Most Recent Proofs - Metamath Proof Explorer

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Most recent proofs These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the set.mm database for the Metamath Proof Explorer (and the Hilbert Space Explorer). The set.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from develop commit 212ee7d9, also available here: set.mm (43MB) or set.mm.bz2 (compressed, 13MB).

The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.

Other links Email: Norm Megill. Mailing list: Metamath Google Group Updated 7-Dec-2021 . Contributing: How can I contribute to Metamath? Syndication: RSS feed (courtesy of Dan Getz) Related wikis: Ghilbert site; Ghilbert Google Group.

Recent news items (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.

(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.

(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.

(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.

(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.

(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.

(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.

(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.

(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.

(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.

(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.

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**Recent Additions to the Metamath Proof Explorer** Notes (last updated 7-Dec-2020 )
Date	Label	Description
Theorem

2-Mar-2025	tfsconcatrnsson 41744	The concatenation of transfinite sequences yields ordinals iff both sequences yield ordinals. Theorem 4 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) ⊆ On ↔ (ran 𝐴 ⊆ On ∧ ran 𝐵 ⊆ On)))

2-Mar-2025	tfsconcatrnss 41743	The concatenation of transfinite sequences yields elements from a class iff both sequences yield elements from that class. (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran (𝐴 + 𝐵) ⊆ 𝑋 ↔ (ran 𝐴 ⊆ 𝑋 ∧ ran 𝐵 ⊆ 𝑋)))

2-Mar-2025	tfsconcatrnss12 41742	The range of the concatenation of transfinite sequences is a superset of the ranges of both sequences. Theorem 3 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (ran 𝐴 ⊆ ran (𝐴 + 𝐵) ∧ ran 𝐵 ⊆ ran (𝐴 + 𝐵)))

2-Mar-2025	tfsconcatrev 41741	If the domain of a transfinite sequence is an ordinal sum, the sequence can be decomposed into two sequences with domains corresponding to the addends. Theorem 2 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ ((𝐹 Fn (𝐶 +_o 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ∃𝑢 ∈ (ran 𝐹 ↑_m 𝐶)∃𝑣 ∈ (ran 𝐹 ↑_m 𝐷)((𝑢 + 𝑣) = 𝐹 ∧ dom 𝑢 = 𝐶 ∧ dom 𝑣 = 𝐷))

2-Mar-2025	isassad 21307	Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ (𝜑 → 𝑉 = (Base‘𝑊)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) & ⊢ (𝜑 → · = ( ·_𝑠 ‘𝑊)) & ⊢ (𝜑 → × = (._r‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑊 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦))) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝐵 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦))) ⇒ ⊢ (𝜑 → 𝑊 ∈ AssAlg)

2-Mar-2025	assasca 21305	The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

2-Mar-2025	isassa 21299	The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = ( ·_𝑠 ‘𝑊) & ⊢ × = (._r‘𝑊) ⇒ ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀𝑟 ∈ 𝐵 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝑟 · 𝑥) × 𝑦) = (𝑟 · (𝑥 × 𝑦)) ∧ (𝑥 × (𝑟 · 𝑦)) = (𝑟 · (𝑥 × 𝑦)))))

2-Mar-2025	df-assa 21296	Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a (commutative) ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣ [(Scalar‘𝑤) / 𝑓]∀𝑟 ∈ (Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[( ·_𝑠 ‘𝑤) / 𝑠][(._r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦)))}

2-Mar-2025	sbcie2s 17044	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.) (Revised by SN, 2-Mar-2025.)
⊢ 𝐴 = (𝐸‘𝑊) & ⊢ 𝐵 = (𝐹‘𝑊) & ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎][(𝐹‘𝑤) / 𝑏]𝜑 ↔ 𝜓))

1-Mar-2025	rp-tfslim 41746	The limit of a sequence of ordinals is the union of its range. (Contributed by RP, 1-Mar-2025.)
⊢ (𝐴 Fn 𝐵 → ∪ 𝑥 ∈ 𝐵 (𝐴‘𝑥) = ∪ ran 𝐴)

1-Mar-2025	tfsnfin 41745	A transfinite sequence is infinite iff its domain is greater than or equal to omega. Theorem 5 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 1-Mar-2025.)
⊢ ((𝐴 Fn 𝐵 ∧ 𝐵 ∈ On) → (¬ 𝐴 ∈ Fin ↔ ω ⊆ 𝐵))

1-Mar-2025	tfsconcat0b 41739	The concatentation with the empty series leaves the finite series unchanged. (Contributed by RP, 1-Mar-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ ω)) → (𝐴 = ∅ ↔ (𝐴 + 𝐵) = 𝐵))

28-Feb-2025	tfsconcat0i 41738	The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 28-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 = ∅ → (𝐴 + 𝐵) = 𝐵))

26-Feb-2025	abid2f 2935	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Feb-2025.)
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴

25-Feb-2025	tfsconcat00 41740	The concatentation of two empty series results in an empty series. (Contributed by RP, 25-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 + 𝐵) = ∅))

25-Feb-2025	tfsconcatb0 41737	The concatentation with the empty series leaves the series unchanged. (Contributed by RP, 25-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐵 = ∅ ↔ (𝐴 + 𝐵) = 𝐴))

25-Feb-2025	evls1maprnss 32457	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴 takes all values in the subring 𝑆. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑃 = (Poly₁‘(𝑅 ↾_s 𝑆)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋)) ⇒ ⊢ (𝜑 → 𝑆 ⊆ ran 𝐹)

25-Feb-2025	evls1maplmhm 32456	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴 is a module homomorphism, when considering the subring algebra. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑃 = (Poly₁‘(𝑅 ↾_s 𝑆)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋)) & ⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑆) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑃 LMHom 𝐴))

25-Feb-2025	lmimdim 32387	Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝑆 LMIso 𝑇)) & ⊢ (𝜑 → 𝑆 ∈ LVec) ⇒ ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇))

25-Feb-2025	evls1vsca 32352	Univariate polynomial evaluation of a scalar product of polynomials. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ × = ( ·_𝑠 ‘𝑊) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴 × 𝑁))‘𝐶) = (𝐴 · ((𝑄‘𝑁)‘𝐶)))

25-Feb-2025	rhmqusker 32278	A surjective ring homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 0 = (0_g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻)) & ⊢ 𝐾 = (^◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐽 ∈ (𝑄 RingIso 𝐻))

25-Feb-2025	lmhmqusker 32273	A surjective module homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 0 = (0_g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻)) & ⊢ 𝐾 = (^◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) ⇒ ⊢ (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))

25-Feb-2025	fldgenssp 32156	The field generated by a set of elements in a division ring is contained in any sub-division-ring which contains those elements. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ 𝑆)

25-Feb-2025	fldgenssv 32153	A generated subfield is a subset of the field's base. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) ⊆ 𝐵)

25-Feb-2025	rndrhmcl 32144	The range of a ring homomorphism from a division ring is a division ring. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ 𝑅 = (𝑁 ↾_s ran 𝐹) & ⊢ 0 = (0_g‘𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → ran 𝐹 ≠ { 0 }) & ⊢ (𝜑 → 𝑀 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing)

25-Feb-2025	ressbasss 17133	The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by SN, 25-Feb-2025.)
⊢ 𝑅 = (𝑊 ↾_s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (Base‘𝑅) ⊆ 𝐵

24-Feb-2025	tfsconcatfo 41736	The concatenation of two transfinite series is onto the union of the ranges. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵):(𝐶 +_o 𝐷)–onto→(ran 𝐴 ∪ ran 𝐵))

24-Feb-2025	tfsconcatrn 41735	The range of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ran (𝐴 + 𝐵) = (ran 𝐴 ∪ ran 𝐵))

24-Feb-2025	tfsconcatfv 41734	The value of the concatenation of two transfinite series. (Contributed by RP, 24-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ (𝐶 +_o 𝐷)) → ((𝐴 + 𝐵)‘𝑋) = if(𝑋 ∈ 𝐶, (𝐴‘𝑋), (𝐵‘(℩𝑑 ∈ 𝐷 (𝐶 +_o 𝑑) = 𝑋))))

23-Feb-2025	tfsconcatfv2 41733	A latter value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐷) → ((𝐴 + 𝐵)‘(𝐶 +_o 𝑋)) = (𝐵‘𝑋))

23-Feb-2025	tfsconcatfv1 41732	An early value of the concatenation of two transfinite series. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ ((((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) ∧ 𝑋 ∈ 𝐶) → ((𝐴 + 𝐵)‘𝑋) = (𝐴‘𝑋))

23-Feb-2025	tfsconcatun 41730	The concatenation of two transfinite series is a union of functions. (Contributed by RP, 23-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) = (𝐴 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐶 +_o 𝐷) ∖ 𝐶) ∧ ∃𝑧 ∈ 𝐷 (𝑥 = (𝐶 +_o 𝑧) ∧ 𝑦 = (𝐵‘𝑧)))}))

23-Feb-2025	tfsconcatlem 41729	Lemma for tfsconcatun 41730. (Contributed by RP, 23-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ ((𝐴 +_o 𝐵) ∖ 𝐴)) → ∃!𝑥∃𝑦 ∈ 𝐵 (𝐶 = (𝐴 +_o 𝑦) ∧ 𝑥 = (𝐹‘𝑦)))

23-Feb-2025	lringuplu 20224	If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝑈 = (Unit‘𝑅)) & ⊢ (𝜑 → + = (+_g‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ LRing) & ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))

23-Feb-2025	lringnz 20223	A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ 1 = (1_r‘𝑅) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ (𝑅 ∈ LRing → 1 ≠ 0 )

23-Feb-2025	lringring 20222	A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)

23-Feb-2025	lringnzr 20221	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)

23-Feb-2025	islring 20220	The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))

23-Feb-2025	df-lring 20219	A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+_g‘𝑟)𝑦) = (1_r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}

23-Feb-2025	01eq0ring 20215	If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })

23-Feb-2025	nzrring 20205	A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)

22-Feb-2025	tfsconcatfn 41731	The concatenation of two transfinite series is a transfinite series. (Contributed by RP, 22-Feb-2025.)
⊢ + = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑎 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((dom 𝑎 +_o dom 𝑏) ∖ dom 𝑎) ∧ ∃𝑧 ∈ dom 𝑏(𝑥 = (dom 𝑎 +_o 𝑧) ∧ 𝑦 = (𝑏‘𝑧)))})) ⇒ ⊢ (((𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → (𝐴 + 𝐵) Fn (𝐶 +_o 𝐷))

22-Feb-2025	ordsssucb 41728	An ordinal number is less than or equal to the successor of an ordinal class iff the ordinal number is either less than or equal to the ordinal class or the ordinal number is equal to the successor of the ordinal class. See also ordsssucim 41796, limsssuc 7791. (Contributed by RP, 22-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

21-Feb-2025	mulsproplem10 27431	Lemma for surreal multiplication. Show that the cut involved in surreal multiplication is a real cut. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → ({𝑖 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑖 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑗 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑗 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) <<s ({𝑘 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑘 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑙 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑙 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}))

21-Feb-2025	mulsproplem9 27430	Lemma for surreal multiplication. Show one of the four inequalities involved in surreal multiplication cuts. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝐴)) & ⊢ (𝜑 → 𝑆 ∈ ( R ‘𝐵)) & ⊢ (𝜑 → 𝑉 ∈ ( R ‘𝐴)) & ⊢ (𝜑 → 𝑊 ∈ ( L ‘𝐵)) ⇒ ⊢ (𝜑 → (((𝑅 ·_s 𝐵) +_s (𝐴 ·_s 𝑆)) -_s (𝑅 ·_s 𝑆)) <s (((𝑉 ·_s 𝐵) +_s (𝐴 ·_s 𝑊)) -_s (𝑉 ·_s 𝑊)))

21-Feb-2025	mulsproplem8 27429	Lemma for surreal multiplication. Show one of the four inequalities involved in surreal multiplication cuts. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝐴)) & ⊢ (𝜑 → 𝑆 ∈ ( R ‘𝐵)) & ⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴)) & ⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵)) ⇒ ⊢ (𝜑 → (((𝑅 ·_s 𝐵) +_s (𝐴 ·_s 𝑆)) -_s (𝑅 ·_s 𝑆)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))

21-Feb-2025	mulsproplem7 27428	Lemma for surreal multiplication. Show one of the four inequalities involved in surreal multiplication cuts. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴)) & ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵)) & ⊢ (𝜑 → 𝑉 ∈ ( R ‘𝐴)) & ⊢ (𝜑 → 𝑊 ∈ ( L ‘𝐵)) ⇒ ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑉 ·_s 𝐵) +_s (𝐴 ·_s 𝑊)) -_s (𝑉 ·_s 𝑊)))

21-Feb-2025	mulsproplem6 27427	Lemma for surreal multiplication. Show one of the four inequalities involved in surreal multiplication cuts. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴)) & ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵)) & ⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴)) & ⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵)) ⇒ ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))

21-Feb-2025	sltsubsub3bd 27413	Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 -_s 𝐶) <s (𝐵 -_s 𝐷) ↔ (𝐷 -_s 𝐶) <s (𝐵 -_s 𝐴)))

21-Feb-2025	sltsubsub2bd 27412	Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 -_s 𝐵) <s (𝐶 -_s 𝐷) ↔ (𝐷 -_s 𝐶) <s (𝐵 -_s 𝐴)))

21-Feb-2025	lltropt 27245	The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.) (Revised by Scott Fenton, 21-Feb-2025.)
⊢ ( L ‘𝐴) <<s ( R ‘𝐴)

20-Feb-2025	ply1degltdim 32405	The space 𝑆 of the univariate polynomials of degree less than 𝑁 has dimension 𝑁. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑆 = (^◡𝐷 “ (-∞[,)𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ 𝐸 = (𝑃 ↾_s 𝑆) ⇒ ⊢ (𝜑 → (dim‘𝐸) = 𝑁)

20-Feb-2025	ply1degltdimlem 32404	Lemma for ply1degltdim 32405. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑆 = (^◡𝐷 “ (-∞[,)𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ 𝐸 = (𝑃 ↾_s 𝑆) & ⊢ 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ⇒ ⊢ (𝜑 → ran 𝐹 ∈ (LBasis‘𝐸))

20-Feb-2025	ply1gsumz 32368	If a polynomial given as a sum of scaled monomials by factors 𝐴 is the zero polynomial, then all factors 𝐴 are zero. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑍 = (0_g‘𝑃) & ⊢ (𝜑 → 𝐴:(0..^𝑁)⟶𝐵) & ⊢ (𝜑 → (𝑃 Σ_g (𝐴 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = 𝑍) ⇒ ⊢ (𝜑 → 𝐴 = ((0..^𝑁) × { 0 }))

20-Feb-2025	gsummoncoe1fzo 32367	A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ ↑ = (._g‘(mulGrp‘𝑃)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ ∗ = ( ·_𝑠 ‘𝑃) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐿 ∈ (0..^𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝑘 = 𝐿 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ (0..^𝑁) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = 𝐶)

20-Feb-2025	ply1degltlss 32366	The space 𝑆 of the univariate polynomials of degree less than 𝑁 forms a vector subspace. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑆 = (^◡𝐷 “ (-∞[,)𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑆 ∈ (LSubSp‘𝑃))

20-Feb-2025	ply1degltel 32365	Characterize elementhood to the set 𝑆 of polynomials of degree less than 𝑁. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑆 = (^◡𝐷 “ (-∞[,)𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑃) ⇒ ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ 𝐵 ∧ (𝐷‘𝐹) ≤ (𝑁 − 1))))

20-Feb-2025	ply1moneq 32364	Two monomials are equal iff their powers are equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ ↑ = (._g‘(mulGrp‘𝑃)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑀 ∈ ℕ₀) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((𝑀 ↑ 𝑋) = (𝑁 ↑ 𝑋) ↔ 𝑀 = 𝑁))

20-Feb-2025	coe1mon 32363	Coefficient vector of a monomial. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ ↑ = (._g‘(mulGrp‘𝑃)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝜑 → (coe₁‘(𝑁 ↑ 𝑋)) = (𝑘 ∈ ℕ₀ ↦ if(𝑘 = 𝑁, 1 , 0 )))

20-Feb-2025	islbs5 32240	An equivalent formulation of the basis predicate in a vector space, using a function 𝐹 for generating the base. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ · = ( ·_𝑠 ‘𝑊) & ⊢ 𝑂 = (0_g‘𝑊) & ⊢ 0 = (0_g‘𝑆) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑆 ∈ NzRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼–1-1→𝐵) ⇒ ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (∀𝑎 ∈ (𝐾 ↑_m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σ_g (𝑎 ∘_f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵)))

20-Feb-2025	suppssnn0 31777	Show that the support of a function is contained in an half-open nonnegative integer range. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝜑 → 𝐹 Fn ℕ₀) & ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑘) → (𝐹‘𝑘) = 𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0..^𝑁))

20-Feb-2025	nn0difffzod 31776	A nonnegative integer that is not in the half-open range from 0 to 𝑁 is at least 𝑁. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ (ℕ₀ ∖ (0..^𝑁))) ⇒ ⊢ (𝜑 → ¬ 𝑀 < 𝑁)

20-Feb-2025	mptiffisupp 31675	Conditions for a mapping function defined with a conditional to have finite support. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍)) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐹 finSupp 𝑍)

20-Feb-2025	ifnebib 31535	The converse of ifbi 4513 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

20-Feb-2025	ifnefals 31534	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)

20-Feb-2025	ifnetrue 31533	Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)

20-Feb-2025	eqelbid 31467	A variable elimination law for equality within a given set 𝐴. See equvel 2454. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶))

20-Feb-2025	mulsproplem5 27426	Lemma for surreal multiplication. The surreal product of a member of the old set of 𝐴 and a member of the old set of 𝐵 is a surreal under the inductive hypothesis. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴))) & ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵))) ⇒ ⊢ (𝜑 → (𝑋 ·_s 𝑌) ∈ No )

20-Feb-2025	mulsproplem4 27425	Lemma for surreal multiplication. The surreal product of a member of the old set of 𝐴 and 𝐵 is a surreal under the inductive hypothesis. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴))) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝑋 ·_s 𝐵) ∈ No )

20-Feb-2025	mulsproplem3 27424	Lemma for surreal multiplication. The surreal product of 𝐴 and a member of the old set of 𝐵 is a surreal under the inductive hypothesis. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵))) ⇒ ⊢ (𝜑 → (𝐴 ·_s 𝑌) ∈ No )

20-Feb-2025	mulsproplem2 27423	Lemma for surreal multiplication. Simplify the induction premise for the common case. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) ⊆ ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) & ⊢ (𝜑 → 𝑃 ∈ No ) & ⊢ (𝜑 → 𝑄 ∈ No ) & ⊢ (𝜑 → 𝑅 ∈ No ) & ⊢ (𝜑 → 𝑆 ∈ No ) & ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) +no (( bday ‘𝑅) +no ( bday ‘𝑆))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))))) ⇒ ⊢ (𝜑 → ((𝑃 ·_s 𝑄) ∈ No ∧ ((𝑃 <s 𝑄 ∧ 𝑅 <s 𝑆) → ((𝑄 ·_s 𝑅) -_s (𝑃 ·_s 𝑅)) <s ((𝑄 ·_s 𝑆) -_s (𝑃 ·_s 𝑆)))))

20-Feb-2025	mulsproplem1 27422	Lemma for surreal multiplication. We begin by instantiating some quantifiers. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ((((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) +no ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻)))) → (((( bday ‘𝑎) +no ( bday ‘𝑏)) +no (( bday ‘𝑐) +no ( bday ‘𝑑))) ⊆ ((( bday ‘𝑒) +no ( bday ‘𝑓)) +no (( bday ‘𝑔) +no ( bday ‘ℎ))) → ((𝑒 ·_s 𝑓) ∈ No ∧ ((𝑒 <s 𝑓 ∧ 𝑔 <s ℎ) → ((𝑓 ·_s 𝑔) -_s (𝑒 ·_s 𝑔)) <s ((𝑓 ·_s ℎ) -_s (𝑒 ·_s ℎ))))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑊 ∈ No ) & ⊢ (𝜑 → 𝑃 ∈ No ) & ⊢ (𝜑 → 𝑄 ∈ No ) & ⊢ (𝜑 → 𝑅 ∈ No ) & ⊢ (𝜑 → 𝑆 ∈ No ) & ⊢ (𝜑 → (((( bday ‘𝑋) +no ( bday ‘𝑌)) +no (( bday ‘𝑍) +no ( bday ‘𝑊))) +no ((( bday ‘𝑃) +no ( bday ‘𝑄)) +no (( bday ‘𝑅) +no ( bday ‘𝑆)))) ∈ (((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) +no ((( bday ‘𝐸) +no ( bday ‘𝐹)) +no (( bday ‘𝐺) +no ( bday ‘𝐻))))) & ⊢ (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝑌)) +no (( bday ‘𝑍) +no ( bday ‘𝑊))) ⊆ ((( bday ‘𝑃) +no ( bday ‘𝑄)) +no (( bday ‘𝑅) +no ( bday ‘𝑆)))) ⇒ ⊢ (𝜑 → ((𝑃 ·_s 𝑄) ∈ No ∧ ((𝑃 <s 𝑄 ∧ 𝑅 <s 𝑆) → ((𝑄 ·_s 𝑅) -_s (𝑃 ·_s 𝑅)) <s ((𝑄 ·_s 𝑆) -_s (𝑃 ·_s 𝑆)))))

20-Feb-2025	naddlid 8635	Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝐴 ∈ On → (∅ +no 𝐴) = 𝐴)

20-Feb-2025	rspc8v 3597	8-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) & ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜂)) & ⊢ (𝑝 = 𝐸 → (𝜂 ↔ 𝜁)) & ⊢ (𝑞 = 𝐹 → (𝜁 ↔ 𝜎)) & ⊢ (𝑟 = 𝐺 → (𝜎 ↔ 𝜌)) & ⊢ (𝑠 = 𝐻 → (𝜌 ↔ 𝜓)) ⇒ ⊢ ((((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) ∧ ((𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌))) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 ∀𝑟 ∈ 𝑋 ∀𝑠 ∈ 𝑌 𝜑 → 𝜓))

20-Feb-2025	4ralbidv 3212	Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))

20-Feb-2025	3ralbidv 3211	Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 20-Feb-2025.)
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜒))

19-Feb-2025	onmcl 41724	If an ordinal is less than a power of omega, the product with a natural number is also less than that power of omega. (Contributed by RP, 19-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑁 ∈ ω) → (𝐴 ∈ (ω ↑_o 𝐵) → (𝐴 ·_o 𝑁) ∈ (ω ↑_o 𝐵)))

19-Feb-2025	oaabsb 41687	The right addend absorbs the sum with an ordinal iff that ordinal times omega is less than or equal to the right addend. (Contributed by RP, 19-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·_o ω) ⊆ 𝐵 ↔ (𝐴 +_o 𝐵) = 𝐵))

19-Feb-2025	rimrcl2 40764	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)

19-Feb-2025	rimrcl1 40763	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)

19-Feb-2025	ply1lvec 32342	In a division ring, the univariate polynomials form a vector space. (Contributed by Thierry Arnoux, 19-Feb-2025.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑃 ∈ LVec)

19-Feb-2025	sdrgunit 20319	A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾_s 𝐴) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑈 = (Unit‘𝑆) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 )))

19-Feb-2025	sdrgbas 20317	Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾_s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 = (Base‘𝑆))

19-Feb-2025	sdrgsubrg 20314	A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))

19-Feb-2025	sdrgdrng 20313	A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾_s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)

19-Feb-2025	sdrgrcl 20312	Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)

19-Feb-2025	isdrngrd 20256	Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a right-inverse 𝐼(𝑥). See isdrngd 20255 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (._r‘𝑅)) & ⊢ (𝜑 → 0 = (0_g‘𝑅)) & ⊢ (𝜑 → 1 = (1_r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing)

19-Feb-2025	isdrngd 20255	Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a left-inverse 𝐼(𝑥). See isdrngrd 20256 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (._r‘𝑅)) & ⊢ (𝜑 → 0 = (0_g‘𝑅)) & ⊢ (𝜑 → 1 = (1_r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing)

18-Feb-2025	selvmul 40822	The "variable selection" function is multiplicative. (Contributed by SN, 18-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = (._r‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ ∙ = (._r‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐹 · 𝐺)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∙ (((𝐼 selectVars 𝑅)‘𝐽)‘𝐺)))

