8260, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do.

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

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

(CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))

Theoremkelac1 36535* Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝜑𝑥𝐼) → 𝑆 ≠ ∅)    &   ((𝜑𝑥𝐼) → 𝐽 ∈ Top)    &   ((𝜑𝑥𝐼) → 𝐶 ∈ (Clsd‘𝐽))    &   ((𝜑𝑥𝐼) → 𝐵:𝑆1-1-onto𝐶)    &   ((𝜑𝑥𝐼) → 𝑈 𝐽)    &   (𝜑 → (∏t‘(𝑥𝐼𝐽)) ∈ Comp)       (𝜑X𝑥𝐼 𝑆 ≠ ∅)

Theoremkelac2lem 36536 Lemma for kelac2 36537 and dfac21 36538: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)

Theoremkelac2 36537* Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
((𝜑𝑥𝐼) → 𝑆𝑉)    &   ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)    &   (𝜑 → (∏t‘(𝑥𝐼 ↦ (topGen‘{𝑆, {𝒫 𝑆}}))) ∈ Comp)       (𝜑X𝑥𝐼 𝑆 ≠ ∅)

Theoremdfac21 36538 Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
(CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))

20.24.37  Finitely generated left modules

Syntaxclfig 36539 Extend class notation with the class of finitely generated left modules.
class LFinGen

Definitiondf-lfig 36540 Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}

Theoremislmodfg 36541* Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐵 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁𝑏) = 𝐵)))

Theoremislssfg 36542* Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝑈(𝑏 ∈ Fin ∧ (𝑁𝑏) = 𝑈)))

Theoremislssfg2 36543* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐵 = (Base‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑁𝑏) = 𝑈))

Theoremislssfgi 36544 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑁 = (LSpan‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑋 = (𝑊s (𝑁𝐵))       ((𝑊 ∈ LMod ∧ 𝐵𝑉𝐵 ∈ Fin) → 𝑋 ∈ LFinGen)

Theoremfglmod 36545 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝑀 ∈ LFinGen → 𝑀 ∈ LMod)

Theoremlsmfgcl 36546 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑈 = (LSubSp‘𝑊)    &    = (LSSum‘𝑊)    &   𝐷 = (𝑊s 𝐴)    &   𝐸 = (𝑊s 𝐵)    &   𝐹 = (𝑊s (𝐴 𝐵))    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷 ∈ LFinGen)    &   (𝜑𝐸 ∈ LFinGen)       (𝜑𝐹 ∈ LFinGen)

20.24.38  Noetherian left modules I

Syntaxclnm 36547 Extend class notation with the class of Noetherian left modules.
class LNoeM

Definitiondf-lnm 36548* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤s 𝑖) ∈ LFinGen}

Theoremislnm 36549* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑆 = (LSubSp‘𝑀)       (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆 (𝑀s 𝑖) ∈ LFinGen))

Theoremislnm2 36550* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑀)    &   𝑆 = (LSubSp‘𝑀)    &   𝑁 = (LSpan‘𝑀)       (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖𝑆𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁𝑔)))

Theoremlnmlmod 36551 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM → 𝑀 ∈ LMod)

Theoremlnmlssfg 36552 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑆 = (LSubSp‘𝑀)    &   𝑅 = (𝑀s 𝑈)       ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)

Theoremlnmlsslnm 36553 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑆 = (LSubSp‘𝑀)    &   𝑅 = (𝑀s 𝑈)       ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LNoeM)

Theoremlnmfg 36554 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
(𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)

Theoremkercvrlsm 36555 The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑈 = (LSubSp‘𝑆)    &    = (LSSum‘𝑆)    &    0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐷𝑈)    &   (𝜑 → (𝐹𝐷) = ran 𝐹)       (𝜑 → (𝐾 𝐷) = 𝐵)

Theoremlmhmfgima 36556 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑇s (𝐹𝐴))    &   𝑋 = (𝑆s 𝐴)    &   𝑈 = (LSubSp‘𝑆)    &   (𝜑𝑋 ∈ LFinGen)    &   (𝜑𝐴𝑈)    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))       (𝜑𝑌 ∈ LFinGen)

Theoremlnmepi 36557 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵) → 𝑇 ∈ LNoeM)

Theoremlmhmfgsplit 36558 If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &   𝑈 = (𝑆s 𝐾)    &   𝑉 = (𝑇s ran 𝐹)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)

Theoremlmhmlnmsplit 36559 If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
0 = (0g𝑇)    &   𝐾 = (𝐹 “ { 0 })    &   𝑈 = (𝑆s 𝐾)    &   𝑉 = (𝑇s ran 𝐹)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)

Theoremlnmlmic 36560 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))

Theorempwssplit4 36561* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐸 = (𝑅s (𝐴𝐵))    &   𝐺 = (Base‘𝐸)    &    0 = (0g𝑅)    &   𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })}    &   𝐹 = (𝑥𝐾 ↦ (𝑥𝐵))    &   𝐶 = (𝑅s 𝐴)    &   𝐷 = (𝑅s 𝐵)    &   𝐿 = (𝐸s 𝐾)       ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))

Theoremfilnm 36562 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐵 = (Base‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)

Theorempwslnmlem0 36563 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s ∅)       (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)

Theorempwslnmlem1 36564* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s {𝑖})       (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)

Theorempwslnmlem2 36565 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑋 = (𝑊s 𝐴)    &   𝑌 = (𝑊s 𝐵)    &   𝑍 = (𝑊s (𝐴𝐵))    &   (𝜑𝑊 ∈ LMod)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑋 ∈ LNoeM)    &   (𝜑𝑌 ∈ LNoeM)       (𝜑𝑍 ∈ LNoeM)