18-Feb-2025	evlmulval 40812	Polynomial evaluation builder for multiplication. (Contributed by SN, 18-Feb-2025.)
⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ ∙ = (._r‘𝑃) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼)) & ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) & ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) ⇒ ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)))

18-Feb-2025	ricfld 40781	A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025.)
⊢ (𝑅 ≃_𝑟 𝑆 → (𝑅 ∈ Field ↔ 𝑆 ∈ Field))

18-Feb-2025	ricdrng 40779	A ring is a division ring if and only if an isomorphic ring is a division ring. (Contributed by SN, 18-Feb-2025.)
⊢ (𝑅 ≃_𝑟 𝑆 → (𝑅 ∈ DivRing ↔ 𝑆 ∈ DivRing))

18-Feb-2025	ricdrng1 40778	A ring isomorphism maps a division ring to a division ring. (Contributed by SN, 18-Feb-2025.)
⊢ ((𝑅 ≃_𝑟 𝑆 ∧ 𝑅 ∈ DivRing) → 𝑆 ∈ DivRing)

17-Feb-2025	imadrhmcl 20320	The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)
⊢ 𝑅 = (𝑁 ↾_s (𝐹 “ 𝑆)) & ⊢ 0 = (0_g‘𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀)) & ⊢ (𝜑 → ran 𝐹 ≠ { 0 }) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing)

16-Feb-2025	sltled 27154	Surreal less-than implies less-than or equal. (Contributed by Scott Fenton, 16-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤s 𝐵)

15-Feb-2025	rexanuz2nf 43848	A simple counterexample related to theorem rexanuz2 15246, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15246 and similar theorems would not hold (see rexanre 15243 and rexanuz 15242). (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ 𝑍 = ℕ₀ & ⊢ (𝜑 ↔ (𝑗 = 0 ∧ 𝑗 ≤ 𝑘)) & ⊢ (𝜓 ↔ 0 < 𝑘) ⇒ ⊢ ¬ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝜑 ∧ 𝜓) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝜑 ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)𝜓))

15-Feb-2025	cvgcaule 43847	A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑗𝐹 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ 𝑍) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑋 ∈ ℝ⁺) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) ≤ 𝑋))

15-Feb-2025	cvgcau 43846	A convergent function is Cauchy. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑗𝐹 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ 𝑍) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) & ⊢ (𝜑 → 𝑋 ∈ ℝ⁺) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑋))

15-Feb-2025	caucvgbf 43845	A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑗𝐹 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ_≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))

15-Feb-2025	fvmpt4d 43626	Value of a function given by the maps-to notation. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝐵 ∈ 𝐶) & ⊢ (𝜑 → 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)

15-Feb-2025	rspced 43505	Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝜒) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)

15-Feb-2025	rexeqif 43504	Equality inference for restricted existential quantifier. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ 𝐴 = 𝐵 ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)

15-Feb-2025	notbicom 43503	Commutative law for the negation of a biconditional. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ ¬ (𝜑 ↔ 𝜓) ⇒ ⊢ ¬ (𝜓 ↔ 𝜑)

15-Feb-2025	nimnbi2 43502	If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ ¬ (𝜓 → 𝜑) ⇒ ⊢ ¬ (𝜑 ↔ 𝜓)

15-Feb-2025	nimnbi 43501	If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ ¬ (𝜑 → 𝜓) ⇒ ⊢ ¬ (𝜑 ↔ 𝜓)

15-Feb-2025	eliund 43500	Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)

15-Feb-2025	naddwordnexlem4 41795	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, there exists a product with omega such that the ordinal sum with 𝐴 is less than or equal to 𝐵 while the natural sum is larger than 𝐵. (Contributed by RP, 15-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀)) & ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁)) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝑀 ∈ ω) & ⊢ (𝜑 → 𝑁 ∈ 𝑀) & ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐷 ⊆ (𝐶 +_o 𝑦)} ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (On ∖ 1_o)((𝐶 +_o 𝑥) = 𝐷 ∧ (𝐴 +_o (ω ·_o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·_o 𝑥))))

15-Feb-2025	ghmqusker 32272	A surjective group homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 15-Feb-2025.)
⊢ 0 = (0_g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) & ⊢ 𝐾 = (^◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻)) ⇒ ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))

15-Feb-2025	ghmquskerco 32270	In the case of theorem ghmqusker 32272, the composition of the natural homomorphism 𝐿 with the constructed homomorphism 𝐽 equals the original homomorphism 𝐹. One says that 𝐹 factors through 𝑄. (Proposed by Saveliy Skresanov, 15-Feb-2025.) (Contributed by Thierry Arnoux, 15-Feb-2025.)
⊢ 0 = (0_g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) & ⊢ 𝐾 = (^◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~_QG 𝐾)) ⇒ ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))

15-Feb-2025	naddword2 8643	Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 15-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +no 𝐴))

15-Feb-2025	eqimsscd 4004	Equality implies inclusion, deduction version. (Contributed by SN, 15-Feb-2025.)
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

14-Feb-2025	oawordex3 41794	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, some ordinal sum of 𝐴 is equal to 𝐵. This is a specialization of oawordex 8509. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀)) & ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁)) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝑀 ∈ ω) & ⊢ (𝜑 → 𝑁 ∈ 𝑀) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +_o 𝑥) = 𝐵)

14-Feb-2025	naddwordnexlem3 41793	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, every natural sum of 𝐴 with a natural number is less that 𝐵. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀)) & ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁)) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝑀 ∈ ω) & ⊢ (𝜑 → 𝑁 ∈ 𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐴 +no 𝑥) ∈ 𝐵)

14-Feb-2025	naddwordnexlem2 41792	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, 𝐵 is larger than 𝐴. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀)) & ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁)) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝑀 ∈ ω) & ⊢ (𝜑 → 𝑁 ∈ 𝑀) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐵)

14-Feb-2025	naddwordnexlem1 41791	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, 𝐵 is equal to or larger than 𝐴. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀)) & ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁)) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝑀 ∈ ω) & ⊢ (𝜑 → 𝑁 ∈ 𝑀) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

14-Feb-2025	naddwordnexlem0 41790	When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, (ω ·_o suc 𝐶) lies between 𝐴 and 𝐵. (Contributed by RP, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 = ((ω ·_o 𝐶) +_o 𝑀)) & ⊢ (𝜑 → 𝐵 = ((ω ·_o 𝐷) +_o 𝑁)) & ⊢ (𝜑 → 𝐶 ∈ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ On) & ⊢ (𝜑 → 𝑀 ∈ ω) & ⊢ (𝜑 → 𝑁 ∈ 𝑀) ⇒ ⊢ (𝜑 → (𝐴 ∈ (ω ·_o suc 𝐶) ∧ (ω ·_o suc 𝐶) ⊆ 𝐵))

14-Feb-2025	ghmquskerlem2 32271	Lemma for ghmqusker 32272. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ 0 = (0_g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) & ⊢ 𝐾 = (^◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))

14-Feb-2025	ghmquskerlem1 32269	Lemma for ghmqusker 32272 (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ 0 = (0_g‘𝐻) & ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) & ⊢ 𝐾 = (^◡𝐹 “ { 0 }) & ⊢ 𝑄 = (𝐺 /_s (𝐺 ~_QG 𝐾)) & ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) & ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺)) ⇒ ⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~_QG 𝐾)) = (𝐹‘𝑋))

14-Feb-2025	ecref 31693	All elements are in their own equivalence class. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ ((𝑅 Er 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ [𝐴]𝑅)

14-Feb-2025	imaexd 31663	The image of a set is a set. Deduction version of imaexg 7857. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐵) ∈ V)

14-Feb-2025	rnexd 31662	The range of a set is a set. Deduction version of rnexd 31662. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ran 𝐴 ∈ V)

14-Feb-2025	negsunif 27393	Uniformity property for surreal negation. If 𝐿 and 𝑅 are any cut that represents 𝐴, then they may be used instead of ( L ‘𝐴) and ( R ‘𝐴) in the definition of negation. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅) & ⊢ (𝜑 → 𝐴 = (𝐿 \|s 𝑅)) ⇒ ⊢ (𝜑 → ( -u_s ‘𝐴) = (( -u_s “ 𝑅) \|s ( -u_s “ 𝐿)))

14-Feb-2025	mins2 27153	The minimum of two surreals is less than or equal to the second. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝐵 ∈ No → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐵)

14-Feb-2025	mins1 27152	The minimum of two surreals is less than or equal to the first. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → if(𝐴 ≤s 𝐵, 𝐴, 𝐵) ≤s 𝐴)

14-Feb-2025	maxs2 27151	A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))

14-Feb-2025	maxs1 27150	A surreal is less than or equal to the maximum of it and another. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ (𝐴 ∈ No → 𝐴 ≤s if(𝐴 ≤s 𝐵, 𝐵, 𝐴))

14-Feb-2025	sletric 27149	Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴))

13-Feb-2025	oaun3 41775	Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +_o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +_o 𝑏)}, {𝑧 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎 +_o 𝑏)}})

13-Feb-2025	oaun2 41774	Ordinal addition as a union of classes. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +_o 𝐵) = ∪ {{𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}, {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = (𝐴 +_o 𝑏)}})

13-Feb-2025	oadif1 41773	Express the set difference of an ordinal sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +_o 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 +_o 𝑏)})

13-Feb-2025	oadif1lem 41772	Express the set difference of a continuous sum and its left addend as a class of sums. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊕ 𝐵) ∈ On) & ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → (𝐴 ⊕ 𝑏) ∈ On) & ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ (𝐴 ⊕ 𝐵))) → ∃𝑏 ∈ 𝐵 (𝐴 ⊕ 𝑏) = 𝑦) & ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑏 ∈ 𝐵 → (𝐴 ⊕ 𝑏) ∈ (𝐴 ⊕ 𝐵))) & ⊢ ((𝐴 ∈ On ∧ 𝑏 ∈ On) → 𝐴 ⊆ (𝐴 ⊕ 𝑏)) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ⊕ 𝐵) ∖ 𝐴) = {𝑥 ∣ ∃𝑏 ∈ 𝐵 𝑥 = (𝐴 ⊕ 𝑏)})

13-Feb-2025	rp-abid 41771	Two ways to express a class. (Contributed by RP, 13-Feb-2025.)
⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

13-Feb-2025	oaun3lem3 41769	The class of all ordinal sums of elements from two ordinals is an ordinal. Lemma for oaun3 41775. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +_o 𝑏)} ∈ On)

13-Feb-2025	oaun3lem2 41768	The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 41775. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +_o 𝑏)} ⊆ (𝐴 +_o 𝐵))

13-Feb-2025	oaun3lem1 41767	The class of all ordinal sums of elements from two ordinals is ordinal. Lemma for oaun3 41775. (Contributed by RP, 13-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +_o 𝑏)})

13-Feb-2025	eqab 2871	One direction of eqabb 2872 is provable from fewer axioms. (Contributed by Wolf Lammen, 13-Feb-2025.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → 𝐴 = {𝑥 ∣ 𝜑})

12-Feb-2025	oaun3lem4 41770	The class of all ordinal sums of elements from two ordinals is less than the successor to the sum. Lemma for oaun3 41775. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +_o 𝑏)} ∈ suc (𝐴 +_o 𝐵))

12-Feb-2025	onsucunitp 41766	The successor to the union of any triple of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → suc ∪ {𝐴, 𝐵, 𝐶} = ∪ {suc 𝐴, suc 𝐵, suc 𝐶})

12-Feb-2025	onsucunipr 41765	The successor to the union of any pair of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc ∪ {𝐴, 𝐵} = ∪ {suc 𝐴, suc 𝐵})

12-Feb-2025	onsucunifi 41763	The successor to the union of any non-empty, finite subset of ordinals is the union of the successors of the elements. (Contributed by RP, 12-Feb-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 suc 𝑥)

12-Feb-2025	eqabb 2872	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbib 2803 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. An example is the conversion of zfauscl 5263 to inex1 5279 (look at the instance of zfauscl 5263 that occurs in the proof of inex1 5279). Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new setvar and wff variables not already in the wff. Examples include dfsymdif2 4215 and cp 9836; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9835. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. Usage of eqabbw 2808 is preferred since it requires fewer axioms. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2025.)
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

11-Feb-2025	sucunisn 41764	The successor to the union of any singleton of a set is the successor of the set. (Contributed by RP, 11-Feb-2025.)
⊢ (𝐴 ∈ 𝑉 → suc ∪ {𝐴} = suc 𝐴)

9-Feb-2025	cantnfub2 41715	Given a finite number of terms of the form ((ω ↑_o (𝐴‘𝑛)) ·_o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑_o suc ∪ ran 𝐴) when (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 9-Feb-2025.)
⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → 𝐴:𝑁–1-1→On) & ⊢ (𝜑 → 𝑀:𝑁⟶ω) & ⊢ 𝐹 = (𝑥 ∈ suc ∪ ran 𝐴 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(^◡𝐴‘𝑥)), ∅)) ⇒ ⊢ (𝜑 → (suc ∪ ran 𝐴 ∈ On ∧ 𝐹 ∈ dom (ω CNF suc ∪ ran 𝐴) ∧ ((ω CNF suc ∪ ran 𝐴)‘𝐹) ∈ (ω ↑_o suc ∪ ran 𝐴)))

9-Feb-2025	selvval2 40820	Value of the "variable selection" function. Use evlsevl 40810 for a simpler definition. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = (((𝐼 eval 𝑇)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))

9-Feb-2025	evladdval 40811	Polynomial evaluation builder for addition. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ ✚ = (+_g‘𝑃) & ⊢ + = (+_g‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼)) & ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) & ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊)) ⇒ ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (𝑉 + 𝑊)))

9-Feb-2025	evlsevl 40810	Evaluation in a subring is the same as evaluation in the ring itself. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑂 = (𝐼 eval 𝑆) & ⊢ 𝑊 = (𝐼 mPoly 𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑄‘𝐹) = (𝑂‘𝐹))

9-Feb-2025	evlsvval 40803	Give a formula for the evaluation of a polynomial. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑈) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝑇 = (𝑆 ↑_s (𝐾 ↑_m 𝐼)) & ⊢ 𝑀 = (mulGrp‘𝑇) & ⊢ ↑ = (._g‘𝑀) & ⊢ · = (._r‘𝑇) & ⊢ 𝐹 = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥})) & ⊢ 𝐺 = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑄‘𝐴) = (𝑇 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σ_g (𝑏 ∘_f ↑ 𝐺))))))

9-Feb-2025	mplsubrgcl 40793	An element of a polynomial algebra over a subring is an element of the polynomial algebra. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑊 = (𝐼 mPoly 𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐶 = (Base‘𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐶)

9-Feb-2025	ply1annprmidl 32462	The set 𝑄 of polynomials annihilating an element 𝐴 is a prime ideal. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub₁ 𝐹) & ⊢ 𝑃 = (Poly₁‘(𝐸 ↾_s 𝐹)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 0 = (0_g‘𝐸) & ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ⇒ ⊢ (𝜑 → 𝑄 ∈ (PrmIdeal‘𝑃))

9-Feb-2025	minplyval 32461	Expand the value of the minimal polynomial (𝑀‘𝐴) for a given element 𝐴. It is defined as the unique monic polynomial of minimal degree which annihilates 𝐴. By ply1annig1p 32460, that polynomial generates the ideal of the annihilators of 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub₁ 𝐹) & ⊢ 𝑃 = (Poly₁‘(𝐸 ↾_s 𝐹)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 0 = (0_g‘𝐸) & ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } & ⊢ 𝐾 = (RSpan‘𝑃) & ⊢ 𝐺 = (idlGen_1p‘(𝐸 ↾_s 𝐹)) & ⊢ 𝑀 = (𝐸 minPoly 𝐹) ⇒ ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

9-Feb-2025	ply1annig1p 32460	The ideal 𝑄 of polynomials annihilating an element 𝐴 is generated by the ideal's canonical generator. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub₁ 𝐹) & ⊢ 𝑃 = (Poly₁‘(𝐸 ↾_s 𝐹)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 0 = (0_g‘𝐸) & ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } & ⊢ 𝐾 = (RSpan‘𝑃) & ⊢ 𝐺 = (idlGen_1p‘(𝐸 ↾_s 𝐹)) ⇒ ⊢ (𝜑 → 𝑄 = (𝐾‘{(𝐺‘𝑄)}))

9-Feb-2025	ply1annidl 32459	The set 𝑄 of polynomials annihilating an element 𝐴 forms an ideal. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑃 = (Poly₁‘(𝑅 ↾_s 𝑆)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } ⇒ ⊢ (𝜑 → 𝑄 ∈ (LIdeal‘𝑃))

9-Feb-2025	ply1annidllem 32458	Write the set 𝑄 of polynomials annihilating an element 𝐴 as the kernel of the ring homomorphism 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑃 = (Poly₁‘(𝑅 ↾_s 𝑆)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 } & ⊢ 𝐹 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴)) ⇒ ⊢ (𝜑 → 𝑄 = (^◡𝐹 “ { 0 }))

9-Feb-2025	evls1fn 32344	Functionality of the subring polynomial evaluation. (Contributed by Thierry Arnoux, 9-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑃 = (Poly₁‘(𝑅 ↾_s 𝑆)) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → 𝑂 Fn 𝑈)

8-Feb-2025	rhmmpl 40799	Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20189. TODO: Currently mhmvlin 21783 would have to be moved up. Investigate the usefulness of surrounding theorems like mndvcl 21777 and the difference between mhmvlin 21783, ofco 7645, and ofco2 21837. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑄 = (𝐼 mPoly 𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))

8-Feb-2025	rhmcomulmpl 40798	Show that the ring homomorphism in rhmmpl 40799 preserves multiplication. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑄 = (𝐼 mPoly 𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Base‘𝑄) & ⊢ · = (._r‘𝑃) & ⊢ ∙ = (._r‘𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐻 ∘ (𝐹 · 𝐺)) = ((𝐻 ∘ 𝐹) ∙ (𝐻 ∘ 𝐺)))

8-Feb-2025	mhmcoaddmpl 40797	Show that the ring homomorphism in rhmmpl 40799 preserves addition. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑄 = (𝐼 mPoly 𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Base‘𝑄) & ⊢ + = (+_g‘𝑃) & ⊢ ✚ = (+_g‘𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)))

8-Feb-2025	rhmmpllem2 40796	Lemma for rhmmpl 40799. A subproof of psrmulcllem 21392. (Contributed by SN, 8-Feb-2025.)
⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) & ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σ_g (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_r ≤ 𝑘} ↦ ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_f − 𝑥))))) ∈ (Base‘𝑅))

8-Feb-2025	rhmmpllem1 40795	Lemma for rhmmpl 40799. A subproof of psrmulcllem 21392. (Contributed by SN, 8-Feb-2025.)
⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅)) & ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_r ≤ 𝑘} ↦ ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_f − 𝑥)))) finSupp (0_g‘𝑅))

8-Feb-2025	evls1maprhm 32455	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝑋 is a ring homomorphism. (Contributed by Thierry Arnoux, 8-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑃 = (Poly₁‘(𝑅 ↾_s 𝑆)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))

8-Feb-2025	evls1addd 32350	Univariate polynomial evaluation of a sum of polynomials. (Contributed by Thierry Arnoux, 8-Feb-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ ⨣ = (+_g‘𝑊) & ⊢ + = (+_g‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑀 ⨣ 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) + ((𝑄‘𝑁)‘𝐶)))

7-Feb-2025	selvadd 40821	The "variable selection" function is additive. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ + = (+_g‘𝑃) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ ✚ = (+_g‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐹 + 𝐺)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ✚ (((𝐼 selectVars 𝑅)‘𝐽)‘𝐺)))

7-Feb-2025	mhmcompl 40794	The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑄 = (𝐼 mPoly 𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐶 = (Base‘𝑄) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆)) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)

7-Feb-2025	mplcrngd 40792	The polynomial ring is a commutative ring. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝑃 ∈ CRing)

7-Feb-2025	mplringd 40791	The polynomial ring is a ring. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑃 ∈ Ring)

7-Feb-2025	coexd 40724	The composition of two sets is a set. (Contributed by SN, 7-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) ∈ V)

7-Feb-2025	psrgrp 21403	The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Grp) ⇒ ⊢ (𝜑 → 𝑆 ∈ Grp)

7-Feb-2025	resrhm2b 20301	Restriction of the codomain of a (ring) homomorphism. resghm2b 19040 analog. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑈 = (𝑇 ↾_s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))

7-Feb-2025	eluzsubi 12805	Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) (Proof shortened by SN, 7-Feb-2025.)
⊢ 𝑀 ∈ ℤ & ⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ_≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ_≥‘𝑀))

7-Feb-2025	eluzaddi 12803	Membership in a later upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007.) Shorten and remove 𝑀 ∈ ℤ hypothesis. (Revised by SN, 7-Feb-2025.)
⊢ 𝐾 ∈ ℤ ⇒ ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ_≥‘(𝑀 + 𝐾)))

7-Feb-2025	eluzsub 12802	Membership in an earlier upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘(𝑀 + 𝐾))) → (𝑁 − 𝐾) ∈ (ℤ_≥‘𝑀))

7-Feb-2025	eluzadd 12801	Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.)
⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ_≥‘(𝑀 + 𝐾)))

7-Feb-2025	naddel12 8651	Natural addition to both sides of ordinal less-than. (Contributed by Scott Fenton, 7-Feb-2025.)
⊢ ((𝐶 ∈ On ∧ 𝐷 ∈ On) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 +no 𝐵) ∈ (𝐶 +no 𝐷)))

7-Feb-2025	rspc4v 3596	4-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2025.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) & ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜃)) & ⊢ (𝑧 = 𝐶 → (𝜃 ↔ 𝜏)) & ⊢ (𝑤 = 𝐷 → (𝜏 ↔ 𝜓)) ⇒ ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 𝜑 → 𝜓))

6-Feb-2025	oalim2cl 41682	The ordinal sum of any ordinal with a limit ordinal on the right is a limit ordinal. (Contributed by RP, 6-Feb-2025.)
⊢ ((𝐴 ∈ On ∧ Lim 𝐵 ∧ 𝐵 ∈ 𝑉) → Lim (𝐴 +_o 𝐵))

6-Feb-2025	sltsubsubbd 27411	Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 6-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 -_s 𝐶) <s (𝐵 -_s 𝐷) ↔ (𝐴 -_s 𝐵) <s (𝐶 -_s 𝐷)))

6-Feb-2025	addsubsassd 27410	Associative-type law for surreal addition and subtraction. (Contributed by Scott Fenton, 6-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) -_s 𝐶) = (𝐴 +_s (𝐵 -_s 𝐶)))

6-Feb-2025	zzlesq 14120	An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
⊢ (𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))

5-Feb-2025	irngnzply1 32452	In the case of a field 𝐸, the roots of nonzero polynomials 𝑝 with coefficients in a subfield 𝐹 are exactly the integral elements over 𝐹. Roots of nonzero polynomials are called algebraic numbers, so this shows that in the case of a field, elements integral over 𝐹 are exactly the algebraic numbers. In this formula, dom 𝑂 represents the polynomials, and 𝑍 the zero polynomial. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub₁ 𝐹) & ⊢ 𝑍 = (0_g‘(Poly₁‘𝐸)) & ⊢ 0 = (0_g‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) ⇒ ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(^◡(𝑂‘𝑝) “ { 0 }))

5-Feb-2025	irngnzply1lem 32451	In the case of a field 𝐸, a root 𝑋 of some nonzero polynomial 𝑃 with coefficients in a subfield 𝐹 is integral over 𝐹. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝐸 evalSub₁ 𝐹) & ⊢ 𝑍 = (0_g‘(Poly₁‘𝐸)) & ⊢ 0 = (0_g‘𝐸) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ 𝐵 = (Base‘𝐸) & ⊢ (𝜑 → 𝑃 ∈ dom 𝑂) & ⊢ (𝜑 → 𝑃 ≠ 𝑍) & ⊢ (𝜑 → ((𝑂‘𝑃)‘𝑋) = 0 ) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

5-Feb-2025	0ringirng 32450	A zero ring 𝑅 has no integral elements. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑈 = (𝑅 ↾_s 𝑆) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → ¬ 𝑅 ∈ NzRing) ⇒ ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∅)

5-Feb-2025	asclply1subcl 32359	Closure of the algebra scalar injection function in a polynomial on a subring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝐴 = (algSc‘𝑉) & ⊢ 𝑈 = (𝑅 ↾_s 𝑆) & ⊢ 𝑉 = (Poly₁‘𝑅) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑃 = (Base‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑍 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴‘𝑍) ∈ 𝑃)

5-Feb-2025	0ringmon1p 32340	There are no monic polynomials over a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝑀 = (Monic_1p‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → (♯‘𝐵) = 1) ⇒ ⊢ (𝜑 → 𝑀 = ∅)

5-Feb-2025	0ringsubrg 32134	A subring of a zero ring is a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → (♯‘𝐵) = 1) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (♯‘𝑆) = 1)

5-Feb-2025	negsubsdi2d 27409	Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → ( -u_s ‘(𝐴 -_s 𝐵)) = (𝐵 -_s 𝐴))

5-Feb-2025	sltsub2d 27408	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 -_s 𝐵) <s (𝐶 -_s 𝐴)))

5-Feb-2025	sltsub1d 27407	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 -_s 𝐶) <s (𝐵 -_s 𝐶)))

5-Feb-2025	subaddsd 27401	Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 -_s 𝐵) = 𝐶 ↔ (𝐵 +_s 𝐶) = 𝐴))

5-Feb-2025	subscld 27397	Closure law for surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 -_s 𝐵) ∈ No )

5-Feb-2025	subsvald 27395	The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 -_s 𝐵) = (𝐴 +_s ( -u_s ‘𝐵)))

5-Feb-2025	negsdi 27389	Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -u_s ‘(𝐴 +_s 𝐵)) = (( -u_s ‘𝐴) +_s ( -u_s ‘𝐵)))

5-Feb-2025	negsidd 27383	Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +_s ( -u_s ‘𝐴)) = 0_s )

5-Feb-2025	adds42d 27360	Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) +_s (𝐶 +_s 𝐷)) = ((𝐴 +_s 𝐶) +_s (𝐷 +_s 𝐵)))

5-Feb-2025	adds4d 27359	Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐷 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) +_s (𝐶 +_s 𝐷)) = ((𝐴 +_s 𝐶) +_s (𝐵 +_s 𝐷)))

5-Feb-2025	addscan1d 27352	Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐶 +_s 𝐴) = (𝐶 +_s 𝐵) ↔ 𝐴 = 𝐵))

5-Feb-2025	addscan2d 27351	Cancellation law for surreal addition. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐶) = (𝐵 +_s 𝐶) ↔ 𝐴 = 𝐵))

5-Feb-2025	sltadd1d 27350	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 +_s 𝐶) <s (𝐵 +_s 𝐶)))

5-Feb-2025	sleadd2d 27348	Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐶 +_s 𝐴) ≤s (𝐶 +_s 𝐵)))

5-Feb-2025	ringcomlem 20014	Lemma for ringcom 20015. This (formerly) part of the proof for ringcom 20015 is also applicable for semirings (without using the commutativity of the addition given per definition of a semiring), see srgcom4lem 19958. (Contributed by Gérard Lang, 4-Dec-2014.) Variant of rglcom4d 19956 for rings. (Revised by AV, 5-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = ((𝑋 + 𝑌) + (𝑋 + 𝑌)))

5-Feb-2025	ringo2times 20010	A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.) Variant of o2timesd 19955 for rings. (Revised by AV, 5-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))

5-Feb-2025	nadd42 8650	Rearragement of terms in a quadruple sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 +no 𝐵) +no (𝐶 +no 𝐷)) = ((𝐴 +no 𝐶) +no (𝐷 +no 𝐵)))

5-Feb-2025	nadd4 8649	Rearragement of terms in a quadruple sum. (Contributed by Scott Fenton, 5-Feb-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On)) → ((𝐴 +no 𝐵) +no (𝐶 +no 𝐷)) = ((𝐴 +no 𝐶) +no (𝐵 +no 𝐷)))

4-Feb-2025	oenord1 41709	When two ordinals (both at least as large as two) are raised to the same power, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 4-Feb-2025.)
⊢ ∃𝑎 ∈ (On ∖ 2_o)∃𝑏 ∈ (On ∖ 2_o)∃𝑐 ∈ (On ∖ 1_o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑_o 𝑐) ∈ (𝑏 ↑_o 𝑐))

4-Feb-2025	omnord1 41698	When the same non-zero ordinal is multiplied on the right, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 4-Feb-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ (On ∖ 1_o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·_o 𝑐) ∈ (𝑏 ·_o 𝑐))

4-Feb-2025	mulslid 27421	Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → ( 1_s ·_s 𝐴) = 𝐴)

4-Feb-2025	mulsrid 27420	Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 ·_s 1_s ) = 𝐴)

4-Feb-2025	muls02 27419	Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → ( 0_s ·_s 𝐴) = 0_s )

4-Feb-2025	muls01 27418	Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 ·_s 0_s ) = 0_s )

4-Feb-2025	mulsval 27417	The value of surreal multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·_s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) \|s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})))

4-Feb-2025	mulsfn 27416	Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ·_s Fn ( No × No )

4-Feb-2025	df-muls 27415	Define surreal multiplication. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ·_s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌(({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘𝑦)𝑎 = (((𝑝𝑚𝑦) +_s (𝑥𝑚𝑞)) -_s (𝑝𝑚𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘𝑦)𝑏 = (((𝑟𝑚𝑦) +_s (𝑥𝑚𝑠)) -_s (𝑟𝑚𝑠))}) \|s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘𝑦)𝑐 = (((𝑡𝑚𝑦) +_s (𝑥𝑚𝑢)) -_s (𝑡𝑚𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘𝑦)𝑑 = (((𝑣𝑚𝑦) +_s (𝑥𝑚𝑤)) -_s (𝑣𝑚𝑤))}))))

4-Feb-2025	sltsub2 27406	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐶 -_s 𝐵) <s (𝐶 -_s 𝐴)))

4-Feb-2025	sltsub1 27405	Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 -_s 𝐶) <s (𝐵 -_s 𝐶)))

4-Feb-2025	npcans 27404	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 -_s 𝐵) +_s 𝐵) = 𝐴)

4-Feb-2025	pncan3s 27403	Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +_s (𝐵 -_s 𝐴)) = 𝐵)

4-Feb-2025	pncans 27402	Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +_s 𝐵) -_s 𝐵) = 𝐴)

4-Feb-2025	right1s 27268	The right set of 1_s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( R ‘ 1_s ) = ∅

4-Feb-2025	left1s 27267	The left set of 1_s is the singleton of 0_s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( L ‘ 1_s ) = { 0_s }

4-Feb-2025	old1 27248	The only surreal older than 1_o is 0_s. (Contributed by Scott Fenton, 4-Feb-2025.)
⊢ ( O ‘1_o) = { 0_s }

3-Feb-2025	cantnf2 41718	For every ordinal, 𝐴, there is a an ordinal exponent 𝑏 such that 𝐴 is less than (ω ↑_o 𝑏) and for every ordinal at least as large as 𝑏 there is a unique Cantor normal form, 𝑓, with zeros for all the unnecessary higher terms, that sums to 𝐴. Theorem 5.3 of [Schloeder] p. 16. (Contributed by RP, 3-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑏 ∈ On ∀𝑐 ∈ (On ∖ 𝑏)∃!𝑓 ∈ dom (ω CNF 𝑐)((𝐴 ∈ (ω ↑_o 𝑏) ∧ 𝑓 finSupp ∅) ∧ (((ω CNF 𝑏)‘(𝑓 ↾ 𝑏)) = 𝐴 ∧ ((ω CNF 𝑐)‘𝑓) = 𝐴)))

3-Feb-2025	cantnfresb 41717	A Cantor normal form which sums to less than a certain power has only zeros for larger components. (Contributed by RP, 3-Feb-2025.)
⊢ (((𝐴 ∈ (On ∖ 2_o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑_o 𝐶) ↔ ∀𝑥 ∈ (𝐵 ∖ 𝐶)(𝐹‘𝑥) = ∅))

3-Feb-2025	subadds 27400	Relationship between addition and subtraction for surreals. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 -_s 𝐵) = 𝐶 ↔ (𝐵 +_s 𝐶) = 𝐴))

3-Feb-2025	subsid 27399	Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 -_s 𝐴) = 0_s )

3-Feb-2025	subsid1 27398	Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 -_s 0_s ) = 𝐴)

3-Feb-2025	subscl 27396	Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -_s 𝐵) ∈ No )

3-Feb-2025	subsval 27394	The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -_s 𝐵) = (𝐴 +_s ( -u_s ‘𝐵)))

3-Feb-2025	negsf1o 27392	Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -u_s : No –1-1-onto→ No

3-Feb-2025	negsfo 27391	Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -u_s : No –onto→ No

3-Feb-2025	negsf 27390	Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ -u_s : No ⟶ No

3-Feb-2025	negs11 27388	Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -u_s ‘𝐴) = ( -u_s ‘𝐵) ↔ 𝐴 = 𝐵))

3-Feb-2025	sleneg 27387	Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ( -u_s ‘𝐵) ≤s ( -u_s ‘𝐴)))

3-Feb-2025	sltneg 27386	Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ( -u_s ‘𝐵) <s ( -u_s ‘𝐴)))

3-Feb-2025	negnegs 27385	A surreal is equal to the negative of its negative. Theorem 4(ii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -u_s ‘( -u_s ‘𝐴)) = 𝐴)

3-Feb-2025	negsex 27384	Every surreal has a negative. Note that this theorem, addscl 27334, addscom 27321, addsass 27356, addsrid 27319, and sltadd1im 27337 are the ordered Abelian group axioms. However, the surreals cannot be said to be an ordered Abelian group because No is a proper class. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ∃𝑥 ∈ No (𝐴 +_s 𝑥) = 0_s )

3-Feb-2025	negsid 27382	Surreal addition of a number and its negative. Theorem 4(iii) of [Conway] p. 17. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (𝐴 +_s ( -u_s ‘𝐴)) = 0_s )

3-Feb-2025	negscut2 27381	The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -u_s “ ( R ‘𝐴)) <<s ( -u_s “ ( L ‘𝐴)))

3-Feb-2025	negscut 27380	The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → (( -u_s ‘𝐴) ∈ No ∧ ( -u_s “ ( R ‘𝐴)) <<s {( -u_s ‘𝐴)} ∧ {( -u_s ‘𝐴)} <<s ( -u_s “ ( L ‘𝐴))))

3-Feb-2025	sltnegim 27379	The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ( -u_s ‘𝐵) <s ( -u_s ‘𝐴)))

3-Feb-2025	negscld 27378	The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → ( -u_s ‘𝐴) ∈ No )

3-Feb-2025	negscl 27377	The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( -u_s ‘𝐴) ∈ No )

3-Feb-2025	negsprop 27376	Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -u_s ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -u_s ‘𝐵) <s ( -u_s ‘𝐴))))

3-Feb-2025	negsproplem7 27375	Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) ⇒ ⊢ (𝜑 → ( -u_s ‘𝐵) <s ( -u_s ‘𝐴))

3-Feb-2025	negsproplem6 27374	Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is the same age as 𝐵. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ( bday ‘𝐴) = ( bday ‘𝐵)) ⇒ ⊢ (𝜑 → ( -u_s ‘𝐵) <s ( -u_s ‘𝐴))

3-Feb-2025	negsproplem5 27373	Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐵 is simpler than 𝐴. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) ⇒ ⊢ (𝜑 → ( -u_s ‘𝐵) <s ( -u_s ‘𝐴))

3-Feb-2025	addslid 27323	Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝐴 ∈ No → ( 0_s +_s 𝐴) = 𝐴)

3-Feb-2025	cuteq0 27214	Condition for a surreal cut to equal zero. (Contributed by Scott Fenton, 3-Feb-2025.)
⊢ (𝜑 → 𝐴 <<s { 0_s }) & ⊢ (𝜑 → { 0_s } <<s 𝐵) ⇒ ⊢ (𝜑 → (𝐴 \|s 𝐵) = 0_s )

2-Feb-2025	negsproplem4 27372	Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is simpler than 𝐵. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵)) ⇒ ⊢ (𝜑 → ( -u_s ‘𝐵) <s ( -u_s ‘𝐴))

2-Feb-2025	negsproplem3 27371	Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → (( -u_s ‘𝐴) ∈ No ∧ ( -u_s “ ( R ‘𝐴)) <<s {( -u_s ‘𝐴)} ∧ {( -u_s ‘𝐴)} <<s ( -u_s “ ( L ‘𝐴))))

2-Feb-2025	negsproplem2 27370	Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → ( -u_s “ ( R ‘𝐴)) <<s ( -u_s “ ( L ‘𝐴)))

2-Feb-2025	negsproplem1 27369	Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -u_s ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -u_s ‘𝑦) <s ( -u_s ‘𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵))) ⇒ ⊢ (𝜑 → (( -u_s ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -u_s ‘𝑌) <s ( -u_s ‘𝑋))))

2-Feb-2025	xpord3indd 8092	Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝜅 → 𝑋 ∈ 𝐴) & ⊢ (𝜅 → 𝑌 ∈ 𝐵) & ⊢ (𝜅 → 𝑍 ∈ 𝐶) & ⊢ (𝜅 → 𝑅 Fr 𝐴) & ⊢ (𝜅 → 𝑅 Po 𝐴) & ⊢ (𝜅 → 𝑅 Se 𝐴) & ⊢ (𝜅 → 𝑆 Fr 𝐵) & ⊢ (𝜅 → 𝑆 Po 𝐵) & ⊢ (𝜅 → 𝑆 Se 𝐵) & ⊢ (𝜅 → 𝑇 Fr 𝐶) & ⊢ (𝜅 → 𝑇 Po 𝐶) & ⊢ (𝜅 → 𝑇 Se 𝐶) & ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) & ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) & ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) & ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) & ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) & ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) & ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) & ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) & ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) ⇒ ⊢ (𝜅 → 𝜆)

2-Feb-2025	xpord3inddlem 8091	Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1^st ‘(1^st ‘𝑥))𝑅(1^st ‘(1^st ‘𝑦)) ∨ (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑦))) ∧ ((2^nd ‘(1^st ‘𝑥))𝑆(2^nd ‘(1^st ‘𝑦)) ∨ (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑦))) ∧ ((2^nd ‘𝑥)𝑇(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))} & ⊢ (𝜅 → 𝑋 ∈ 𝐴) & ⊢ (𝜅 → 𝑌 ∈ 𝐵) & ⊢ (𝜅 → 𝑍 ∈ 𝐶) & ⊢ (𝜅 → 𝑅 Fr 𝐴) & ⊢ (𝜅 → 𝑅 Po 𝐴) & ⊢ (𝜅 → 𝑅 Se 𝐴) & ⊢ (𝜅 → 𝑆 Fr 𝐵) & ⊢ (𝜅 → 𝑆 Po 𝐵) & ⊢ (𝜅 → 𝑆 Se 𝐵) & ⊢ (𝜅 → 𝑇 Fr 𝐶) & ⊢ (𝜅 → 𝑇 Po 𝐶) & ⊢ (𝜅 → 𝑇 Se 𝐶) & ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃)) & ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃)) & ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏)) & ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃)) & ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏)) & ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌)) & ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇)) & ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆)) & ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑)) ⇒ ⊢ (𝜅 → 𝜆)

2-Feb-2025	ralxp3 8075	Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)

2-Feb-2025	issetlem 2812	Lemma for elisset 2814 and isset 3459. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3459. (Revised by WL, 2-Feb-2025.)
⊢ 𝑥 ∈ 𝑉 ⇒ ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴)

1-Feb-2025	smfinfdmmbl 45210	If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their infimum function has the domain in the sigma-algebra. This is the fifth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 1-Feb-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

1-Feb-2025	smfinfdmmbllem 45209	If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their infimum function has the domain in the sigma-algebra. This is the fifth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 1-Feb-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

1-Feb-2025	finfdm2 45208	The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45179. Note that this definition of the inf function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fifth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 1-Feb-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) ⇒ ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

1-Feb-2025	finfdm 45207	The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45179. Note that this definition of the inf function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fifth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 1-Feb-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) ⇒ ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

1-Feb-2025	archd 43499	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by Glauco Siliprandi, 1-Feb-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝐴 < 𝑛)

1-Feb-2025	onexoegt 41636	For any ordinal, there is always a larger power of omega. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ↑_o 𝑥))

1-Feb-2025	onexlimgt 41635	For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥 ∧ 𝐴 ∈ 𝑥))

1-Feb-2025	onexomgt 41633	For any ordinal, there is always a larger product of omega. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·_o 𝑥))

1-Feb-2025	onexgt 41632	For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)
⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)

1-Feb-2025	zaddcomlem 40978	Lemma for zaddcom 40979. (Contributed by SN, 1-Feb-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −_ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ₀) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

1-Feb-2025	nn0addcom 40977	Addition is commutative for nonnegative integers. Proven without ax-mulcom 11124. (Contributed by SN, 1-Feb-2025.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

1-Feb-2025	grpcominv2 40757	If two elements commute, then they commute with each other's inverses (case of the second element commuting with the inverse of the first element). (Contributed by SN, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) ⇒ ⊢ (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌))

1-Feb-2025	ringcom 20015	Commutativity of the additive group of a ring. (See also lmodcom 20425.) This proof requires the existence of a multiplicative identity, and the existence of additive inverses. Therefore, this proof is not applicable for semirings. (Contributed by Gérard Lang, 4-Dec-2014.) (Proof shortened by AV, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

1-Feb-2025	ringadd2 20011	A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) (Proof shortened by AV, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ · = (._r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋))

1-Feb-2025	srgcom4 19959	Restricted commutativity of the addition in semirings (without using the commutativity of the addition given per definition of a semiring). (Contributed by AV, 1-Feb-2025.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑋 + 𝑌)) + 𝑌) = ((𝑋 + (𝑌 + 𝑋)) + 𝑌))

1-Feb-2025	srgcom4lem 19958	Lemma for srgcom4 19959. This (formerly) part of the proof for ringcom 20015 is applicable for semirings (without using the commutativity of the addition given per definition of a semiring). (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = ((𝑋 + 𝑌) + (𝑋 + 𝑌)))

1-Feb-2025	srgo2times 19957	A semiring element plus itself is two times the element. "Two" in an arbitrary (unital) semiring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.) Variant of o2timesd 19955 for semirings. (Revised by AV, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))

1-Feb-2025	rglcom4d 19956	Restricted commutativity of the addition in a ring-like structure. This (formerly) part of the proof for ringcom 20015 depends on the closure of the addition, the (left and right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) & ⊢ (𝜑 → 1 ∈ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑋) + (𝑌 + 𝑌)) = ((𝑋 + 𝑌) + (𝑋 + 𝑌)))

1-Feb-2025	o2timesd 19955	An element of a ring-like structure plus itself is two times the element. "Two" in such a structure is the sum of the unity element with itself. This (formerly) part of the proof for ringcom 20015 depends on the (right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) & ⊢ (𝜑 → 1 ∈ 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 𝑋) = (( 1 + 1 ) · 𝑋))

1-Feb-2025	mulgnn0cld 18911	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 18906. (Contributed by SN, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ · = (._g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Mnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ∈ 𝐵)

1-Feb-2025	el2xpss 7974	Version of elrel 5759 for triple Cartesian products. (Contributed by Scott Fenton, 1-Feb-2025.)
⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)

31-Jan-2025	cantnfub 41714	Given a finite number of terms of the form ((ω ↑_o (𝐴‘𝑛)) ·_o (𝑀‘𝑛)) with distinct exponents, we may order them from largest to smallest and find the sum is less than (ω ↑_o 𝑋) when (𝐴‘𝑛) is less than 𝑋 and (𝑀‘𝑛) is less than ω. Lemma 5.2 of [Schloeder] p. 15. (Contributed by RP, 31-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ On) & ⊢ (𝜑 → 𝑁 ∈ ω) & ⊢ (𝜑 → 𝐴:𝑁–1-1→𝑋) & ⊢ (𝜑 → 𝑀:𝑁⟶ω) & ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ ran 𝐴, (𝑀‘(^◡𝐴‘𝑥)), ∅)) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom (ω CNF 𝑋) ∧ ((ω CNF 𝑋)‘𝐹) ∈ (ω ↑_o 𝑋)))

31-Jan-2025	finsubmsubg 40758	A submonoid of a finite group is a subgroup. This does not extend to infinite groups, as the submonoid ℕ₀ of the group (ℤ, + ) shows. Note also that the union of a submonoid and its inverses need not be a submonoid, as the submonoid (ℕ₀ ∖ {1}) of the group (ℤ, + ) shows: 3 is in that submonoid, -2 is the inverse of 2, but 1 is not in their union. Or simply, the subgroup generated by (ℕ₀ ∖ {1}) is ℤ, not (ℤ ∖ {1, -1}). (Contributed by SN, 31-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))

31-Jan-2025	0subgALT 19364	A shorter proof of 0subg 18967 using df-od 19324. (Contributed by SN, 31-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 = (0_g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))

31-Jan-2025	finodsubmsubg 19363	A submonoid whose elements have finite order is a subgroup. (Contributed by SN, 31-Jan-2025.)
⊢ 𝑂 = (od‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝑂‘𝑎) ∈ ℕ) ⇒ ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))

31-Jan-2025	odm1inv 19349	The (order-1)th multiple of an element is its inverse. (Contributed by SN, 31-Jan-2025.)
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ · = (._g‘𝐺) & ⊢ 𝐼 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → (((𝑂‘𝐴) − 1) · 𝐴) = (𝐼‘𝐴))

31-Jan-2025	0subg 18967	The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
⊢ 0 = (0_g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))

31-Jan-2025	xpord3pred 8089	Calculate the predecsessor class for the triple order. (Contributed by Scott Fenton, 31-Jan-2025.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1^st ‘(1^st ‘𝑥))𝑅(1^st ‘(1^st ‘𝑦)) ∨ (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑦))) ∧ ((2^nd ‘(1^st ‘𝑥))𝑆(2^nd ‘(1^st ‘𝑦)) ∨ (2^nd ‘(1^st ‘𝑥)) = (2^nd ‘(1^st ‘𝑦))) ∧ ((2^nd ‘𝑥)𝑇(2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))} ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑋, 𝑌, 𝑍⟩) = ((((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) × (Pred(𝑇, 𝐶, 𝑍) ∪ {𝑍})) ∖ {⟨𝑋, 𝑌, 𝑍⟩}))

31-Jan-2025	otelxp 5681	Ordered triple membership in a triple Cartesian product. (Contributed by Scott Fenton, 31-Jan-2025.)
⊢ (⟨𝐴, 𝐵, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))

31-Jan-2025	otthne 5448	Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 31-Jan-2025.)
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))

30-Jan-2025	cantnftermord 41713	For terms of the form of a power of omega times a non-zero natural number, ordering of the exponents implies ordering of the terms. Lemma 5.1 of [Schloeder] p. 15. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ (ω ∖ 1_o) ∧ 𝐷 ∈ (ω ∖ 1_o))) → (𝐴 ∈ 𝐵 → ((ω ↑_o 𝐴) ·_o 𝐶) ∈ ((ω ↑_o 𝐵) ·_o 𝐷)))

30-Jan-2025	oenass 41712	Ordinal exponentiation is not associative. Remark 4.6 of [Schloeder] p. 14. (Contributed by RP, 30-Jan-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑_o (𝑏 ↑_o 𝑐)) = ((𝑎 ↑_o 𝑏) ↑_o 𝑐)

30-Jan-2025	oenassex 41711	Ordinal two raised to two to the zeroth power is not the same as two squared then raised to the zeroth power. (Contributed by RP, 30-Jan-2025.)
⊢ ¬ (2_o ↑_o (2_o ↑_o ∅)) = ((2_o ↑_o 2_o) ↑_o ∅)

30-Jan-2025	oaomoencom 41710	Ordinal addition, multiplication, and exponentiation do not generally commute. Theorem 4.1 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 +_o 𝑏) = (𝑏 +_o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ·_o 𝑏) = (𝑏 ·_o 𝑎) ∧ ∃𝑎 ∈ On ∃𝑏 ∈ On ¬ (𝑎 ↑_o 𝑏) = (𝑏 ↑_o 𝑎))