Theorempwslnm 36566 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝐼)       ((𝑊 ∈ LNoeM ∧ 𝐼 ∈ Fin) → 𝑌 ∈ LNoeM)

20.24.40  Every set admits a group structure iff choice

Theoremunxpwdom3 36567* Weaker version of unxpwdom 8257 where a function is required only to be cancellative, not an injection. 𝐷 and 𝐵 are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into 𝐴, each row must hit an element of 𝐵; by column injectivity, each row can be identified in at least one way by the 𝐵 element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐷𝑋)    &   ((𝜑𝑎𝐶𝑏𝐷) → (𝑎 + 𝑏) ∈ (𝐴𝐵))    &   (((𝜑𝑎𝐶) ∧ (𝑏𝐷𝑐𝐷)) → ((𝑎 + 𝑏) = (𝑎 + 𝑐) ↔ 𝑏 = 𝑐))    &   (((𝜑𝑑𝐷) ∧ (𝑎𝐶𝑐𝐶)) → ((𝑐 + 𝑑) = (𝑎 + 𝑑) ↔ 𝑐 = 𝑎))    &   (𝜑 → ¬ 𝐷𝐴)       (𝜑𝐶* (𝐷 × 𝐵))

Theorempwfi2f1o 36568* The pw2f1o 7830 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
𝑆 = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅}    &   𝐹 = (𝑥𝑆 ↦ (𝑥 “ {1𝑜}))       (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))

Theorempwfi2en 36569* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
𝑆 = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅}       (𝐴𝑉𝑆 ≈ (𝒫 𝐴 ∩ Fin))

Theoremfrlmpwfi 36570 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Proof shortened by AV, 14-Jun-2020.)
𝑅 = (ℤ/nℤ‘2)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)       (𝐼𝑉𝐵 ≈ (𝒫 𝐼 ∩ Fin))

Theoremgicabl 36571 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
(𝐺𝑔 𝐻 → (𝐺 ∈ Abel ↔ 𝐻 ∈ Abel))

Theoremimasgim 36572 A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &   (𝜑𝐹:𝑉1-1-onto𝐵)    &   (𝜑𝑅 ∈ Grp)       (𝜑𝐹 ∈ (𝑅 GrpIso 𝑈))

Theorembasfn 36573 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
Base Fn V

Theoremisnumbasgrplem1 36574 A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Abel ∧ 𝐶𝐵) → 𝐶 ∈ (Base “ Abel))

Theoremharn0 36575 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆𝑉 → (har‘𝑆) ≠ ∅)

Theoremnuminfctb 36576 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆)

Theoremisnumbasgrplem2 36577 If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Theoremisnumbasgrplem3 36578 Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.)
((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel))

Theoremisnumbasabl 36579 A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Abel))

Theoremisnumbasgrp 36580 A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
(𝑆 ∈ dom card ↔ (𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp))

Theoremdfacbasgrp 36581 A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.)
(CHOICE ↔ (Base “ Grp) = (V ∖ {∅}))

20.24.41  Noetherian rings and left modules II

Syntaxclnr 36582 Extend class notation with the class of left Noetherian rings.
class LNoeR

Definitiondf-lnr 36583 A ring is left-Noetherian iff it is Noetherian as a left module over itself. (Contributed by Stefan O'Rear, 24-Jan-2015.)
LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}

Theoremislnr 36584 Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))

Theoremlnrring 36585 Left-Noetherian rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR → 𝐴 ∈ Ring)

Theoremlnrlnm 36586 Left-Noetherian rings have Noetherian associated modules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝐴 ∈ LNoeR → (ringLMod‘𝐴) ∈ LNoeM)

Theoremislnr2 36587* 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 ∧ ∀𝑖𝑈𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁𝑔)))

Theoremislnr3 36588 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)))

Theoremlnr2i 36589* 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 ∧ 𝐼𝑈) → ∃𝑔 ∈ (𝒫 𝐼 ∩ Fin)𝐼 = (𝑁𝑔))

Theoremlpirlnr 36590 Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ LNoeR)

Theoremlnrfrlm 36591 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)

Theoremlnrfg 36592 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)

Theoremlnrfgtr 36593 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 = (𝑀s 𝑃)       ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃𝑈) → 𝑁 ∈ LFinGen)

20.24.42  Hilbert's Basis Theorem

Syntaxcldgis 36594 The leading ideal sequence used in the Hilbert Basis Theorem.
class ldgIdlSeq

Definitiondf-ldgis 36595* 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 36603. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))

Theoremhbtlem1 36596* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝐷 = ( deg1𝑅)       ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})

Theoremhbtlem2 36597 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝑇 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) ∈ 𝑇)

Theoremhbtlem7 36598 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   𝑇 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝑆𝐼):ℕ0𝑇)

Theoremhbtlem4 36599 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑈)    &   (𝜑𝑋 ∈ ℕ0)    &   (𝜑𝑌 ∈ ℕ0)    &   (𝜑𝑋𝑌)       (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐼)‘𝑌))

Theoremhbtlem3 36600 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
𝑃 = (Poly1𝑅)    &   𝑈 = (LIdeal‘𝑃)    &   𝑆 = (ldgIdlSeq‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑈)    &   (𝜑𝐽𝑈)    &   (𝜑𝐼𝐽)    &   (𝜑𝑋 ∈ ℕ0)       (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐽)‘𝑋))

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