30-Jan-2025	oenord1ex 41708	When ordinals two and three are both raised to the power of omega, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ ¬ (2_o ∈ 3_o ↔ (2_o ↑_o ω) ∈ (3_o ↑_o ω))

30-Jan-2025	nnoeomeqom 41705	Any natural number at least as large as two raised to the power of omega is omega. Lemma 3.25 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 1_o ∈ 𝐴) → (𝐴 ↑_o ω) = ω)

30-Jan-2025	oeord2com 41704	When the same base at least as large as two is raised to ordinal powers, , ordering of the power is equivalent to the ordering of the exponents. Theorem 3.24 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1_o ∈ 𝐴) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 ↑_o 𝐵) ∈ (𝐴 ↑_o 𝐶)))

30-Jan-2025	oeord2i 41703	Ordinal exponentiation of the same base at least as large as two preserves the ordering of the exponents. Lemma 3.23 of [Schloeder] p. 11. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1_o ∈ 𝐴) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑_o 𝐵) ∈ (𝐴 ↑_o 𝐶)))

30-Jan-2025	oeord2lim 41702	Given a limit ordinal, the power of any base at least as large as two raised to an ordinal less than that limit ordinal is less than the power of that base raised to the limit ordinal . Lemma 3.22 of [Schloeder] p. 10. See oeordi 8539. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1_o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑_o 𝐵) ∈ (𝐴 ↑_o 𝐶)))

30-Jan-2025	rp-oelim2 41701	The power of an ordinal at least as large as two with a limit ordinal on thr right is a limit ordinal. Lemma 3.21 of [Schloeder] p. 10. See oelimcl 8552. (Contributed by RP, 30-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1_o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑_o 𝐵))

30-Jan-2025	2omomeqom 41696	Ordinal two times omega is omega. Lemma 3.17 of [Schloeder] p. 10. (Contributed by RP, 30-Jan-2025.)
⊢ (2_o ·_o ω) = ω

30-Jan-2025	ressply1sub 32358	A restricted polynomial algebra has the same subtraction operation. (Contributed by Thierry Arnoux, 30-Jan-2025.)
⊢ 𝑆 = (Poly₁‘𝑅) & ⊢ 𝐻 = (𝑅 ↾_s 𝑇) & ⊢ 𝑈 = (Poly₁‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑃 = (𝑆 ↾_s 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋(-_g‘𝑈)𝑌) = (𝑋(-_g‘𝑃)𝑌))

30-Jan-2025	ressply1invg 32357	An element of a restricted polynomial algebra has the same group inverse. (Contributed by Thierry Arnoux, 30-Jan-2025.)
⊢ 𝑆 = (Poly₁‘𝑅) & ⊢ 𝐻 = (𝑅 ↾_s 𝑇) & ⊢ 𝑈 = (Poly₁‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑃 = (𝑆 ↾_s 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((inv_g‘𝑈)‘𝑋) = ((inv_g‘𝑃)‘𝑋))

29-Jan-2025	oege2 41700	Any power of an ordinal at least as large as two is greater-than-or-equal to the term on the right. Lemma 3.20 of [Schloeder] p. 10. See oeworde 8545. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 1_o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑_o 𝐵))

29-Jan-2025	oege1 41699	Any non-zero ordinal power is greater-than-or-equal to the term on the left. Lemma 3.19 of [Schloeder] p. 10. See oewordi 8543. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑_o 𝐵))

29-Jan-2025	omnord1ex 41697	When omega is multiplied on the right to ordinals one and two, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 29-Jan-2025.)
⊢ ¬ (1_o ∈ 2_o ↔ (1_o ·_o ω) ∈ (2_o ·_o ω))

29-Jan-2025	omord2com 41695	When the same non-zero ordinal is multiplied on the left, ordering of the products is equivalent to the ordering of the ordinals on the right. Theorem 3.16 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∈ 𝐶 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·_o 𝐵) ∈ (𝐴 ·_o 𝐶)))

29-Jan-2025	omord2i 41694	Ordinal multiplication of the same non-zero number on the left preserves the ordering of the numbers on the right. Lemma 3.15 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ·_o 𝐵) ∈ (𝐴 ·_o 𝐶)))

29-Jan-2025	omord2lim 41693	Given a limit ordinal, the product of any non-zero ordinal with an ordinal less than that limit ordinal is less than the product of the non-zero ordinal with the limit ordinal . Lemma 3.14 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ·_o 𝐵) ∈ (𝐴 ·_o 𝐶)))

29-Jan-2025	omlim2 41692	The non-zero product with an limit ordinal on the right is a limit ordinal. Lemma 3.13 of [Schloeder] p. 9. (Contributed by RP, 29-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐴 ≠ ∅) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ·_o 𝐵))

29-Jan-2025	omge2 41691	Any non-zero ordinal product is greater-than-or-equal to the term on the right. Lemma 3.12 of [Schloeder] p. 9. See omword2 8526. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·_o 𝐵))

29-Jan-2025	omge1 41690	Any non-zero ordinal product is greater-than-or-equal to the term on the left. Lemma 3.11 of [Schloeder] p. 8. See omword1 8525. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·_o 𝐵))

29-Jan-2025	oaordnr 41689	When the same ordinal is added on the right, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 +_o 𝑐) ∈ (𝑏 +_o 𝑐))

29-Jan-2025	oaordnrex 41688	When omega is added on the right to ordinals zero and one, ordering of the sums is not equivalent to the ordering of the ordinals on the left. Remark 3.9 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ¬ (∅ ∈ 1_o ↔ (∅ +_o ω) ∈ (1_o +_o ω))

29-Jan-2025	1oaomeqom 41686	Ordinal one plus omega is equal to omega. See oaabs 8599 for the sum of any natural number on the left and ordinal at least as large as omega on the right. Lemma 3.8 of [Schloeder] p. 8. See oaabs2 8600 where a power of omega is the upper bound of the left and a lower bound on the right. (Contributed by RP, 29-Jan-2025.)
⊢ (1_o +_o ω) = ω

29-Jan-2025	oaord3 41685	When the same ordinal is added on the left, ordering of the sums is equivalent to the ordering of the ordinals on the right. Theorem 3.7 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 +_o 𝐵) ∈ (𝐴 +_o 𝐶)))

29-Jan-2025	oaordi3 41684	Ordinal addition of the same number on the left preserves the ordering of the numbers on the right. Lemma 3.6 of [Schloeder] p. 8. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 +_o 𝐵) ∈ (𝐴 +_o 𝐶)))

29-Jan-2025	oaltublim 41683	Given 𝐶 is a limit ordinal, the sum of any ordinal with an ordinal less than 𝐶 is less than the sum of the first ordinal with 𝐶. Lemma 3.5 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐴 +_o 𝐵) ∈ (𝐴 +_o 𝐶))

29-Jan-2025	onsucwordi 41681	The successor operation preserves the less-than-or-equal relationship between ordinals. Lemma 3.1 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → suc 𝐴 ⊆ suc 𝐵))

29-Jan-2025	nnamecl 41680	Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8563, nnmcl 8564, nnecl 8565. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +_o 𝐵) ∈ ω ∧ (𝐴 ·_o 𝐵) ∈ ω ∧ (𝐴 ↑_o 𝐵) ∈ ω))

29-Jan-2025	oasubex 41679	While subtraction can't be a binary operation on ordinals, for any pair of ordinals there exists an ordinal that can be added to the lessor (or equal) one which will sum to the greater. Theorem 2.19 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +_o 𝑐) = 𝐴))

29-Jan-2025	oe0rif 41678	Ordinal zero raised to any non-zero ordinal power is zero and zero to the zeroth power is one. Lemma 2.18 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ∈ On → (∅ ↑_o 𝐴) = if(∅ ∈ 𝐴, ∅, 1_o))

29-Jan-2025	om1om1r 41677	Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8494 and om1r 8495 for individual statements. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ∈ On → ((1_o ·_o 𝐴) = (𝐴 ·_o 1_o) ∧ (𝐴 ·_o 1_o) = 𝐴))

29-Jan-2025	onintunirab 41619	The intersection of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 29-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

29-Jan-2025	unielss 41610	Two ways to say the union of a class is an element of a subclass. (Contributed by RP, 29-Jan-2025.)
⊢ (𝐴 ⊆ 𝐵 → (∪ 𝐵 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥))

29-Jan-2025	grpcominv1 40756	If two elements commute, then they commute with each other's inverses (case of the first element commuting with the inverse of the second element). (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) ⇒ ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))

29-Jan-2025	grpasscan2d 40755	An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

29-Jan-2025	sn-grprinvd 40754	The right inverse of a group element. Deduction associated with grprinv 18815. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )

29-Jan-2025	grplinvd 40753	The left inverse of a group element. Deduction associated with grplinv 18814. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )

29-Jan-2025	grpassd 40752	A group operation is associative. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

29-Jan-2025	sn-grpridd 40751	The identity element of a group is a right identity. Deduction associated with grprid 18795. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋)

29-Jan-2025	sn-grplidd 40750	The identity element of a group is a left identity. Deduction associated with grplid 18794. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)

29-Jan-2025	riotaeqbidva 31488	Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3319 analog.) (Contributed by Thierry Arnoux, 29-Jan-2025.)
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))

29-Jan-2025	grpinvcld 18813	A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵)

28-Jan-2025	onuniintrab 41618	The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Closed form of uniordint 7741. (Contributed by RP, 28-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

28-Jan-2025	irngssv 32449	An integral element is an element of the base set. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑈 = (𝑅 ↾_s 𝑆) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 IntgRing 𝑆) ⊆ 𝐵)

28-Jan-2025	irngss 32448	All elements of a subring 𝑆 are integral over 𝑆. This is only true in the case of a nonzero ring, since there are no integral elements in a zero ring (see 0ringirng 32450). (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑈 = (𝑅 ↾_s 𝑆) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))

28-Jan-2025	elirng 32447	Property for an element 𝑋 of a field 𝑅 to be integral over a subring 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑈 = (𝑅 ↾_s 𝑆) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )))

28-Jan-2025	irngval 32446	The elements of a field 𝑅 integral over a subset 𝑆. In the case of a subfield, those are the algebraic numbers over the field 𝑆 within the field 𝑅. That is, the numbers 𝑋 which are roots of monic polynomials 𝑃(𝑋) with coefficients in 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)
⊢ 𝑂 = (𝑅 evalSub₁ 𝑆) & ⊢ 𝑈 = (𝑅 ↾_s 𝑆) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic_1p‘𝑈)(^◡(𝑂‘𝑓) “ { 0 }))

28-Jan-2025	df-irng 32445	Define the subring of elements of a ring 𝑟 integral over a subset 𝑠. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Thierry Arnoux, 28-Jan-2025.)
⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic_1p‘(𝑟 ↾_s 𝑠))(^◡((𝑟 evalSub₁ 𝑠)‘𝑓) “ {(0_g‘𝑟)}))

27-Jan-2025	limexissupab 41676	An ordinal which is a limit ordinal is equal to the supremum of the class of all its elements. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup({𝑥 ∣ 𝑥 ∈ 𝐴}, On, E ))

27-Jan-2025	limiun 41675	A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6383. (Contributed by RP, 27-Jan-2025.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)

27-Jan-2025	limexissup 41674	An ordinal which is a limit ordinal is equal to its supremum. Lemma 2.13 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 = sup(𝐴, On, E ))

27-Jan-2025	onsssupeqcond 41673	If for every element of a set of ordinals there is an element of a subset which is at least as large, then the union of the set and the subset is the same. Lemma 2.12 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐵 ⊆ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏) → ∪ 𝐴 = ∪ 𝐵))

27-Jan-2025	onsupsucismax 41672	If the union of a set of ordinals is a successor ordinal, then that union is the maximum element of the set. This is not a bijection because sets where the maximum element is zero or a limit ordinal exist. Lemma 2.11 of [Schloeder] p. 5. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑏 ∈ On ∪ 𝐴 = suc 𝑏 → ∪ 𝐴 ∈ 𝐴))

27-Jan-2025	onfisupcl 41642	Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9244. (Contributed by RP, 27-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴))

27-Jan-2025	onsupnub 41641	An upper bound of a set of ordinals is not less than the supremum. (Contributed by RP, 27-Jan-2025.)
⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵)

27-Jan-2025	onsuplub 41640	The supremum of a set of ordinals is the least upper bound. (Contributed by RP, 27-Jan-2025.)
⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧))

27-Jan-2025	onsupeqnmax 41639	Condition when the supremum of a class of ordinals is not the maximum element of that class. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (∪ 𝐴 = ∪ ∪ 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴)))

27-Jan-2025	onsupnmax 41620	If the union of a class of ordinals is not the maximum element of that class, then the union is a limit ordinal or empty. But this isn't a biconditional since 𝐴 could be a non-empty set where a limit ordinal or the empty set happens to be the largest element. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

27-Jan-2025	onsupneqmaxlim0 41616	If the supremum of a class of ordinals is not in that class, then the supremum is a limit ordinal or empty. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

27-Jan-2025	onmaxnelsup 41615	Two ways to say the maximum element of a class of ordinals is also the supremum of that class. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ On → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))

27-Jan-2025	ssunib 41612	Two ways to say a class is a subclass of a union. (Contributed by RP, 27-Jan-2025.)
⊢ (𝐴 ⊆ ∪ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ∈ 𝑦)

27-Jan-2025	unielid 41611	Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.)
⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)

27-Jan-2025	uniel 41609	Two ways to say a union is an element of a class. (Contributed by RP, 27-Jan-2025.)
⊢ (∪ 𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝑦))

26-Jan-2025	vtoclegft 3543	Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3516.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑)

26-Jan-2025	vtoclg 3526	Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2171. (Revised by SN, 20-Apr-2024.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝜓)

26-Jan-2025	vtoclf 3517	Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2393. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓

25-Jan-2025	onsupmaxb 41631	The union of a class of ordinals is an element is an element of that class if and only if there is a maximum element of that class under the epsilon relation, which is to say that the domain of the restricted epsilon relation is not the whole class. (Contributed by RP, 25-Jan-2025.)
⊢ (𝐴 ⊆ On → (dom ( E ∩ (𝐴 × 𝐴)) = 𝐴 ↔ ¬ ∪ 𝐴 ∈ 𝐴))

25-Jan-2025	zmulcom 40983	Multiplication is commutative for integers. Proven without ax-mulcom 11124. From this result and grpcominv1 40756, we can show that rationals commute under multiplication without using ax-mulcom 11124. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

25-Jan-2025	zmulcomlem 40982	Lemma for zmulcom 40983. (Contributed by SN, 25-Jan-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −_ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ₀) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

25-Jan-2025	nn0mulcom 40981	Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11124. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

25-Jan-2025	renegmulnnass 40980	Move multiplication by a natural number inside and outside negation. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ((0 −_ℝ 𝐴) · 𝑁) = (0 −_ℝ (𝐴 · 𝑁)))

25-Jan-2025	zaddcom 40979	Addition is commutative for integers. Proven without ax-mulcom 11124. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

25-Jan-2025	reelznn0nn 40976	elznn0nn 12522 restated using df-resub 40893. (Contributed by SN, 25-Jan-2025.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ (0 −_ℝ 𝑁) ∈ ℕ)))

25-Jan-2025	sn-nnne0 40975	nnne0 12196 without ax-mulcom 11124. (Contributed by SN, 25-Jan-2025.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)

25-Jan-2025	sn-addgt0d 40974	The sum of positive numbers is positive. Proof of addgt0d 11739 without ax-mulcom 11124. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → 0 < (𝐴 + 𝐵))

25-Jan-2025	sn-addlt0d 40973	The sum of negative numbers is negative. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 < 0) & ⊢ (𝜑 → 𝐵 < 0) ⇒ ⊢ (𝜑 → (𝐴 + 𝐵) < 0)

25-Jan-2025	sn-ltaddneg 40969	ltaddneg 11379 without ax-mulcom 11124. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))

25-Jan-2025	remulneg2d 40941	Product with negative is negative of product. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 · (0 −_ℝ 𝐵)) = (0 −_ℝ (𝐴 · 𝐵)))

25-Jan-2025	wl-issetft 36107	A closed form of issetf 3460. The proof here is a modification of a subproof in vtoclgft 3510, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.)
⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))

25-Jan-2025	sbhypf 3508	Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3887. (Contributed by Raph Levien, 10-Apr-2004.) (Proof shortened by Wolf Lammen, 25-Jan-2025.)
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))

24-Jan-2025	smfsupdmmbl 45206	If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their supremum function has the domain in the sigma-algebra. This is the fourth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

24-Jan-2025	smfsupdmmbllem 45205	If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their supremum function has the domain in the sigma-algebra. This is the fourth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ_≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})) & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

24-Jan-2025	fsupdm2 45204	The domain of the sup function is defined in Proposition 121F (b) of [Fremlin1], p. 38. Note that this definition of the sup function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fourth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})) ⇒ ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

24-Jan-2025	fsupdm 45203	The domain of the sup function is defined in Proposition 121F (b) of [Fremlin1], p. 38. Note that this definition of the sup function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fourth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})) ⇒ ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

24-Jan-2025	saliinclf 44687	SAlg sigma-algebra is closed under countable indexed intersection. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝑆 & ⊢ Ⅎ𝑘𝐾 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐾 ≼ ω) & ⊢ (𝜑 → 𝐾 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → ∩ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)

24-Jan-2025	saliunclf 44683	SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝑆 & ⊢ Ⅎ𝑘𝐾 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐾 ≼ ω) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆)

24-Jan-2025	nnxr 43629	A natural number is an extended real. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ^*)

24-Jan-2025	rnmptssdff 43625	The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶)

24-Jan-2025	rnmptssff 43624	The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

24-Jan-2025	rn1st 43623	The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 22840, with a not-free hypothesis replacing a disjoint variable constraint. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)

24-Jan-2025	fvmpt2df 43622	Deduction version of fvmpt2 6964. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)

24-Jan-2025	iindif2f 43497	Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws". (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) = (𝐵 ∖ ∪ 𝑥 ∈ 𝐴 𝐶))

24-Jan-2025	r19.28zf 43496	Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

24-Jan-2025	r19.3rzf 43495	Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

24-Jan-2025	iinss2d 43494	Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

24-Jan-2025	iunssdf 43493	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐶 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

24-Jan-2025	rabidd 43492	An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → 𝜒) ⇒ ⊢ (𝜑 → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜒})

24-Jan-2025	onsupeqmax 41638	Condition when the supremum of a set of ordinals is the maximum element of that set. (Contributed by RP, 24-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))

24-Jan-2025	evls1muld 32351	Univariate polynomial evaluation of a product of polynomials. (Contributed by Thierry Arnoux, 24-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ × = (._r‘𝑊) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑀 × 𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) · ((𝑄‘𝑁)‘𝐶)))

24-Jan-2025	evls1varpwval 32347	Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 21765. (Contributed by Thierry Arnoux, 24-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑋 = (var₁‘𝑈) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ ∧ = (._g‘(mulGrp‘𝑊)) & ⊢ ↑ = (._g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑋))‘𝐶) = (𝑁 ↑ 𝐶))

24-Jan-2025	evls1expd 32346	Univariate polynomial evaluation builder for an exponential. See also evl1expd 21748. (Contributed by Thierry Arnoux, 24-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ ∧ = (._g‘(mulGrp‘𝑊)) & ⊢ ↑ = (._g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝑁 ∧ 𝑀))‘𝐶) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐶)))

23-Jan-2025	oninfex2 41637	The infimum of a non-empty class of ordinals exists. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ V)

23-Jan-2025	oninfcl2 41630	The infimum of a non-empty class of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} ∈ On)

23-Jan-2025	oninfunirab 41629	The infimum of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦})

23-Jan-2025	oninfint 41628	The infimum of a non-empty class of ordinals is the intersection of that class. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = ∩ 𝐴)

23-Jan-2025	onuniintrab2 41627	The union of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

23-Jan-2025	onsupex3 41626	The supremum of a set of ordinals exists. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ V)

23-Jan-2025	onsupcl3 41625	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} ∈ On)

23-Jan-2025	onsupintrab2 41624	The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

23-Jan-2025	onsupintrab 41623	The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of [Schloeder] p. 5. (Contributed by RP, 23-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

23-Jan-2025	onsupcl2 41617	The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On)

23-Jan-2025	rp-unirabeq 41614	Equality theorem for infimum of non-empty classes of ordinals. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 = 𝐵 → ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑥 ⊆ 𝑦} = ∪ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦})

23-Jan-2025	rp-intrabeq 41613	Equality theorem for supremum of sets of ordinals. (Contributed by RP, 23-Jan-2025.)
⊢ (𝐴 = 𝐵 → ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥})

23-Jan-2025	ressply1evl 32349	Evaluation of a univariate subring polynomial is the same as the evaluation in the bigger ring. (Contributed by Thierry Arnoux, 23-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐸 = (eval₁‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) ⇒ ⊢ (𝜑 → 𝑄 = (𝐸 ↾ 𝐵))

23-Jan-2025	evls1fpws 32348	Evaluation of a univariate subring polynomial as a function in a power series. (Contributed by Thierry Arnoux, 23-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ · = (._r‘𝑆) & ⊢ ↑ = (._g‘(mulGrp‘𝑆)) & ⊢ 𝐴 = (coe₁‘𝑀) ⇒ ⊢ (𝜑 → (𝑄‘𝑀) = (𝑥 ∈ 𝐾 ↦ (𝑆 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))

23-Jan-2025	cofcutrtime2d 27291	If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( R ‘𝑋) is coinitial with 𝐵. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) & ⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 \|s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧)

23-Jan-2025	cofcutrtime1d 27290	If 𝑋 is a timely cut of 𝐴 and 𝐵, then ( L ‘𝑋) is cofinal with 𝐴. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋))) & ⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 \|s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦)

23-Jan-2025	cofcutr1d 27287	If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋). First half of theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 \|s 𝐵)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦)

23-Jan-2025	cofcut2d 27285	If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝒫 No ) & ⊢ (𝜑 → 𝐷 ∈ 𝒫 No ) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) & ⊢ (𝜑 → ∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢) & ⊢ (𝜑 → ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟) ⇒ ⊢ (𝜑 → (𝐴 \|s 𝐵) = (𝐶 \|s 𝐷))

23-Jan-2025	cofcut1d 27283	If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷, then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 23-Jan-2025.)
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) & ⊢ (𝜑 → 𝐶 <<s {(𝐴 \|s 𝐵)}) & ⊢ (𝜑 → {(𝐴 \|s 𝐵)} <<s 𝐷) ⇒ ⊢ (𝜑 → (𝐴 \|s 𝐵) = (𝐶 \|s 𝐷))

23-Jan-2025	ceqsal 3480	A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid df-clab 2709. (Revised by Wolf Lammen, 23-Jan-2025.)
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)

22-Jan-2025	onsupuni2 41622	The supremum of a set of ordinals is the union of that set. (Contributed by RP, 22-Jan-2025.)
⊢ (𝐴 ∈ 𝒫 On → sup(𝐴, On, E ) = ∪ 𝐴)

22-Jan-2025	subsfn 27366	Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ -_s Fn ( No × No )

22-Jan-2025	adds32d 27358	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) +_s 𝐶) = ((𝐴 +_s 𝐶) +_s 𝐵))

22-Jan-2025	addsassd 27357	Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) +_s 𝐶) = (𝐴 +_s (𝐵 +_s 𝐶)))

22-Jan-2025	addsass 27356	Surreal addition is associative. Part of theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 22-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +_s 𝐵) +_s 𝐶) = (𝐴 +_s (𝐵 +_s 𝐶)))

22-Jan-2025	ceqsexv 3495	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

22-Jan-2025	ceqsex 3493	Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
⊢ Ⅎ𝑥𝜓 & ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)

21-Jan-2025	ressply1mon1p 32356	The monic polynomials of a restricted polynomial algebra. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ 𝑆 = (Poly₁‘𝑅) & ⊢ 𝐻 = (𝑅 ↾_s 𝑇) & ⊢ 𝑈 = (Poly₁‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑀 = (Monic_1p‘𝑅) & ⊢ 𝑁 = (Monic_1p‘𝐻) ⇒ ⊢ (𝜑 → 𝑁 = (𝐵 ∩ 𝑀))

21-Jan-2025	evls1scafv 32345	Value of the univariate polynomial evaluation for scalars. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝑋 ∈ 𝑅) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐶) = 𝑋)

21-Jan-2025	13an22anass 31458	Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

21-Jan-2025	addsasslem2 27355	Lemma for addition associativity. Expand the other form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 +_s (𝐵 +_s 𝐶)) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +_s (𝐵 +_s 𝐶))} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +_s (𝑚 +_s 𝐶))}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = (𝐴 +_s (𝐵 +_s 𝑛))}) \|s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = (𝑝 +_s (𝐵 +_s 𝐶))} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = (𝐴 +_s (𝑞 +_s 𝐶))}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = (𝐴 +_s (𝐵 +_s 𝑟))})))

21-Jan-2025	addsasslem1 27354	Lemma for addition associativity. Expand one form of the triple sum. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) +_s 𝐶) = ((({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = ((𝑙 +_s 𝐵) +_s 𝐶)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = ((𝐴 +_s 𝑚) +_s 𝐶)}) ∪ {𝑤 ∣ ∃𝑛 ∈ ( L ‘𝐶)𝑤 = ((𝐴 +_s 𝐵) +_s 𝑛)}) \|s (({𝑎 ∣ ∃𝑝 ∈ ( R ‘𝐴)𝑎 = ((𝑝 +_s 𝐵) +_s 𝐶)} ∪ {𝑏 ∣ ∃𝑞 ∈ ( R ‘𝐵)𝑏 = ((𝐴 +_s 𝑞) +_s 𝐶)}) ∪ {𝑐 ∣ ∃𝑟 ∈ ( R ‘𝐶)𝑐 = ((𝐴 +_s 𝐵) +_s 𝑟)})))

21-Jan-2025	addsunif 27353	Uniformity theorem for surreal addition. This theorem states that we can use any cuts that define 𝐴 and 𝐵 in the definition of surreal addition. Theorem 3.2 of [Gonshor] p. 15. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐿 <<s 𝑅) & ⊢ (𝜑 → 𝑀 <<s 𝑆) & ⊢ (𝜑 → 𝐴 = (𝐿 \|s 𝑅)) & ⊢ (𝜑 → 𝐵 = (𝑀 \|s 𝑆)) ⇒ ⊢ (𝜑 → (𝐴 +_s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +_s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +_s 𝑚)}) \|s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +_s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +_s 𝑠)})))

21-Jan-2025	sltadd2d 27349	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 +_s 𝐴) <s (𝐶 +_s 𝐵)))

21-Jan-2025	sleadd1d 27347	Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 +_s 𝐶) ≤s (𝐵 +_s 𝐶)))

21-Jan-2025	addscan1 27346	Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +_s 𝐴) = (𝐶 +_s 𝐵) ↔ 𝐴 = 𝐵))

21-Jan-2025	addscan2 27345	Cancellation law for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +_s 𝐶) = (𝐵 +_s 𝐶) ↔ 𝐴 = 𝐵))

21-Jan-2025	sltadd1 27344	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐴 +_s 𝐶) <s (𝐵 +_s 𝐶)))

21-Jan-2025	sltadd2 27343	Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐶 +_s 𝐴) <s (𝐶 +_s 𝐵)))

21-Jan-2025	sleadd2 27342	Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐶 +_s 𝐴) ≤s (𝐶 +_s 𝐵)))

21-Jan-2025	sleadd1 27341	Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 +_s 𝐶) ≤s (𝐵 +_s 𝐶)))

21-Jan-2025	sleadd2im 27340	Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +_s 𝐴) ≤s (𝐶 +_s 𝐵) → 𝐴 ≤s 𝐵))

21-Jan-2025	sleadd1im 27339	Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +_s 𝐶) ≤s (𝐵 +_s 𝐶) → 𝐴 ≤s 𝐵))

21-Jan-2025	sltadd2im 27338	Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐶 +_s 𝐴) <s (𝐶 +_s 𝐵)))

21-Jan-2025	sltadd1im 27337	Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐴 +_s 𝐶) <s (𝐵 +_s 𝐶)))

21-Jan-2025	addsfo 27336	Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ +_s :( No × No )–onto→ No

21-Jan-2025	addsf 27335	Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ +_s :( No × No )⟶ No

21-Jan-2025	addscl 27334	Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +_s 𝐵) ∈ No )

21-Jan-2025	addscld 27333	Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → (𝑋 +_s 𝑌) ∈ No )

21-Jan-2025	addscut 27332	Demonstrate the cut properties of surreal addition. This gives us closure together with a pair of set-less-than relationships for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → ((𝑋 +_s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s {(𝑋 +_s 𝑌)} ∧ {(𝑋 +_s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})))

21-Jan-2025	addsprop 27331	Inductively show that surreal addition is closed and compatible with less-than. This proof follows from induction on the birthdays of the surreal numbers involved. This pattern occurs throughout surreal development. Theorem 3.1 of [Gonshor] p. 14. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → ((𝑋 +_s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +_s 𝑋) <s (𝑍 +_s 𝑋))))

21-Jan-2025	addsproplem7 27330	Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) ⇒ ⊢ (𝜑 → (𝑌 +_s 𝑋) <s (𝑍 +_s 𝑋))

21-Jan-2025	addsproplem6 27329	Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) & ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍)) ⇒ ⊢ (𝜑 → (𝑌 +_s 𝑋) <s (𝑍 +_s 𝑋))

21-Jan-2025	addsproplem5 27328	Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) & ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌)) ⇒ ⊢ (𝜑 → (𝑌 +_s 𝑋) <s (𝑍 +_s 𝑋))

21-Jan-2025	addsproplem4 27327	Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → 𝑌 <s 𝑍) & ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍)) ⇒ ⊢ (𝜑 → (𝑌 +_s 𝑋) <s (𝑍 +_s 𝑋))

21-Jan-2025	addsproplem3 27326	Lemma for surreal addition properties. Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → ((𝑋 +_s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s {(𝑋 +_s 𝑌)} ∧ {(𝑋 +_s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})))

21-Jan-2025	addsproplem2 27325	Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝑋 ∈ No ) & ⊢ (𝜑 → 𝑌 ∈ No ) ⇒ ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))

21-Jan-2025	addsproplem1 27324	Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8350 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥))))) & ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (( bday ‘𝐴) +no ( bday ‘𝐶))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍)))) ⇒ ⊢ (𝜑 → ((𝐴 +_s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +_s 𝐴) <s (𝐶 +_s 𝐴))))

21-Jan-2025	addsval2 27318	The value of surreal addition with different choices for each bound variable. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +_s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +_s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +_s 𝑚)}) \|s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +_s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +_s 𝑠)})))

21-Jan-2025	naddword1 8642	Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +no 𝐵))

20-Jan-2025	ressply10g 32355	A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝑆 = (Poly₁‘𝑅) & ⊢ 𝐻 = (𝑅 ↾_s 𝑇) & ⊢ 𝑈 = (Poly₁‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) & ⊢ 𝑍 = (0_g‘𝑆) ⇒ ⊢ (𝜑 → 𝑍 = (0_g‘𝑈))

20-Jan-2025	ply1ascl0 32354	The zero scalar as a polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝑊 = (Poly₁‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑂 = (0_g‘𝑅) & ⊢ 0 = (0_g‘𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐴‘𝑂) = 0 )

20-Jan-2025	ressdeg1 32353	The degree of a univariate polynomial in a structure restriction. (Contributed by Thierry Arnoux, 20-Jan-2025.)
⊢ 𝐻 = (𝑅 ↾_s 𝑇) & ⊢ 𝐷 = ( deg₁ ‘𝑅) & ⊢ 𝑈 = (Poly₁‘𝐻) & ⊢ 𝐵 = (Base‘𝑈) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝐷‘𝑃) = (( deg₁ ‘𝐻)‘𝑃))

20-Jan-2025	nadd32 8648	Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ((𝐴 +no 𝐶) +no 𝐵))

20-Jan-2025	naddass 8647	Natural ordinal addition is associative. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))

20-Jan-2025	naddasslem2 8646	Lemma for naddass 8647. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no (𝐵 +no 𝐶)) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥)})

20-Jan-2025	naddasslem1 8645	Lemma for naddass 8647. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})

20-Jan-2025	naddunif 8644	Uniformity theorem for natural addition. If 𝐴 is the upper bound of 𝑋 and 𝐵 is the upper bound of 𝑌, then (𝐴 +no 𝐵) can be expressed in terms of 𝑋 and 𝑌. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) & ⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥}) & ⊢ (𝜑 → 𝐵 = ∩ {𝑦 ∈ On ∣ 𝑌 ⊆ 𝑦}) ⇒ ⊢ (𝜑 → (𝐴 +no 𝐵) = ∩ {𝑧 ∈ On ∣ (( +no “ (𝑋 × {𝐵})) ∪ ( +no “ ({𝐴} × 𝑌))) ⊆ 𝑧})

20-Jan-2025	naddf 8632	Function statement for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ +no :(On × On)⟶On

20-Jan-2025	naddov3 8631	Alternate expression for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) ⊆ 𝑥})

20-Jan-2025	cofonr 8625	Inverse cofinality law for ordinals. Contrast with cofcutr 27286 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥}) ⇒ ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)

20-Jan-2025	cofon2 8624	Cofinality theorem for ordinals. If 𝐴 and 𝐵 are mutually cofinal, then their upper bounds agree. Compare cofcut2 27284 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 On) & ⊢ (𝜑 → 𝐵 ∈ 𝒫 On) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) ⇒ ⊢ (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = ∩ {𝑏 ∈ On ∣ 𝐵 ⊆ 𝑏})

20-Jan-2025	cofon1 8623	Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and the upper bound of 𝐴 dominates 𝐵, then their upper bounds are equal. Compare with cofcut1 27282 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 On) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) & ⊢ (𝜑 → 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧}) ⇒ ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} = ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})

20-Jan-2025	coflton 8622	Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27280 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝜑 → 𝐴 ⊆ On) & ⊢ (𝜑 → 𝐵 ⊆ On) & ⊢ (𝜑 → 𝐶 ⊆ On) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤) ⇒ ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)

20-Jan-2025	imaeqalov 7598	Substitute an operation value into a universal quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓))

20-Jan-2025	imaeqexov 7597	Substitute an operation value into an existential quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.)
⊢ (𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓))

19-Jan-2025	onsupuni 41621	The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.)
⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)

19-Jan-2025	r1sssucd 41602	Deductive form of r1sssuc 9728. (Contributed by Noam Pasman, 19-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ On) ⇒ ⊢ (𝜑 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘suc 𝐴))

19-Jan-2025	bj-adjfrombun 35590	Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝑥 ∪ {𝑦}) ∈ V

19-Jan-2025	bj-prfromadj 35589	Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝑥, 𝑦} ∈ V

19-Jan-2025	bj-snfromadj 35588	Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝑥} ∈ V

19-Jan-2025	bj-adjg1 35587	Existence of the result of the adjunction (generalized only in the first term since this suffices for current applications). (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)

19-Jan-2025	ax-bj-adj 35586	Axiom of adjunction. (Contributed by BJ, 19-Jan-2025.)
⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))

19-Jan-2025	df-minply 32454	Define the minimal polynomial builder function. (Contributed by Thierry Arnoux, 19-Jan-2025.)
⊢ minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen_1p‘(𝑒 ↾_s 𝑓))‘{𝑝 ∈ dom (𝑒 evalSub₁ 𝑓) ∣ (((𝑒 evalSub₁ 𝑓)‘𝑝)‘𝑥) = (0_g‘𝑒)})))

18-Jan-2025	oaomoecl 41671	The operations of addition, multiplication, and exponentiation are closed. Remark 2.8 of [Schloeder] p. 5. See oacl 8486, omcl 8487, oecl 8488. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +_o 𝐵) ∈ On ∧ (𝐴 ·_o 𝐵) ∈ On ∧ (𝐴 ↑_o 𝐵) ∈ On))

18-Jan-2025	oe0suclim 41670	Closed form expression of the value of ordinal exponentiation for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.6 of [Schloeder] p. 4. See oe0 8473, oesuc 8478, oe0m1 8472, and oelim 8485. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ↑_o 𝐵) = 1_o) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ↑_o 𝐵) = ((𝐴 ↑_o 𝐶) ·_o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ↑_o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑_o 𝑐), ∅))))

18-Jan-2025	om0suclim 41669	Closed form expression of the value of ordinal multiplication for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.5 of [Schloeder] p. 4. See om0 8468, omsuc 8477, and omlim 8484. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ·_o 𝐵) = ∅) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ·_o 𝐵) = ((𝐴 ·_o 𝐶) +_o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ·_o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 ·_o 𝑐))))

18-Jan-2025	oa0suclim 41668	Closed form expression of the value of ordinal addition for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.3 of [Schloeder] p. 4. See oa0 8467, oasuc 8475, and oalim 8483. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +_o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +_o 𝐵) = suc (𝐴 +_o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +_o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +_o 𝑐))))

18-Jan-2025	onov0suclim 41667	Compactly express rules for binary operations on ordinals. (Contributed by RP, 18-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ⊗ ∅) = 𝐷) & ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊗ suc 𝐶) = 𝐸) & ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ⊗ 𝐵) = 𝐹) ⇒ ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))

18-Jan-2025	onsucf1o 41665	The successor operation is a bijective function between the ordinals and the class of succesor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

18-Jan-2025	onsucrn 41664	The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) ⇒ ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}

18-Jan-2025	onsucf1olem 41663	The successor operation is bijective between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)

18-Jan-2025	onsucf1lem 41662	For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the succesor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
⊢ (𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)

18-Jan-2025	bj-clex 35575	Two ways of stating that a class is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) ⇒ ⊢ (𝐴 ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))

18-Jan-2025	bj-abex 35574	Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
⊢ ({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))

17-Jan-2025	dflim7 41666	A limit ordinal is a non-zero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7789. (Contributed by RP, 17-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅))

17-Jan-2025	orddif0suc 41661	For any distinct pair of ordinals, if the set difference between the greater and the successor of the lesser is empty, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 17-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))

17-Jan-2025	prjspnn0 41018	A projective point is nonempty. (Contributed by SN, 17-Jan-2025.)
⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾) & ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) & ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)}) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐴 ≠ ∅)

17-Jan-2025	prjspnssbas 41017	A projective point spans a subset of the (nonzero) affine points. (Contributed by SN, 17-Jan-2025.)
⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾) & ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) & ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)}) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐾 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑃 ⊆ 𝒫 𝐵)

17-Jan-2025	flddrngd 40780	A field is a division ring. (Contributed by SN, 17-Jan-2025.)
⊢ (𝜑 → 𝑅 ∈ Field) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing)

17-Jan-2025	rictr 40769	Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝑅 ≃_𝑟 𝑆 ∧ 𝑆 ≃_𝑟 𝑇) → 𝑅 ≃_𝑟 𝑇)

17-Jan-2025	rimco 40766	The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 RingIso 𝑇))

17-Jan-2025	bj-elpwgALT 35598	Alternate proof of elpwg 4568. See comment for bj-velpwALT 35597. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

17-Jan-2025	bj-velpwALT 35597	This theorem bj-velpwALT 35597 and the next theorem bj-elpwgALT 35598 are alternate proofs of velpw 4570 and elpwg 4568 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3529 instead of proving first the general case using elab2g 3635 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2171. In other cases, that order is better (e.g., vsnex 5391 proved before snexg 5392). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)

17-Jan-2025	intidg 5419	The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) Put in closed form and avoid ax-nul 5268. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})

17-Jan-2025	snelpwg 5404	A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 5268. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))

16-Jan-2025	ordnexbtwnsuc 41660	For any distinct pair of ordinals, if there is no ordinal between the lesser and the greater, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))

16-Jan-2025	onsucss 41659	If one ordinal is less than another, then the successor of the first is less than or equal to the second. Lemma 1.13 of [Schloeder] p. 2. See ordsucss 7758. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))

16-Jan-2025	limnsuc 41658	A limit ordinal is not an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

16-Jan-2025	dflim6 41657	A limit ordinal is a non-zero ordinal which is not a succesor ordinal. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))

16-Jan-2025	onsucelab 41656	The successor of every ordinal is an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

16-Jan-2025	ondif1i 41655	Ordinal zero is less than every non-zero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8452. (Contributed by RP, 16-Jan-2025.)
⊢ (𝐴 ∈ (On ∖ 1_o) → ∅ ∈ 𝐴)

16-Jan-2025	ordne0gt0 41654	Ordinal zero is less than every non-zero ordinal. Theorem 1.10 of [Schloeder] p. 2. Closely related to ord0eln0 6377. (Contributed by RP, 16-Jan-2025.)
⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)

16-Jan-2025	ordeldif1o 41653	Membership in the difference of ordinal and ordinal one. (Contributed by RP, 16-Jan-2025.)
⊢ (Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1_o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅)))

16-Jan-2025	ordeldifsucon 41652	Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))

15-Jan-2025	ordeldif 41651	Membership in the difference of ordinals. (Contributed by RP, 15-Jan-2025.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))

15-Jan-2025	oneltri 41650	The elementhood relation on the ordinals is complete, so we have triality. Theorem 1.9(iii) of [Schloeder] p. 1. See ordtri3or 6354. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))

15-Jan-2025	oneptri 41649	The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵))

15-Jan-2025	oneltr 41648	The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6368. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

15-Jan-2025	oneptr 41647	The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))

15-Jan-2025	epirron 41646	The strict order on the ordinals is irreflexive. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ (𝐴 ∈ On → ¬ 𝐴 E 𝐴)

15-Jan-2025	epsoon 41645	The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7714 and weso 5629. (Contributed by RP, 15-Jan-2025.)
⊢ E Or On

15-Jan-2025	onepsuc 41644	Every ordinal is less than its successor, relationship version. Lemma 1.7 of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
⊢ (𝐴 ∈ On → 𝐴 E suc 𝐴)

15-Jan-2025	onelord 41643	Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6347 and eloni 6332. (Contributed by RP, 15-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

15-Jan-2025	1fldgenq 32160	The field of rational numbers ℚ is generated by 1 in ℂ_fld, that is, ℚ is the prime field of ℂ_fld. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ (ℂ_fld fldGen {1}) = ℚ

15-Jan-2025	fldgenidfld 32155	The subfield generated by a subfield is the subfield itself. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹)) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) = 𝑆)

15-Jan-2025	fldgenss 32154	Generated subfields preserve subset ordering. ( see lspss 20502 and spanss 30353) (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ (𝐹 fldGen 𝑆))

15-Jan-2025	fldgenssid 32151	The field generated by a set of elements contains those elements. See lspssid 20503. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆))

15-Jan-2025	sdrginvcl 32146	A sub-division-ring is closed under the ring inverse operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ 𝐼 = (inv_r‘𝑅) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubDRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐴)

15-Jan-2025	sdrgdvcl 32145	A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
⊢ / = (/_r‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴)

15-Jan-2025	cshwsexa 14724	The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof shortened by SN, 15-Jan-2025.)
⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V

15-Jan-2025	sels 5400	If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5402. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5261, ax-nul 5268, ax-pow 5325. (Revised by BTernaryTau, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)

15-Jan-2025	snexg 5392	A singleton built on a set is a set. Special case of snex 5393 which does not require ax-nul 5268 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5393 and shorten proof. (Revised by BJ, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

15-Jan-2025	vsnex 5391	A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
⊢ {𝑥} ∈ V

15-Jan-2025	iunid 5025	An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.)
⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴

15-Jan-2025	rabeqc 3417	A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.)
⊢ (𝑥 ∈ 𝐴 → 𝜑) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

14-Jan-2025	resisoeq45d 41814	Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)
⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

14-Jan-2025	isoeq145d 41813	Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))

14-Jan-2025	sdomne0 41807	A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.)
⊢ (𝐵 ≺ 𝐴 → 𝐴 ≠ ∅)

14-Jan-2025	omcl3g 41727	Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3_o ∨ (𝐶 = (ω ↑_o (ω ↑_o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·_o 𝐵) ∈ 𝐶)

13-Jan-2025	mbfmbfm 32947	A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)
⊢ (𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽))) ⇒ ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

13-Jan-2025	isanmbfm 32945	The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) ⇒ ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM)

13-Jan-2025	fvssunirn 6880	The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by SN, 13-Jan-2025.)
⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

13-Jan-2025	fvn0fvelrn 6878	If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by SN, 13-Jan-2025.)
⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)

12-Jan-2025	dfno2 41822	A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.)
⊢ No = {𝑓 ∈ 𝒫 (On × {1_o, 2_o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}

12-Jan-2025	omcl2 41726	Closure law for ordinal multiplication. (Contributed by RP, 12-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑_o (ω ↑_o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·_o 𝐵) ∈ 𝐶)

12-Jan-2025	omabs2 41725	Ordinal multiplication by a larger ordinal is absorbed when the larger ordinal is either 2 or ω raised to some power of ω. (Contributed by RP, 12-Jan-2025.)
⊢ (((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ 𝐵 = 2_o ∨ (𝐵 = (ω ↑_o (ω ↑_o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·_o 𝐵) = 𝐵)

12-Jan-2025	bj-axadj 35585	Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 35586). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))

12-Jan-2025	bj-prex 35584	Existence of unordered pairs proved from ax-bj-sn 35577 and ax-bj-bun 35581. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝐴, 𝐵} ∈ V

12-Jan-2025	bj-prexg 35583	Existence of unordered pairs formed on sets, proved from ax-bj-sn 35577 and ax-bj-bun 35581. Contrary to bj-prex 35584, this proof is intuitionistically valid and does not require ax-nul 5268. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

12-Jan-2025	bj-unexg 35582	Existence of binary unions of sets, proved from ax-bj-bun 35581. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)

12-Jan-2025	ax-bj-bun 35581	Axiom of binary union. (Contributed by BJ, 12-Jan-2025.)
⊢ ∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))

12-Jan-2025	bj-axbun 35580	Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 35581). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))

12-Jan-2025	bj-snex 35579	A singleton is a set. See also snex 5393, snexALT 5343. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 35577. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ {𝐴} ∈ V

12-Jan-2025	bj-snexg 35578	A singleton built on a set is a set. Contrary to bj-snex 35579, this proof is intuitionistically valid and does not require ax-nul 5268. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5393 and prove it from ax-bj-sn 35577. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

12-Jan-2025	ax-bj-sn 35577	Axiom of singleton. (Contributed by BJ, 12-Jan-2025.)
⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)

12-Jan-2025	bj-axsn 35576	Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 35577). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))

12-Jan-2025	isrim 20181	An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))

12-Jan-2025	rabbiia 3409	Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}

12-Jan-2025	reuanid 3362	Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
⊢ (∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑)

12-Jan-2025	rmoanid 3361	Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑)

11-Jan-2025	primefldgen1 32159	The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 }))

11-Jan-2025	fldgenfld 32158	A generated subfield is a field. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ Field) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ↾_s (𝐹 fldGen 𝑆)) ∈ Field)

11-Jan-2025	fldgenid 32157	The subfield of a field 𝐹 generated by the whole base set of 𝐹 is 𝐹 itself. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝐵) = 𝐵)

11-Jan-2025	fldgensdrg 32152	A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))

11-Jan-2025	fldgenval 32150	Value of the field generating function: (𝐹 fldGen 𝑆) is the smallest sub-division-ring of 𝐹 containing 𝑆. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})

11-Jan-2025	resrng 21062	The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) (Proof shortened by Thierry Arnoux, 11-Jan-2025.)
⊢ ℝ_fld ∈ *-Ring

11-Jan-2025	fldsdrgfld 20321	A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ ((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRing‘𝐹)) → (𝐹 ↾_s 𝐴) ∈ Field)

11-Jan-2025	sucdom 9186	Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-pow 5325. (Revised by BTernaryTau, 4-Dec-2024.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ (𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵))

11-Jan-2025	onuniorsuc 7777	An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.) Put in closed form. (Revised by BJ, 11-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))

11-Jan-2025	sucexeloni 7749	If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7751 does not require ax-un 7677. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

11-Jan-2025	ordsuci 7748	The successor of an ordinal class is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) Extract and adapt from a subproof of onsuc 7751. (Revised by BTernaryTau, 6-Jan-2025.) (Proof shortened by BJ, 11-Jan-2025.)
⊢ (Ord 𝐴 → Ord suc 𝐴)

10-Jan-2025	riccrng 40771	A ring is commutative if and only if an isomorphic ring is commutative. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃_𝑟 𝑆 → (𝑅 ∈ CRing ↔ 𝑆 ∈ CRing))

10-Jan-2025	riccrng1 40770	Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.)
⊢ ((𝑅 ≃_𝑟 𝑆 ∧ 𝑅 ∈ CRing) → 𝑆 ∈ CRing)

10-Jan-2025	ricsym 40768	Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃_𝑟 𝑆 → 𝑆 ≃_𝑟 𝑅)

10-Jan-2025	brrici 40767	Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃_𝑟 𝑆)

10-Jan-2025	rimcnv 40765	The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → ^◡𝐹 ∈ (𝑆 RingIso 𝑅))

10-Jan-2025	imacrhmcl 40762	The image of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)
⊢ 𝐶 = (𝑁 ↾_s (𝐹 “ 𝑆)) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → 𝑀 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀)) ⇒ ⊢ (𝜑 → 𝐶 ∈ CRing)

10-Jan-2025	rimrhm 20185	A ring isomorphism is a homomorphism. Compare gimghm 19068. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))

10-Jan-2025	isrim0 20172	A ring isomorphism is a homomorphism whose converse is also a homomorphism. Compare isgim2 19069. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ^◡𝐹 ∈ (𝑆 RingHom 𝑅)))

10-Jan-2025	ressbasss2 17135	The base set of a restriction to 𝐴 is a subset of 𝐴. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾_s 𝐴) ⇒ ⊢ (Base‘𝑅) ⊆ 𝐴

10-Jan-2025	ressbasssg 17131	The base set of a restriction to 𝐴 is a subset of 𝐴 and the base set 𝐵 of the original structure. (Contributed by SN, 10-Jan-2025.)
⊢ 𝑅 = (𝑊 ↾_s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)

10-Jan-2025	xpfi 9268	The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5325. (Revised by BTernaryTau, 10-Jan-2025.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

10-Jan-2025	elfvunirn 6879	A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6884). (Contributed by Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised by SN, 10-Jan-2025.)
⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)

9-Jan-2025	ply1fermltlchr 32361	Fermat's little theorem for polynomials in a commutative ring 𝐹 of characteristic 𝑃 prime: we have the polynomial equation (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴). (Contributed by Thierry Arnoux, 9-Jan-2025.)
⊢ 𝑊 = (Poly₁‘𝐹) & ⊢ 𝑋 = (var₁‘𝐹) & ⊢ + = (+_g‘𝑊) & ⊢ 𝑁 = (mulGrp‘𝑊) & ⊢ ↑ = (._g‘𝑁) & ⊢ 𝐶 = (algSc‘𝑊) & ⊢ 𝐴 = (𝐶‘((ℤRHom‘𝐹)‘𝐸)) & ⊢ 𝑃 = (chr‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ CRing) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝐸 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))

9-Jan-2025	df-fldgen 32149	Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 32152). If the base structure is a field, this is a subfield (see fldgenfld 32158 and fldsdrgfld 20321). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.)
⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})

8-Jan-2025	dflim5 41722	A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)
⊢ (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1_o)𝐴 = (ω ·_o 𝑥)))

8-Jan-2025	succlg 41721	Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·_o 𝐶) ∧ 𝐶 ∈ (On ∖ 1_o)))) → suc 𝐴 ∈ 𝐵)

8-Jan-2025	omlimcl2 41634	The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·_o 𝐴))

8-Jan-2025	infn0 9258	An infinite set is not empty. For a shorter proof using ax-un 7677, see infn0ALT 9259. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7677. (Revised by BTernaryTau, 8-Jan-2025.)
⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)

8-Jan-2025	cbvrexdva2 3322	Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 8-Jan-2025.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))

7-Jan-2025	nnawordexg 41720	If an ordinal, 𝐵, is in a half-open interval between some 𝐴 and the next limit ordinal, 𝐵 is the sum of the 𝐴 and some natural number. This weakens the antecedent of nnawordex 8589. (Contributed by RP, 7-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +_o ω)) → ∃𝑥 ∈ ω (𝐴 +_o 𝑥) = 𝐵)

7-Jan-2025	oawordex2 41719	If 𝐶 is between 𝐴 (inclusive) and (𝐴 +_o 𝐵) (exclusive), there is an ordinal which equals 𝐶 when summed to 𝐴. This is a slightly different statement than oawordex 8509 or oawordeu 8507. (Contributed by RP, 7-Jan-2025.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +_o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +_o 𝑥) = 𝐶)

7-Jan-2025	aks6d1c2p2 40622	Injective condition for countability argument assuming that 𝑁 is not a prime power. (Contributed by metakunt, 7-Jan-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) & ⊢ (𝜑 → 𝑄 ∈ ℙ) & ⊢ (𝜑 → 𝑄 ∥ 𝑁) & ⊢ (𝜑 → 𝑃 ≠ 𝑄) ⇒ ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)–1-1→ℕ)

7-Jan-2025	aks6d1c2p1 40621	In the AKS-theorem the subset defined by 𝐸 takes values in the positive integers. (Contributed by metakunt, 7-Jan-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) & ⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) ⇒ ⊢ (𝜑 → 𝐸:(ℕ₀ × ℕ₀)⟶ℕ)

7-Jan-2025	fldhmf1 40620	A field homomorphism is injective. This follows immediately from the definition of the ring homomorphism that sends the multiplicative identity to the multiplicative identity. (Contributed by metakunt, 7-Jan-2025.)
⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝐿 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (𝐾 RingHom 𝐿)) & ⊢ 𝐴 = (Base‘𝐾) & ⊢ 𝐵 = (Base‘𝐿) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)

7-Jan-2025	infsdomnn 9256	An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5325. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴)

7-Jan-2025	nnsdomg 9253	Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of Infinity, we include it as part of the antecedent. See nnsdom 9599 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ ((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)

7-Jan-2025	findcard3 9236	Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.) Avoid ax-pow 5325. (Revised by BTernaryTau, 7-Jan-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ (𝑦 ∈ Fin → (∀𝑥(𝑥 ⊊ 𝑦 → 𝜑) → 𝜒)) ⇒ ⊢ (𝐴 ∈ Fin → 𝜏)

6-Jan-2025	enp1i 9230	Proof induction for en2 9232 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ Ord 𝑀 & ⊢ 𝑁 = suc 𝑀 & ⊢ ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑) & ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ⇒ ⊢ (𝐴 ≈ 𝑁 → ∃𝑥𝜓)

6-Jan-2025	dif1ennn 9112	If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. See also dif1ennnALT 9228. (Contributed by BTernaryTau, 6-Jan-2025.)
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

6-Jan-2025	dif1en 9111	If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5325. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

6-Jan-2025	ord3 8434	Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.)
⊢ Ord 3_o

6-Jan-2025	ordsuc 7753	A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7677. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

5-Jan-2025	et-sqrtnegnre 45234	The square root of a negative number is not a real number. (Contributed by Ender Ting, 5-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ¬ (√‘𝐴) ∈ ℝ)

5-Jan-2025	smfdivdmmbl2 45202	If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator. It is required only for the function at the denominator. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑉) & ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) & ⊢ 𝐷 = {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0} & ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ 𝐷) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

5-Jan-2025	smfpimne2 45201	Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value is in the subspace sigma-algebra induced by its domain. Notice that 𝐴 is not assumed to be an extended real. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≠ 𝐴} ∈ (𝑆 ↾_t 𝐷))

5-Jan-2025	smfpimne 45200	Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value in the extended reals is in the subspace of sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≠ 𝐴} ∈ (𝑆 ↾_t 𝐷))

5-Jan-2025	smfdivdmmbl 45199	If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator (it is needed only for the function at the denominator). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ 𝑊) & ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ (SMblFn‘𝑆)) & ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ 𝐷 ≠ 0} ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐸) ∈ 𝑆)

5-Jan-2025	smfdmmblpimne 45198	If a measurable function w.r.t. to a sigma-algebra has domain in the sigma-algebra, the set of elements that are not mapped to a given real, is in the sigma-algebra (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} ⇒ ⊢ (𝜑 → 𝐷 ∈ 𝑆)

5-Jan-2025	smffmptf 45165	A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)

5-Jan-2025	fmptdff 43621	A version of fmptd 7067 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

5-Jan-2025	fvmptelcdmf 43620	The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)

5-Jan-2025	fmptff 43619	Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

5-Jan-2025	dmmpt1 43618	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐵 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) = 𝐵)

5-Jan-2025	ssrabdf 43447	Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓) ⇒ ⊢ (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})

5-Jan-2025	ofoacom 41754	Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑_m 𝐴) ∧ 𝐺 ∈ (ω ↑_m 𝐴))) → (𝐹 ∘_f +_o 𝐺) = (𝐺 ∘_f +_o 𝐹))

5-Jan-2025	ofoaass 41753	Component-wise addition of ordinal-yielding functions is associative. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑_m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑_m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑_m 𝐴))) → ((𝐹 ∘_f +_o 𝐺) ∘_f +_o 𝐻) = (𝐹 ∘_f +_o (𝐺 ∘_f +_o 𝐻)))

5-Jan-2025	ofoaid2 41752	Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑_m 𝐴)) → ((𝐴 × {∅}) ∘_f +_o 𝐹) = 𝐹)

5-Jan-2025	ofoaid1 41751	Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑_m 𝐴)) → (𝐹 ∘_f +_o (𝐴 × {∅})) = 𝐹)

5-Jan-2025	ofoacl 41750	Closure law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑_o 𝐵))) ∧ (𝐹 ∈ (𝐶 ↑_m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑_m 𝐴))) → (𝐹 ∘_f +_o 𝐺) ∈ (𝐶 ↑_m 𝐴))

5-Jan-2025	ofoafo 41749	Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑_o 𝐵))) → ( ∘_f +_o ↾ ((𝐶 ↑_m 𝐴) × (𝐶 ↑_m 𝐴))):((𝐶 ↑_m 𝐴) × (𝐶 ↑_m 𝐴))–onto→(𝐶 ↑_m 𝐴))

5-Jan-2025	ofoaf 41748	Addition operator for functions from sets into power of omega results in a function from the intersection of sets to that power of omega. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑_o 𝐷))) → ( ∘_f +_o ↾ ((𝐸 ↑_m 𝐴) × (𝐸 ↑_m 𝐵))):((𝐸 ↑_m 𝐴) × (𝐸 ↑_m 𝐵))⟶(𝐸 ↑_m 𝐶))

5-Jan-2025	ofoafg 41747	Addition operator for functions from sets into ordinals results in a function from the intersection of sets into an ordinal. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +_o 𝐸))) → ( ∘_f +_o ↾ ((𝐷 ↑_m 𝐴) × (𝐸 ↑_m 𝐵))):((𝐷 ↑_m 𝐴) × (𝐸 ↑_m 𝐵))⟶(𝐹 ↑_m 𝐶))

5-Jan-2025	oacl2g 41723	Closure law for ordinal addition. Here we show that ordinal addition is closed within the empty set or any ordinal power of omega. (Contributed by RP, 5-Jan-2025.)
⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑_o 𝐷) ∧ 𝐷 ∈ On))) → (𝐴 +_o 𝐵) ∈ 𝐶)

5-Jan-2025	rexdif1en 9109	If a set is equinumerous to a nonzero ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals and avoid ax-un 7677. (Revised by BTernaryTau, 5-Jan-2025.)
⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)

5-Jan-2025	dif1enlem 9107	Lemma for rexdif1en 9109 and dif1en 9111. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7677. (Revised by BTernaryTau, 5-Jan-2025.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(^◡𝐹‘𝑀)}) ≈ 𝑀)

5-Jan-2025	imbibi 392	The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.)
⊢ (((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))

4-Jan-2025	en1eqsn 9225	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 4-Jan-2025.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1_o) → 𝐵 = {𝐴})

4-Jan-2025	f1finf1o 9222	Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) Avoid ax-pow 5325. (Revised by BTernaryTau, 4-Jan-2025.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))

3-Jan-2025	omltoe 41801	Exponentiation eventually dominates multiplication. (Contributed by RP, 3-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1_o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 ·_o 𝐴) ∈ (𝐵 ↑_o 𝐴)))

3-Jan-2025	oe2 41800	Two ways to square an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 ·_o 𝐴) = (𝐴 ↑_o 2_o))

3-Jan-2025	oaltom 41799	Multiplication eventually dominates addition. (Contributed by RP, 3-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1_o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 +_o 𝐴) ∈ (𝐵 ·_o 𝐴)))

3-Jan-2025	om2 41798	Two ways to double an ordinal. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∈ On → (𝐴 +_o 𝐴) = (𝐴 ·_o 2_o))

3-Jan-2025	insucid 41797	The intersection of a class and its successor is itself. (Contributed by RP, 3-Jan-2025.)
⊢ (𝐴 ∩ suc 𝐴) = 𝐴

3-Jan-2025	ordsssucim 41796	If an ordinal is less than or equal to the successor of another, then the first is either less than or equal to the second or the first is equal to the successor of the second. Theorem 1 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 See also ordsssucb 41728 for a biimplication when 𝐴 is a set. (Contributed by RP, 3-Jan-2025.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

3-Jan-2025	naddcnfass 41762	Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘_f +_o 𝐺) ∘_f +_o 𝐻) = (𝐹 ∘_f +_o (𝐺 ∘_f +_o 𝐻)))

3-Jan-2025	naddcnfid2 41761	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘_f +_o 𝐹) = 𝐹)

3-Jan-2025	naddcnfid1 41760	Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘_f +_o (𝑋 × {∅})) = 𝐹)

3-Jan-2025	glbconN 37912	De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) New df-riota 7318. (Revised by SN, 3-Jan-2025.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑈 = (lub‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ⊥ = (oc‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))

3-Jan-2025	nfra2w 3280	Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 43264. Version of nfra2 3347 with a disjoint variable condition not requiring ax-13 2370. (Contributed by Alan Sare, 31-Dec-2011.) Reduce axiom usage. (Revised by Gino Giotto, 24-Sep-2024.) (Proof shortened by Wolf Lammen, 3-Jan-2025.)
⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

2-Jan-2025	naddcnfcom 41759	Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘_f +_o 𝐺) = (𝐺 ∘_f +_o 𝐹))

2-Jan-2025	naddcnfcl 41758	Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ (((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘_f +_o 𝐺) ∈ 𝑆)

2-Jan-2025	naddcnffo 41757	Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘_f +_o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)

2-Jan-2025	naddcnffn 41756	Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘_f +_o ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))

2-Jan-2025	naddcnff 41755	Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
⊢ ((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘_f +_o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)

2-Jan-2025	isinf 9211	Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.) Avoid ax-pow 5325. (Revised by BTernaryTau, 2-Jan-2025.)
⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

2-Jan-2025	ominf 9209	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 2-Jan-2025.)
⊢ ¬ ω ∈ Fin

2-Jan-2025	dtru 5398	Given any set (the "𝑦 " in the statement), not all sets are equal to it. The same statement without disjoint variable condition is false since it contradicts stdpc6 2031. The same comments and revision history concerning axiom usage as in exneq 5397 apply. (Contributed by NM, 7-Nov-2006.) Extract exneq 5397 as an intermediate result. (Revised by BJ, 2-Jan-2025.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦

2-Jan-2025	exneq 5397	Given any set (the "𝑦 " in the statement), there exists a set not equal to it. The same statement without disjoint variable condition is false, since we do not have ∃𝑥¬ 𝑥 = 𝑥. This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext 2702, ax-sep 5261, or ax-pow 5325 nor auxiliary logical axiom schemes ax-10 2137 to ax-13 2370. See dtruALT 5348 for a shorter proof using more axioms, and dtruALT2 5330 for a proof using ax-pow 5325 instead of ax-pr 5389. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-8 2108. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5389 instead of ax-pow 5325. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5398. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥 ¬ 𝑥 = 𝑦

2-Jan-2025	exexneq 5396	There exist two different sets. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-8 2108. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5389 instead of ax-pow 5325. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5398. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦

2-Jan-2025	ralcom13 3275	Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.) (Proof shortened by Wolf Lammen, 2-Jan-2025.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)

1-Jan-2025	naddonnn 41789	Natural addition with a natural number on the right results in a value equal to that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +_o 𝐵) = (𝐴 +no 𝐵))

1-Jan-2025	naddgeoa 41788	Natural addition results in a value greater than or equal than that of ordinal addition. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +_o 𝐵) ⊆ (𝐴 +no 𝐵))

1-Jan-2025	naddass1 41787	Natural addition of ordinal numbers is associative when the third element is 1. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) +no 1_o) = (𝐴 +no (𝐵 +no 1_o)))

1-Jan-2025	naddsuc2 41786	Natural addition with successor. (Contributed by RP, 1-Jan-2025.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no suc 𝐵) = suc (𝐴 +no 𝐵))

1-Jan-2025	snssg 4749	The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

1-Jan-2025	snssb 4748	Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))

31-Dec-2024	nadd1suc 41785	Natural addition with 1 is same as successor. (Contributed by RP, 31-Dec-2024.)
⊢ (𝐴 ∈ On → (𝐴 +no 1_o) = suc 𝐴)

31-Dec-2024	mpets 37377	Member Partition-Equivalence Theorem in its shortest possible form: it shows that member partitions and comember equivalence relations are literally the same. Cf. pet 37386, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ MembParts = CoMembErs

31-Dec-2024	cpet 37373	The conventional form of Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have been calling disjoint or partition what we call element disjoint or member partition, see also cpet2 37372. Cf. mpet 37374, mpet2 37375 and mpet3 37371 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 37386 and pet2 37385 for Partition-Equivalence Theorem with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ ( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

31-Dec-2024	eqvrelcossid 37329	The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ I

31-Dec-2024	eqvrelcoss0 37323	The cosets by the null class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ EqvRel ≀ ∅

31-Dec-2024	eldisjn0elb 37280	Two forms of disjoint elements when the empty set is not an element of the class. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (^◡ E ↾ 𝐴) ∧ (dom (^◡ E ↾ 𝐴) / (^◡ E ↾ 𝐴)) = 𝐴))

31-Dec-2024	suceqsneq 36767	One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))

30-Dec-2024	muldmmbl2 45197	If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → dom 𝐹 ∈ 𝑆) & ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) & ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

30-Dec-2024	muldmmbl 45196	If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)

30-Dec-2024	adddmmbl2 45195	If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → dom 𝐹 ∈ 𝑆) & ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) & ⊢ 𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → dom 𝐻 ∈ 𝑆)

30-Dec-2024	adddmmbl 45194	If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) ∈ 𝑆)

30-Dec-2024	cpet2 37372	The conventional form of the Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have called disjoint or partition what we call element disjoint or member partition, see also cpet 37373. Together with cpet 37373, mpet 37374 mpet2 37375, this is what we used to think of as the partition equivalence theorem (but cf. pet2 37385 with general 𝑅). (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

30-Dec-2024	fences3 37365	Implication of eqvrelqseqdisj2 37364 and n0eldmqseq 37184, see comment of fences 37379. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

30-Dec-2024	eldisjim2 37320	Alternate form of eldisjim 37319. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ ( ElDisj 𝐴 → EqvRel ∼ 𝐴)

30-Dec-2024	eqvreldmqs2 37211	Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (( EqvRel ≀ (^◡ E ↾ 𝐴) ∧ (dom ≀ (^◡ E ↾ 𝐴) / ≀ (^◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

30-Dec-2024	n0elim 37185	Implication of that the empty set is not an element of a class. (Contributed by Peter Mazsa, 30-Dec-2024.)
⊢ (¬ ∅ ∈ 𝐴 → (dom (^◡ E ↾ 𝐴) / (^◡ E ↾ 𝐴)) = 𝐴)

30-Dec-2024	pr2ne 9949	If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2_o ↔ 𝐴 ≠ 𝐵))

30-Dec-2024	enpr2 9947	An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9000. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2_o)

30-Dec-2024	1sdom 9199	A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8981.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7677. (Revised by BTernaryTau, 30-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (1_o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

30-Dec-2024	rex2dom 9197	A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2_o ≼ 𝐴)

29-Dec-2024	dffun2 6511	Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

29-Dec-2024	cnvsym 6071	Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

29-Dec-2024	cotrg 6066	Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6069 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6069. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

28-Dec-2024	onunisuc 6432	An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6442. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)

28-Dec-2024	unisucg 6400	A transitive class is equal to the union of its successor, closed form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) Generalize from unisuc 6401. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

28-Dec-2024	unisucs 6399	The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6401. (Revised by BJ, 28-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

28-Dec-2024	dftr5 5231	An alternate way of defining a transitive class. Definition 1.1 of [Schloeder] p. 1. (Contributed by NM, 20-Mar-2004.) Avoid ax-11 2154. (Revised by BTernaryTau, 28-Dec-2024.)
⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)

28-Dec-2024	dftr2c 5230	Variant of dftr2 5229 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2154. (Contributed by BTernaryTau, 28-Dec-2024.)
⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))

28-Dec-2024	unissb 4905	Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) Avoid ax-11 2154. (Revised by BTernaryTau, 28-Dec-2024.)
⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

28-Dec-2024	alcomw 2047	Weak version of alcom 2156 and biconditional form of alcomiw 2046. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024.)
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)

25-Dec-2024	partimeq 37344	Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 37214. (Contributed by Peter Mazsa, 25-Dec-2024.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))

23-Dec-2024	rmxyelqirr 41291	The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.)
⊢ ((𝐴 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ₀ ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

23-Dec-2024	1sdom2dom 9198	Strict dominance over 1 is the same as dominance over 2. (Contributed by BTernaryTau, 23-Dec-2024.)
⊢ (1_o ≺ 𝐴 ↔ 2_o ≼ 𝐴)

23-Dec-2024	enpr2d 9000	A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-un 7677. (Revised by BTernaryTau, 23-Dec-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2_o)

23-Dec-2024	en2prd 8999	Two unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐶 ≠ 𝐷) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})

23-Dec-2024	tz6.12-1 6870	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)

23-Dec-2024	tz6.12c 6869	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

23-Dec-2024	relssdmrn 6225	A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) (Proof shortened by SN, 23-Dec-2024.)
⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))

23-Dec-2024	cnvsymOLD 6072	Obsolete proof of cnvsym 6071 as of 29-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (^◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))

23-Dec-2024	exel 5395	There exist two sets, one a member of the other. This theorem looks similar to el 5399, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1811, ax-6 1971, and ax-pr 5389. This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq 5396. (Contributed by SN, 23-Dec-2024.)
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦

23-Dec-2024	rexlimivw 3144	Weaker version of rexlimiv 3141. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2024.)
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

23-Dec-2024	rexlimiva 3140	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.) Shorten dependent theorems. (Revised by Wolf lammen, 23-Dec-2024.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ⇒ ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

22-Dec-2024	ssabdv 40713	Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ 𝜓})

22-Dec-2024	ss2ab1 40712	Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4021 using fewer axioms. (Contributed by SN, 22-Dec-2024.)
⊢ (∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓})

22-Dec-2024	mainpart 37378	Partition with general 𝑅 also imply member partition. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
⊢ (𝑅 Part 𝐴 → MembPart 𝐴)

22-Dec-2024	partimcomember 37370	Partition with general 𝑅 (in addition to the member partition cf. mpet 37374 and mpet2 37375) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴)

22-Dec-2024	abssdv 4030	Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) (Proof shortened by SN, 22-Dec-2024.)
⊢ (𝜑 → (𝜓 → 𝑥 ∈ 𝐴)) ⇒ ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)

22-Dec-2024	r19.29 3113	Restricted quantifier version of 19.29 1876. See also r19.29r 3115. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

22-Dec-2024	r19.35 3107	Restricted quantifier version of 19.35 1880. (Contributed by NM, 20-Sep-2003.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

22-Dec-2024	ralcom3 3096	A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))

21-Dec-2024	et-equeucl 45233	Alternative proof that equality is left-Euclidean, using ax7 2019 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.)
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))

21-Dec-2024	salrestss 44722	A sigma-algebra restricted to one of its elements is a subset of the original sigma-algebra. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐸 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑆 ↾_t 𝐸) ⊆ 𝑆)

21-Dec-2024	pimxrneun 43844	The preimage of a set of extended reals that does not contain a value 𝐶 is the union of the preimage of the elements smaller than 𝐶 and the preimage of the subset of elements larger than 𝐶. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ^) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))

21-Dec-2024	mpteq2dfa 43617	Slightly more general equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

21-Dec-2024	dmmptif 43616	Domain of the mapping operation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝐴 & ⊢ 𝐵 ∈ V & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ dom 𝐹 = 𝐴

21-Dec-2024	fnmptif 43615	Functionality and domain of an ordered-pair class abstraction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝐴 & ⊢ 𝐵 ∈ V & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ 𝐹 Fn 𝐴

21-Dec-2024	dmmptdff 43565	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐵 & ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → dom 𝐴 = 𝐵)

21-Dec-2024	toprestsubel 43491	A subset is open in the topology it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾_t 𝐴))

21-Dec-2024	restsubel 43490	A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ 𝑉) & ⊢ (𝜑 → ∪ 𝐽 ∈ 𝐽) & ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐽) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝐽 ↾_t 𝐴))

21-Dec-2024	restopnssd 43489	A topology restricted to an open set is a subset of the original topology. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) ⇒ ⊢ (𝜑 → (𝐽 ↾_t 𝐴) ⊆ 𝐽)

21-Dec-2024	restopn3 43488	If 𝐴 is open, then 𝐴 is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ (𝐽 ↾_t 𝐴))

21-Dec-2024	ss2rabdf 43487	Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

21-Dec-2024	inopnd 43486	The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ (𝜑 → 𝐴 ∈ 𝐽) & ⊢ (𝜑 → 𝐵 ∈ 𝐽) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽)

20-Dec-2024	smfpimgtxrmpt 45146	Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾_t 𝐴))

20-Dec-2024	smfpimgtxrmptf 45145	Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾_t 𝐴))

20-Dec-2024	smfpimltxrmpt 45120	Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾_t 𝐴))

20-Dec-2024	smfpimltxrmptf 45119	Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ (𝜑 → 𝑅 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾_t 𝐴))

20-Dec-2024	pimgtmnf 45084	Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)

20-Dec-2024	pimgtmnff 45083	Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)

20-Dec-2024	pimltpnf 45065	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)

20-Dec-2024	pimltpnff 45064	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)

20-Dec-2024	nadd1rabon 41784	The set of ordinals which have a natural sum less than some ordinal is an ordinal number. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶} ∈ On)

20-Dec-2024	nadd1rabex 41783	The class of ordinals which have a natural sum less than some ordinal is a set. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶} ∈ V)

20-Dec-2024	nadd1rabord 41782	The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶})

20-Dec-2024	nadd1rabtr 41781	The set of ordinals which have a natural sum less than some ordinal is transitive. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥 ∈ 𝐴 ∣ (𝑥 +no 𝐵) ∈ 𝐶})

20-Dec-2024	nadd2rabon 41780	The set of ordinals which have a natural sum less than some ordinal is an ordinal number. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ∈ On)

20-Dec-2024	nadd2rabex 41779	The class of ordinals which have a natural sum less than some ordinal is a set. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ∈ V)

20-Dec-2024	nadd2rabord 41778	The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})

20-Dec-2024	nadd2rabtr 41777	The set of ordinals which have a natural sum less than some ordinal is transitive. (Contributed by RP, 20-Dec-2024.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})

19-Dec-2024	naddov4 41776	Alternate expression for natural addition. (Contributed by RP, 19-Dec-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥}))

19-Dec-2024	isarep1 6595	Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)

19-Dec-2024	funimaexg 6592	Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.) Shorten proof and avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)

19-Dec-2024	funmo 6521	A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)

19-Dec-2024	dffun3 6515	Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

19-Dec-2024	dffun6 6514	Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.)
⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))

19-Dec-2024	dffun2OLD 6512	Obsolete version of dffun2 6511 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))

19-Dec-2024	cotrgOLD 6067	Obsolete version of cotrg 6066 as of 29-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6069. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))

19-Dec-2024	difopab 5791	Difference of two ordered-pair class abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof shortened by SN, 19-Dec-2024.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}

15-Dec-2024	smfpimgtxr 45141	Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾_t 𝐷))

15-Dec-2024	smfpimltxr 45108	Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = dom 𝐹 & ⊢ (𝜑 → 𝐴 ∈ ℝ^*) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾_t 𝐷))

15-Dec-2024	pimltpnf2 45074	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

15-Dec-2024	pimltpnf2f 45073	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

15-Dec-2024	pimgtpnf2 45067	Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

15-Dec-2024	pimltmnf2 45059	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)

15-Dec-2024	pimltmnf2f 45058	Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)

14-Dec-2024	fzuntgd 41852	Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ (𝐿 + 1)) & ⊢ (𝜑 → 𝐿 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

14-Dec-2024	fzunt1d 41851	Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝐿 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ 𝐿) & ⊢ (𝜑 → 𝐿 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

14-Dec-2024	fzuntd 41850	Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)
⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

14-Dec-2024	fzunt 41849	Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.)
⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

13-Dec-2024	nlim4 41839	4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 4_o

13-Dec-2024	nlim3 41838	3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ ¬ Lim 3_o

13-Dec-2024	nlim2NEW 41837	2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 2_o

13-Dec-2024	nlim1NEW 41836	1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
⊢ ¬ Lim 1_o

13-Dec-2024	nlimsuc 41835	A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)

13-Dec-2024	wksonproplem 28715	Lemma for theorems for properties of walks between two vertices, e.g., trlsonprop 28719. (Contributed by AV, 16-Jan-2021.) Remove is-walk hypothesis. (Revised by SN, 13-Dec-2024.)
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃))) & ⊢ 𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝)})) ⇒ ⊢ (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))

13-Dec-2024	mptmpoopabovd 8019	The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) & ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)})) ⇒ ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})

13-Dec-2024	mptmpoopabbrd 8018	The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) & ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃)) & ⊢ (𝑔 = 𝐺 → (𝜒 ↔ 𝜏)) & ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)})) ⇒ ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})

13-Dec-2024	fvmptopab 7416	The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) Add disjoint variable condition on 𝐹, 𝑥, 𝑦 to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.)
⊢ (𝑧 = 𝑍 → (𝜑 ↔ 𝜓)) & ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)}) ⇒ ⊢ (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}

13-Dec-2024	opabresex2 7414	Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable conditions betweem 𝑊, 𝐺 and 𝑥, 𝑦 to remove hypotheses. (Revised by SN, 13-Dec-2024.)
⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

13-Dec-2024	nfralw 3292	Bound-variable hypothesis builder for restricted quantification. Version of nfral 3345 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 1-Sep-1999.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 13-Dec-2024.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

12-Dec-2024	sdom1 9193	A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 12-Dec-2024.)
⊢ (𝐴 ≺ 1_o ↔ 𝐴 = ∅)

11-Dec-2024	wksv 28630	The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.)
⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V

11-Dec-2024	abrexexg 7898	Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path axrep6g 5255, axrep6 5254, ax-rep 5247. See also abrexex2g 7902. There are partial converses under additional conditions, see for instance abnexg 7695. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2137, ax-11 2154, ax-12 2171, ax-pr 5389, ax-un 7677 and shorten proof. (Revised by SN, 11-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)

11-Dec-2024	ssrel 5743	A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2171. (Revised by SN, 11-Dec-2024.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

11-Dec-2024	elopaelxp 5726	Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.) Avoid ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by SN, 11-Dec-2024.)
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

11-Dec-2024	elopabr 5523	Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

11-Dec-2024	elopabw 5488	Membership in a class abstraction of ordered pairs. Weaker version of elopab 5489 with a sethood antecedent, avoiding ax-sep 5261, ax-nul 5268, and ax-pr 5389. Originally a subproof of elopab 5489. (Contributed by SN, 11-Dec-2024.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

11-Dec-2024	axrep6g 5255	axrep6 5254 in class notation. It is equivalent to both ax-rep 5247 and abrexexg 7898, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)

11-Dec-2024	dfiun2g 4995	Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 11-Dec-2024.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

11-Dec-2024	r19.21v 3172	Restricted quantifier version of 19.21v 1942. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 11-Dec-2024.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))

10-Dec-2024	sltn0 27277	If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.)
⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅))

10-Dec-2024	cbvreuw 3381	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3397 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2137. (Revised by Wolf Lammen, 10-Dec-2024.)
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

8-Dec-2024	1sdom2 9191	Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un 7677, see 1sdom2ALT 9192. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7677. (Revised by BTernaryTau, 8-Dec-2024.)
⊢ 1_o ≺ 2_o

8-Dec-2024	rexcom 3271	Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2024.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

7-Dec-2024	0sdom1dom 9189	Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7677, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7677. (Revised by BTernaryTau, 7-Dec-2024.)
⊢ (∅ ≺ 𝐴 ↔ 1_o ≼ 𝐴)

7-Dec-2024	ssct 9002	Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 7-Dec-2024.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)

7-Dec-2024	domssr 8946	If 𝐶 is a superset of 𝐵 and 𝐵 dominates 𝐴, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶)

7-Dec-2024	domssl 8945	If 𝐴 is a subset of 𝐵 and 𝐶 dominates 𝐵, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

7-Dec-2024	f1dom4g 8912	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8919 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

7-Dec-2024	f1oen4g 8911	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8918 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)

5-Dec-2024	sb8f 2349	Substitution of variable in universal quantifier. Version of sb8 2515 with a disjoint variable condition, not requiring ax-10 2137 or ax-13 2370. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2137. (Revised by SN, 5-Dec-2024.)
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

5-Dec-2024	sb8v 2348	Substitution of variable in universal quantifier. Version of sb8f 2349 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2171. (Contributed by SN, 5-Dec-2024.)
⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

4-Dec-2024	sucdom2 9157	Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5325. (Revised by BTernaryTau, 4-Dec-2024.)
⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

4-Dec-2024	undom 9010	Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5325. (Revised by BTernaryTau, 4-Dec-2024.)
⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

3-Dec-2024	eqvreldisj1 37359	The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj2 37360, eqvreldisj3 37361). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 3-Dec-2024.)
⊢ ( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))

3-Dec-2024	fvprc 6839	A function's value at a proper class is the empty set. See fvprcALT 6840 for a proof that uses ax-pow 5325 instead of ax-pr 5389. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 3-Aug-2024.) (Proof shortened by BTernaryTau, 3-Dec-2024.)
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)

3-Dec-2024	f1un 6809	The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.)
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))

3-Dec-2024	dtruOLD 5403	Obsolete proof of dtru 5398 as of 01-Jan-2025. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5389 instead of ax-pow 5325. (Revised by BTernaryTau, 3-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦

2-Dec-2024	onomeneq 9179	An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.) Avoid ax-pow 5325. (Revised by BTernaryTau, 2-Dec-2024.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))

2-Dec-2024	el 5399	Any set is an element of some other set. See elALT 5402 for a shorter proof using more axioms, and see elALT2 5329 for a proof that uses ax-9 2116 and ax-pow 5325 instead of ax-pr 5389. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5389 instead of ax-9 2116 and ax-pow 5325. (Revised by BTernaryTau, 2-Dec-2024.)
⊢ ∃𝑦 𝑥 ∈ 𝑦

1-Dec-2024	frrlem16 9703	Lemma for general well-founded recursion. Establish a subset relation. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))

1-Dec-2024	snnen2o 9188	A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ ¬ {𝐴} ≈ 2_o

1-Dec-2024	2onn 8593	The ordinal 2 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7677, see 2onnALT 8594. (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7677. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ 2_o ∈ ω

1-Dec-2024	1onn 8591	The ordinal 1 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7677, see 1onnALT 8592. Lemma 2.2 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7677. (Revised by BTernaryTau, 1-Dec-2024.)
⊢ 1_o ∈ ω

1-Dec-2024	2ellim 8450	A limit ordinal contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Lim 𝐴 → 2_o ∈ 𝐴)

1-Dec-2024	1ellim 8449	A limit ordinal contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Lim 𝐴 → 1_o ∈ 𝐴)

1-Dec-2024	ord2eln012 8448	An ordinal that is not 0, 1, or 2 contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Ord 𝐴 → (2_o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1_o ∧ 𝐴 ≠ 2_o)))

1-Dec-2024	ord1eln01 8447	An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ (Ord 𝐴 → (1_o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1_o)))

1-Dec-2024	nlim2 8441	2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ ¬ Lim 2_o

1-Dec-2024	nlim1 8440	1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ ¬ Lim 1_o

1-Dec-2024	f1cdmsn 7233	If a one-to-one function with a nonempty domain has a singleton as its codomain, its domain must also be a singleton. (Contributed by BTernaryTau, 1-Dec-2024.)
⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})

30-Nov-2024	2on 8431	Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Avoid ax-un 7677. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ 2_o ∈ On

30-Nov-2024	1on 8429	Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7677. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ 1_o ∈ On

30-Nov-2024	onsuc 7751	The successor of an ordinal number is an ordinal number. Closed form of onsuci 7779. Forward implication of onsucb 7757. Proposition 7.24 of [TakeutiZaring] p. 41. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) (Proof shortened by BTernaryTau, 30-Nov-2024.)
⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)

30-Nov-2024	sucexeloniOLD 7750	Obsolete version of sucexeloni 7749 as of 6-Jan-2025. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

30-Nov-2024	epweon 7714	The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7677, see epweonALT 7715. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7677. (Revised by BTernaryTau, 30-Nov-2024.)
⊢ E We On

30-Nov-2024	elex2 2811	If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2116, ax-ext 2702, df-clab 2709. (Revised by Wolf Lammen, 30-Nov-2024.)
⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)

29-Nov-2024	nndomog 9167	Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 9184 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9184. (Revised by RP, 5-Nov-2023.) Avoid ax-pow 5325. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))

29-Nov-2024	sdom0 9059	The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ¬ 𝐴 ≺ ∅

29-Nov-2024	0sdomg 9055	A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))

29-Nov-2024	dom0 9053	A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅)

29-Nov-2024	0domg 9051	Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)

29-Nov-2024	en0r 8967	The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)
⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅)

29-Nov-2024	brdomi 8905	Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)

29-Nov-2024	brdomg 8903	Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract brdom2g 8902 as an intermediate result. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

29-Nov-2024	brdom2g 8902	Dominance relation. This variation of brdomg 8903 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 8903. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))

29-Nov-2024	peano1 7830	Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7830 through peano5 7835 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.) Avoid ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ∅ ∈ ω

28-Nov-2024	phpeqd 9166	Corollary of the Pigeonhole Principle using equality. Strengthening of php 9161 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-pow. (Revised by BTernaryTau, 28-Nov-2024.)
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵)

27-Nov-2024	frmin 9694	Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6306 and tz7.5 6343. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

26-Nov-2024	php3 9163	Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5325. (Revised by BTernaryTau, 26-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

25-Nov-2024	domsdomtrfi 9156	Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 9063). (Contributed by BTernaryTau, 25-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)

25-Nov-2024	sdomdomtrfi 9155	Transitivity of strict dominance and dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike sdomdomtr 9061). (Contributed by BTernaryTau, 25-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶)

25-Nov-2024	predres 6298	Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.)
⊢ Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)

25-Nov-2024	predprc 6297	The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.)
⊢ (¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)

25-Nov-2024	predrelss 6296	Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.)
⊢ (𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))

24-Nov-2024	ssdomfi2 9151	A set dominates its finite subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8947). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵)

24-Nov-2024	domtrfir 9148	Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 8954). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

24-Nov-2024	domtrfi 9147	Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 8954). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

24-Nov-2024	domtrfil 9146	Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 8954). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

24-Nov-2024	f1domfi2 9136	If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g 8916). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

24-Nov-2024	rabid2 3437	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2024.)
⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)

24-Nov-2024	clelsb2 2860	Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2123). (Contributed by Jim Kingdon, 22-Nov-2018.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 24-Nov-2024.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

23-Nov-2024	natglobalincr 45236	Local monotonicity on half-open integer range implies global monotonicity. Inference form. (Contributed by Ender Ting, 23-Nov-2024.)
⊢ ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1)) & ⊢ 𝑇 ∈ ℤ ⇒ ⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)

23-Nov-2024	prjcrv0 41029	The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑌 = ((0...𝑁) mPoly 𝐾) & ⊢ 0 = (0_g‘𝑌) & ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐾 ∈ Field) ⇒ ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)

23-Nov-2024	prjcrvval 41028	Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
⊢ 𝐻 = ((0...𝑁) mHomP 𝐾) & ⊢ 𝐸 = ((0...𝑁) eval 𝐾) & ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾) & ⊢ 0 = (0_g‘𝐾) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐾 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻) ⇒ ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})

23-Nov-2024	prjcrvfval 41027	Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
⊢ 𝐻 = ((0...𝑁) mHomP 𝐾) & ⊢ 𝐸 = ((0...𝑁) eval 𝐾) & ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾) & ⊢ 0 = (0_g‘𝐾) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐾 ∈ Field) ⇒ ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))

23-Nov-2024	df-prjcrv 41026	Define the projective curve function. This takes a homogeneous polynomial and outputs the homogeneous coordinates where the polynomial evaluates to zero (the "zero set"). (In other words, scalar multiples are collapsed into the same projective point. See mhphf4 40832 and prjspvs 41006). (Contributed by SN, 23-Nov-2024.)
⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ₀, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛_n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0_g‘𝑘)}}))

23-Nov-2024	mhphf4 40832	A homogeneous polynomial defines a homogeneous function; this is mhphf3 40831 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝐻 = (𝐼 mHomP 𝑆) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐹 = (𝑆 freeLMod 𝐼) & ⊢ 𝑀 = (Base‘𝐹) & ⊢ ∙ = ( ·_𝑠 ‘𝐹) & ⊢ · = (._r‘𝑆) & ⊢ ↑ = (._g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐿 ∈ 𝐾) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) & ⊢ (𝜑 → 𝐴 ∈ 𝑀) ⇒ ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

23-Nov-2024	mhphf3 40831	A homogeneous polynomial defines a homogeneous function; this is mhphf2 40830 with the finite support restriction (frlmpws 21193, frlmbas 21198) on the assignments 𝐴 from variables to values. See comment of mhphf2 40830. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝐻 = (𝐼 mHomP 𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐹 = (𝑆 freeLMod 𝐼) & ⊢ 𝑀 = (Base‘𝐹) & ⊢ ∙ = ( ·_𝑠 ‘𝐹) & ⊢ · = (._r‘𝑆) & ⊢ ↑ = (._g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) & ⊢ (𝜑 → 𝐴 ∈ 𝑀) ⇒ ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

23-Nov-2024	evl0 40801	The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑂 = (0_g‘𝑅) & ⊢ 0 = (0_g‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → (𝑄‘ 0 ) = ((𝐵 ↑_m 𝐼) × {𝑂}))

23-Nov-2024	mplascl0 40800	The zero scalar as a polynomial. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ 𝑂 = (0_g‘𝑅) & ⊢ 0 = (0_g‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → (𝐴‘𝑂) = 0 )

23-Nov-2024	abbi1sn 40716	Originally part of uniabio 6468. Convert a theorem about df-iota 6453 to one about dfiota2 6454, without ax-10 2137, ax-11 2154, ax-12 2171. Although, eu6 2567 uses ax-10 2137 and ax-12 2171. (Contributed by SN, 23-Nov-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

23-Nov-2024	recvs 24546	The field of the real numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) (Proof shortened by SN, 23-Nov-2024.)
⊢ 𝑅 = (ringLMod‘ℝ_fld) ⇒ ⊢ 𝑅 ∈ ℂVec

23-Nov-2024	fldcrngd 20237	A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
⊢ (𝜑 → 𝑅 ∈ Field) ⇒ ⊢ (𝜑 → 𝑅 ∈ CRing)

23-Nov-2024	iotaval 6472	Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 23-Nov-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

23-Nov-2024	nfrabw 3441	A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3444 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}

23-Nov-2024	moel 3373	"At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)

23-Nov-2024	rmobidva 3366	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) Avoid ax-6 1971, ax-7 2011, ax-12 2171. (Revised by Wolf Lammen, 23-Nov-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

22-Nov-2024	tworepnotupword 45245	Concatenation of identical singletons is never an increasing sequence. (Contributed by Ender Ting, 22-Nov-2024.)
⊢ 𝐴 ∈ V ⇒ ⊢ ¬ (⟨“𝐴”⟩ ++ ⟨“𝐴”⟩) ∈ UpWord 𝑆

22-Nov-2024	singoutnupword 45242	Singleton with character out of range 𝑆 is not an increasing sequence for that range. (Contributed by Ender Ting, 22-Nov-2024.)
⊢ 𝐴 ∈ V ⇒ ⊢ (¬ 𝐴 ∈ 𝑆 → ¬ ⟨“𝐴”⟩ ∈ UpWord 𝑆)

22-Nov-2024	natlocalincr 45235	Global monotonicity on half-open range implies local monotonicity. Inference form. (Contributed by Ender Ting, 22-Nov-2024.)
⊢ ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘) < (𝐵‘𝑡)) ⇒ ⊢ ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))

22-Nov-2024	et-ltneverrefl 45232	Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11273. (New usage is discouraged.)
⊢ ¬ 𝐴 < 𝐴

22-Nov-2024	domnsymfi 9154	If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 9050). (Contributed by BTernaryTau, 22-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴)

21-Nov-2024	upwordsseti 45244	Strictly increasing sequences with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.)
⊢ 𝑆 ∈ V ⇒ ⊢ UpWord 𝑆 ∈ V

21-Nov-2024	upwordsing 45243	Singleton is an increasing sequence for any compatible range. (Contributed by Ender Ting, 21-Nov-2024.)
⊢ 𝐴 ∈ 𝑆 ⇒ ⊢ ⟨“𝐴”⟩ ∈ UpWord 𝑆

21-Nov-2024	singoutnword 45241	Singleton with character out of range 𝑉 is not a word for that range. (Contributed by Ender Ting, 21-Nov-2024.)
⊢ 𝐴 ∈ V ⇒ ⊢ (¬ 𝐴 ∈ 𝑉 → ¬ ⟨“𝐴”⟩ ∈ Word 𝑉)

21-Nov-2024	nfreuw 3385	Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3404 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2702. (Revised by Wolf Lammen, 21-Nov-2024.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑

21-Nov-2024	nfrmow 3384	Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3403 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2702. (Revised by Wolf Lammen, 21-Nov-2024.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑

21-Nov-2024	eeor 2329	Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

21-Nov-2024	aaan 2327	Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

20-Nov-2024	php2 9162	Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 20-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)

20-Nov-2024	2ralor 3217	Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Wolf Lammen, 20-Nov-2024.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))

20-Nov-2024	sbrim 2300	Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Avoid ax-10 2137. (Revised by Gino Giotto, 20-Nov-2024.)
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

19-Nov-2024	upwordisword 45240	Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.)
⊢ (𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)

19-Nov-2024	upwordnul 45239	Empty set is an increasing sequence for every range. (Contributed by Ender Ting, 19-Nov-2024.)
⊢ ∅ ∈ UpWord 𝑆

19-Nov-2024	df-upword 45238	Strictly increasing sequence is a sequence, adjacent elements of which increase. (Contributed by Ender Ting, 19-Nov-2024.)
⊢ UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}

19-Nov-2024	moeu2 36896	Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.)
⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))

19-Nov-2024	fri 5598	A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (inference form). (Contributed by BJ, 16-Nov-2024.) (Proof shortened by BJ, 19-Nov-2024.)
⊢ (((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

18-Nov-2024	mopickr 36897	"At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references 2.21 (pm2.21 123) and 14.26 (eupickbi 2631) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.)
⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))

18-Nov-2024	php 9161	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of phplem1 9158, phplem2 9159, nneneq 9160, and this final piece of the proof. (Contributed by NM, 29-May-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 18-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)

18-Nov-2024	wfr3 8288	The principle of Well-Ordered Recursion, part 3 of 3. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in wfr1 8286 and wfr2 8287 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)

18-Nov-2024	wfr1 8286	The Principle of Well-Ordered Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a set-like well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-ordered recursion". The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)

18-Nov-2024	wfr2a 8285	A weak version of wfr2 8287 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.) (Proof shortened by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

18-Nov-2024	wfrresex 8284	Show without using the axiom of replacement that the restriction of the well-ordered recursion generator to a predecessor class is a set. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)

18-Nov-2024	csbwrecsg 8257	Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.) (Revised by Scott Fenton, 18-Nov-2024.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

18-Nov-2024	fprresex 8246	The restriction of a function defined by well-founded recursion to the predecessor of an element of its domain is a set. Avoids the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)

18-Nov-2024	fprfung 8245	A "function" defined by well-founded recursion is indeed a function when the relation is a partial order. Avoids the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

18-Nov-2024	frrdmss 8243	Show without using the axiom of replacement that the domain of the well-founded recursion generator is a subclass of 𝐴. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ dom 𝐹 ⊆ 𝐴

18-Nov-2024	frrrel 8242	Show without using the axiom of replacement that the well-founded recursion generator gives a relation. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ Rel 𝐹

18-Nov-2024	fpr2 8240	Law of well-founded recursion over a partial order, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.) (Proof shortened by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

18-Nov-2024	fpr2a 8238	Weak version of fpr2 8240 which is useful for proofs that avoid the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

18-Nov-2024	csbfrecsg 8220	Move class substitution in and out of the well-founded recursive function generator. (Contributed by Scott Fenton, 18-Nov-2024.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, 𝐹) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))

18-Nov-2024	drnf1v 2368	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2441 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

18-Nov-2024	dral1v 2365	Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2437 with a disjoint variable condition, which does not require ax-13 2370. Remark: the corresponding versions for dral2 2436 and drex2 2440 are instances of albidv 1923 and exbidv 1924 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2172. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

18-Nov-2024	equsexv 2259	An equivalence related to implicit substitution. Version of equsex 2416 with a disjoint variable condition, which does not require ax-13 2370. See equsexvw 2008 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2258. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)

17-Nov-2024	bj-rdg0gALT 35615	Alternate proof of rdg0g 8378. More direct since it bypasses tz7.44-1 8357 and rdg0 8372 (and vtoclg 3526, vtoclga 3535). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)

17-Nov-2024	wfrfun 8283	The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 17-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)

17-Nov-2024	wfrdmcl 8282	The predecessor class of an element of the well-ordered recursion generator's domain is a subset of its domain. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

17-Nov-2024	wfrdmss 8281	The domain of the well-ordered recursion generator is a subclass of 𝐴. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ dom 𝐹 ⊆ 𝐴

17-Nov-2024	wfrrel 8280	The well-ordered recursion generator generates a relation. Avoids the axiom of replacement. (Contributed by Scott Fenton, 8-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ Rel 𝐹

17-Nov-2024	nfwrecs 8252	Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)

17-Nov-2024	wrecseq123 8250	General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

17-Nov-2024	frrdmcl 8244	Show without using the axiom of replacement that for a "function" defined by well-founded recursion, the predecessor class of an element of its domain is a subclass of its domain. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)

17-Nov-2024	wfis2fg 6315	Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ Ⅎ𝑦𝜓 & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

17-Nov-2024	wfisg 6312	Well-Ordered Induction Schema. If a property passes from all elements less than 𝑦 of a well-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

17-Nov-2024	wfi 6309	The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)

17-Nov-2024	tz6.26 6306	All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

17-Nov-2024	cbvmptv 5223	Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid auxiliary axioms . See cbvmptvg 5225 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Nov-2024.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

17-Nov-2024	cbvopab1v 5189	Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) Reduce axiom usage. (Revised by Gino Giotto, 17-Nov-2024.)
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

16-Nov-2024	frd 5597	A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.)
⊢ (𝜑 → 𝑅 Fr 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)

16-Nov-2024	dffr6 5596	Alternate definition of df-fr 5593. See dffr5 34413 for a definition without dummy variables (but note that their equivalence uses ax-sep 5261). (Contributed by BJ, 16-Nov-2024.)
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)

15-Nov-2024	1strbas 17111	The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩} ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))

15-Nov-2024	1strstr1 17110	A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
⊢ 𝐺 = {⟨(Base‘ndx), 𝐵⟩} ⇒ ⊢ 𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩

14-Nov-2024	aks4d1 40619	Lemma 4.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf, existence of a polynomially bounded number by the digit size of 𝑁 that asserts the polynomial subspace that we need to search to guarantee that 𝑁 is prime. Eventually we want to show that the polynomial searching space is bounded by degree 𝐵. (Contributed by metakunt, 14-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3)) & ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5)) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵)((𝑁 gcd 𝑟) = 1 ∧ ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑟)‘𝑁)))

14-Nov-2024	aks4d1p9 40618	Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ⇒ ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))

14-Nov-2024	aks4d1lem1 40592	Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3)) & ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5)) ⇒ ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))

13-Nov-2024	aks4d1p8d3 40616	The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.)
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑁) ⇒ ⊢ (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1)

13-Nov-2024	aks4d1p8d2 40615	Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)
⊢ (𝜑 → 𝑅 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑄 ∈ ℙ) & ⊢ (𝜑 → 𝑃 ∥ 𝑅) & ⊢ (𝜑 → 𝑄 ∥ 𝑅) & ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁) & ⊢ (𝜑 → 𝑄 ∥ 𝑁) ⇒ ⊢ (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)

12-Nov-2024	prstcocval 47211	Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ⊥ = (oc‘𝐾)) ⇒ ⊢ (𝜑 → ⊥ = (oc‘𝐶))

12-Nov-2024	prstcleval 47208	Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) & ⊢ (𝜑 → 𝐾 ∈ Proset ) & ⊢ (𝜑 → ≤ = (le‘𝐾)) ⇒ ⊢ (𝜑 → ≤ = (le‘𝐶))

12-Nov-2024	zlmtset 32634	Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐽 = (TopSet‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))

12-Nov-2024	setsmsbas 23865	The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) ⇒ ⊢ (𝜑 → 𝑋 = (Base‘𝐾))

12-Nov-2024	matvsca 21801	The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·_𝑠 ‘𝐺) = ( ·_𝑠 ‘𝐴))

12-Nov-2024	matsca 21799	The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))

12-Nov-2024	sravsca 20707	The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (._r‘𝑊) = ( ·_𝑠 ‘𝐴))

12-Nov-2024	srasca 20705	The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (𝑊 ↾_s 𝑆) = (Scalar‘𝐴))

12-Nov-2024	odubas 18194	Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝐷 = (ODual‘𝑂) & ⊢ 𝐵 = (Base‘𝑂) ⇒ ⊢ 𝐵 = (Base‘𝐷)

12-Nov-2024	slotsdifocndx 17313	The index of the slot for the orthocomplementation is not the index of other slots. Formerly part of proof for prstcocval 47211. (Contributed by AV, 12-Nov-2024.)
⊢ ((oc‘ndx) ≠ (comp‘ndx) ∧ (oc‘ndx) ≠ (Hom ‘ndx))

12-Nov-2024	slotsdifplendx2 17312	The index of the slot for the "less than or equal to" ordering is not the index of other slots. Formerly part of proof for prstcleval 47208. (Contributed by AV, 12-Nov-2024.)
⊢ ((le‘ndx) ≠ (comp‘ndx) ∧ (le‘ndx) ≠ (Hom ‘ndx))

12-Nov-2024	slotsdifipndx 17230	The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 20705 and sravsca 20707. (Contributed by AV, 12-Nov-2024.)
⊢ (( ·_𝑠 ‘ndx) ≠ (·_𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·_𝑖‘ndx))

12-Nov-2024	ssdomfi 9150	A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8947). (Contributed by BTernaryTau, 12-Nov-2024.)
⊢ (𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵))

11-Nov-2024	mpteq1df 43582	An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by SN, 11-Nov-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

11-Nov-2024	mhphf2 40830	A homogeneous polynomial defines a homogeneous function; this is mhphf 40829 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 21209 but without the finite support restriction (frlmpws 21193, frlmbas 21198) on the assignments 𝐴 from variables to values. TODO?: Polynomials (df-mpl 21350) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) & ⊢ 𝐻 = (𝐼 mHomP 𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ ∙ = ( ·_𝑠 ‘((ringLMod‘𝑆) ↑_s 𝐼)) & ⊢ · = (._r‘𝑆) & ⊢ ↑ = (._g‘(mulGrp‘𝑆)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐿 ∈ 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼)) ⇒ ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

11-Nov-2024	zlmds 32632	Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊))

11-Nov-2024	setsmsds 23867	The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) & ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)) ⇒ ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾))

11-Nov-2024	thlle 21139	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝐾 = (toHL‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 𝐼 = (toInc‘𝐶) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ≤ = (le‘𝐾)

11-Nov-2024	thlbas 21137	Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝐾 = (toHL‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ 𝐶 = (Base‘𝐾)

11-Nov-2024	cnfldfunALT 20846	The field of complex numbers is a function. Alternate proof of cnfldfun 20845 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Fun ℂ_fld

11-Nov-2024	fldidom 20812	A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.)
⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn)

11-Nov-2024	slotsdifdsndx 17289	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 20846. (Contributed by AV, 11-Nov-2024.)
⊢ ((*_𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))

11-Nov-2024	plendxnocndx 17279	The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle 21139. (Contributed by AV, 11-Nov-2024.)
⊢ (le‘ndx) ≠ (oc‘ndx)

11-Nov-2024	basendxnocndx 17278	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas 21137. (Contributed by AV, 11-Nov-2024.)
⊢ (Base‘ndx) ≠ (oc‘ndx)

11-Nov-2024	slotsdifplendx 17270	The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 20846. (Contributed by AV, 11-Nov-2024.)
⊢ ((*_𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))

11-Nov-2024	tsetndxnstarvndx 17254	The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfunALT 20846. (Contributed by AV, 11-Nov-2024.)
⊢ (TopSet‘ndx) ≠ (*_𝑟‘ndx)

11-Nov-2024	nneneq 9160	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5325. (Revised by BTernaryTau, 11-Nov-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))

11-Nov-2024	ofeqd 7624	Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → ∘_f 𝑅 = ∘_f 𝑆)

11-Nov-2024	iunopab 5521	Move indexed union inside an ordered-pair class abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) Avoid ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by SN, 11-Nov-2024.)
⊢ ∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}

11-Nov-2024	mpteq2ia 5213	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

11-Nov-2024	mpteq2dva 5210	Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) Remove dependency on ax-10 2137. (Revised by SN, 11-Nov-2024.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

11-Nov-2024	mpteq2da 5208	Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))

11-Nov-2024	mpteq1i 5206	An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) Remove all disjoint variable conditions. (Revised by SN, 11-Nov-2024.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)

11-Nov-2024	mpteq1 5203	An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

11-Nov-2024	mpteq12dva 5199	An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2137, ax-12 2171. (Revised by SN, 11-Nov-2024.)
⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

11-Nov-2024	mpteq12df 5196	An equality inference for the maps-to notation. Compare mpteq12dv 5201. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof shortened by SN, 11-Nov-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ (𝜑 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

11-Nov-2024	mpteq12da 5195	An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Remove dependency on ax-10 2137. (Revised by SN, 11-Nov-2024.)
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

10-Nov-2024	aks4d1p8 40617	Show that 𝑁 and 𝑅 are coprime for AKS existence theorem, with eliminated hypothesis. (Contributed by metakunt, 10-Nov-2024.) (Proof sketch by Thierry Arnoux.)
⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3)) & ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) & ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5)) & ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < ) ⇒ ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)

10-Nov-2024	aks4d1p8d1 40614	If a prime divides one number 𝑀, but not another number 𝑁, then it divides the quotient of 𝑀 and the gcd of 𝑀 and 𝑁. (Contributed by Thierry Arnoux, 10-Nov-2024.)
⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∥ 𝑀) & ⊢ (𝜑 → ¬ 𝑃 ∥ 𝑁) ⇒ ⊢ (𝜑 → 𝑃 ∥ (𝑀 / (𝑀 gcd 𝑁)))

10-Nov-2024	slotsdifunifndx 17296	The index of the slot for the uniform set is not the index of other slots. Formerly part of proof for cnfldfunALT 20846. (Contributed by AV, 10-Nov-2024.)
⊢ (((+_g‘ndx) ≠ (UnifSet‘ndx) ∧ (._r‘ndx) ≠ (UnifSet‘ndx) ∧ (*_𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))

9-Nov-2024	bj-flddrng 35833	Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)

9-Nov-2024	bj-dfid2ALT 35609	Alternate version of dfid2 5538. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5536 instead to make the semantics of the construction df-opab 5173 clearer. (New usage is discouraged.)
⊢ I = {⟨𝑥, 𝑥⟩ ∣ ⊤}

9-Nov-2024	ttgval 27880	Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.)
⊢ 𝐺 = (toTG‘𝐻) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ − = (-_g‘𝐻) & ⊢ · = ( ·_𝑠 ‘𝐻) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})))

9-Nov-2024	lngndxnitvndx 27448	The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 27880. (Contributed by AV, 9-Nov-2024.)
⊢ (LineG‘ndx) ≠ (Itv‘ndx)

9-Nov-2024	rescabs 17732	Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 9-Nov-2024.)
⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) & ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → ((𝐶 ↾_cat 𝐻) ↾_cat 𝐽) = (𝐶 ↾_cat 𝐽))

7-Nov-2024	ressbas 17129	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝑅 = (𝑊 ↾_s 𝐴) & ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))

7-Nov-2024	setsnid 17092	Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 7-Nov-2024.)
⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ 𝐷 ⇒ ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))

6-Nov-2024	sn-iotalemcor 40715	Corollary of sn-iotalem 40714. Compare sb8iota 6465. (Contributed by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})

6-Nov-2024	sn-iotalem 40714	An unused lemma showing that many equivalences involving df-iota 6453 are potentially provable without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}

6-Nov-2024	hlhilsmul 40480	Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ · = (._r‘𝐸) ⇒ ⊢ (𝜑 → · = (._r‘𝑅))

6-Nov-2024	hlhilsplus 40478	Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ + = (+_g‘𝐸) ⇒ ⊢ (𝜑 → + = (+_g‘𝑅))

6-Nov-2024	hlhilsbase 40476	The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐶 = (Base‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (Base‘𝑅))

6-Nov-2024	hlhilslem 40474	Lemma for hlhilsbase 40476 etc. (Contributed by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊) & ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ 𝐹 = Slot (𝐹‘ndx) & ⊢ (𝐹‘ndx) ≠ (*_𝑟‘ndx) & ⊢ 𝐶 = (𝐹‘𝐸) ⇒ ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅))

6-Nov-2024	oppradd 20072	Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
⊢ 𝑂 = (opp_r‘𝑅) & ⊢ + = (+_g‘𝑅) ⇒ ⊢ + = (+_g‘𝑂)

6-Nov-2024	opprbas 20070	Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
⊢ 𝑂 = (opp_r‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂)

6-Nov-2024	opprlem 20068	Lemma for opprbas 20070 and oppradd 20072. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
⊢ 𝑂 = (opp_r‘𝑅) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (._r‘ndx) ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

6-Nov-2024	symgvalstruct 19192	The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (Proof shortened by AV, 6-Nov-2024.)
⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} & ⊢ 𝑀 = (𝐴 ↑_m 𝐴) & ⊢ + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔)) & ⊢ 𝐽 = (∏_t‘(𝐴 × {𝒫 𝐴})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+_g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

6-Nov-2024	frmdplusg 18678	The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) (Proof shortened by AV, 6-Nov-2024.)
⊢ 𝑀 = (freeMnd‘𝐼) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+_g‘𝑀) ⇒ ⊢ + = ( ++ ↾ (𝐵 × 𝐵))

6-Nov-2024	iotaex 6474	Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) ∈ V

6-Nov-2024	iotassuni 6473	The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 6-Nov-2024.)
⊢ (℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}

6-Nov-2024	iotanul2 6471	Version of iotanul 6479 using df-iota 6453 instead of dfiota2 6454. (Contributed by SN, 6-Nov-2024.)
⊢ (¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)

6-Nov-2024	iotauni2 6470	Version of iotauni 6476 using df-iota 6453 instead of dfiota2 6454. (Contributed by SN, 6-Nov-2024.)
⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

6-Nov-2024	iotaval2 6469	Version of iotaval 6472 using df-iota 6453 instead of dfiota2 6454. (Contributed by SN, 6-Nov-2024.)
⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)

6-Nov-2024	eqimssd 4003	Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

5-Nov-2024	dfid2 5538	Alternate definition of the identity relation. Instance of dfid3 5539 not requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce axiom usage. (Revised by Gino Giotto, 4-Nov-2024.) (Proof shortened by BJ, 5-Nov-2024.) Use df-id 5536 instead to make the semantics of the constructor df-opab 5173 clearer. (New usage is discouraged.)
⊢ I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}

5-Nov-2024	r19.30 3119	Restricted quantifier version of 19.30 1884. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 5-Nov-2024.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

4-Nov-2024	phplem2 9159	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5325. (Revised by BTernaryTau, 4-Nov-2024.)
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))

4-Nov-2024	sbthfi 9153	Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9044). (Contributed by BTernaryTau, 4-Nov-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

4-Nov-2024	sbthfilem 9152	Lemma for sbthfi 9153. (Contributed by BTernaryTau, 4-Nov-2024.)
⊢ 𝐴 ∈ V & ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))} & ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (^◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)

4-Nov-2024	r19.12 3295	Restricted quantifier version of 19.12 2320. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2370, ax-ext 2702. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

4-Nov-2024	r19.29vva 3203	A commonly used pattern based on r19.29 3113, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) ⇒ ⊢ (𝜑 → 𝜒)

4-Nov-2024	reximdvai 3158	Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))

4-Nov-2024	r19.29d2r 3133	Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

4-Nov-2024	ralrexbid 3105	Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))

4-Nov-2024	exexw 2054	Existential quantification over a given variable is idempotent. Weak version of bj-exexbiex 35241, requiring fewer axioms. (Contributed by Gino Giotto, 4-Nov-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑥𝜑)

3-Nov-2024	znmul 20984	The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤ_ring) & ⊢ 𝑈 = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ₀ → (._r‘𝑈) = (._r‘𝑌))

3-Nov-2024	znadd 20982	The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤ_ring) & ⊢ 𝑈 = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ₀ → (+_g‘𝑈) = (+_g‘𝑌))

3-Nov-2024	znbas2 20980	The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤ_ring) & ⊢ 𝑈 = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ₀ → (Base‘𝑈) = (Base‘𝑌))

3-Nov-2024	znbaslem 20978	Lemma for znbas 20987. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤ_ring) & ⊢ 𝑈 = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (le‘ndx) ⇒ ⊢ (𝑁 ∈ ℕ₀ → (𝐸‘𝑈) = (𝐸‘𝑌))

3-Nov-2024	zlmmulr 20960	Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (._r‘𝐺) ⇒ ⊢ · = (._r‘𝑊)

3-Nov-2024	zlmplusg 20958	Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ + = (+_g‘𝐺) ⇒ ⊢ + = (+_g‘𝑊)

3-Nov-2024	zlmbas 20956	Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 = (Base‘𝑊)

3-Nov-2024	zlmlem 20954	Lemma for zlmbas 20956 and zlmplusg 20958. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) & ⊢ (𝐸‘ndx) ≠ ( ·_𝑠 ‘ndx) ⇒ ⊢ (𝐸‘𝐺) = (𝐸‘𝑊)

3-Nov-2024	nelb 3220	A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
⊢ (¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)

3-Nov-2024	rexbi 3103	Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

2-Nov-2024	psrvscafval 21395	The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ ∙ = ( ·_𝑠 ‘𝑆) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ · = (._r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ⇒ ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘_f · 𝑓))

2-Nov-2024	zlmsca 20962	Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → ℤ_ring = (Scalar‘𝑊))

2-Nov-2024	rexab 3655	Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))

2-Nov-2024	ralab 3652	Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))

1-Nov-2024	mnringvscad 42626	The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( ·_𝑠 ‘𝑉) = ( ·_𝑠 ‘𝐹))

1-Nov-2024	mnringscad 42624	The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝐹))

1-Nov-2024	mnringaddgd 42619	The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (+_g‘𝑉) = (+_g‘𝐹))

1-Nov-2024	mnringbased 42613	The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ 𝐵 = (Base‘𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐵 = (Base‘𝐹))

1-Nov-2024	mnringnmulrd 42611	Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.)
⊢ 𝐹 = (𝑅 MndRing 𝑀) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (._r‘ndx) & ⊢ 𝐴 = (Base‘𝑀) & ⊢ 𝑉 = (𝑅 freeLMod 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑀 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))

1-Nov-2024	opsrsca 21497	The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))

1-Nov-2024	opsrvsca 21495	The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑂))

1-Nov-2024	opsrmulr 21493	The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇) & ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼)) ⇒ ⊢ (𝜑 → (._r‘𝑆) = (._r‘𝑂))

Older news:

(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.

(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.

(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabh�user 4.6 for the Metamath Proof Explorer. See also his older videos.

(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"

(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.

(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.

(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.

(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)

(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.

(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.

(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.

(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.

(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.

(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.

(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)

(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.

(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".

(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.

(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]

(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.

(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.

(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.

(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).

(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.

(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.

(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.

(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.

(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.

(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.

(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.

(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)

(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.

(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.

(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.

(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."

(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.

(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.

(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.

(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.

(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.

(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.

(11-Nov-2017) Alan Sare updated his completeusersproof program.

(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.

(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)

(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.

(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.

(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.

(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.

(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).

(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.

(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.

(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.

(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.

(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.

(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.

(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.

(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.

(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.

(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.

(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.

(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica

(12-Aug-2016) A Gitter chat room has been created for Metamath.

(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project

(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.

(4-Aug-2016) Mario gave two presentations at CICM 2016.

(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.

(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.

(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.

(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.

(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).

(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.

(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)

(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.

(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html

(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.

(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.

(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.

(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".

(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.

(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.

(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.

(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.

(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.

(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"

(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.

(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.

(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.

(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.

(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.

(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.

(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.

(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.

(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).

(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.

(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.

(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.

(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.

(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.

(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.

(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.

(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.

(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.

(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.

(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)

(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.

(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.

(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.

(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.

(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.

(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.

(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.

(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.

(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.

(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.

(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.

(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.

(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.

(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.

(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.

(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.

(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.

(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio

(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.

(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.

(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes

(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.

(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which tests all but 5 definitions in set.mm for soundness. You can turn on the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.

(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.

(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.

(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincar� in Paris. The slides for this talk are here.

(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.

(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.

(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.

(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!

(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.

(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.

(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)

(4-Feb-2014) Mario Carneiro writes:

The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?
One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.
That said, how can this new feature be optimized to help you when writing proofs?

(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.

(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.

(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.

(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.

(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.

(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.

(8-Oct-2013) Today's Notes entry describes some proof repair techniques.

(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.

(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.

(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."

(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.

(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)

(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.

(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.

(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.

(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.

(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"

(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.

(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.

(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.

(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.

(20-Sep-2012) Mel O'Cat updated https://us.metamath.org/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.

(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.

(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).

(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."

(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.

(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).

Front	Back	Detail

(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.

(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101) release: https://us.metamath.org/ocat/mmj2/mmj2.zip https://us.metamath.org/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)! See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems. See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if there are errors in the command line arguments -- and the last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes. Good luck. And thanks for all of your help!

(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.

(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),

(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.

(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)

(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.

(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.

(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.

To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.

To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.

I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.

Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.

(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.

(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.

(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.

(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.

Old	New	Description
C.	_C	binomial coefficient
E	_E	epsilon relation
e	_e	Euler's constant
I	_I	identity relation
i	_i	imaginary unit
V	_V	universal class
(_	C_	subset
(.	C.	proper subset
P~	~P	power class
H~	~H	Hilbert space

(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.

(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.

(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)

(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.

(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.

(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.

(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.

(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.

(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.

(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)

(28-May-2009) AsteroidMeta is back on-line. Note the URL change.

(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.

(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)

(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.

(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."

(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.

(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.

(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.

(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.

(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).

(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.

(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.

(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.

(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.

(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.

(30-Dec-2007) "How to build a library of formalized mathematics".

(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.

(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)

(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.

(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.

(5-Nov-2007) A Usenet challenge. :)

(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.

(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.

(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.

(29-Apr-2007) Blog mentions of Metamath: here and here.

(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.

(15-Mar-2007) A picture of Penny the cat guarding the us.metamath.org server and making the rounds.

(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.

(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.

(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.

(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.

(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.

(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.

(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.

(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.

(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.

(29-Oct-2006) ocat has announced that the long-awaited 1-Nov-2006 release of mmj2 is available now.
     The new "Unify+Get Hints" is quite useful, and any proof can be generated as follows. With "?" in the Hyp field and Ref field blank, select "Unify+Get Hints". Select a hint from the list and put it in the Ref field. Edit any $n dummy variables to become the desired wffs. Rinse and repeat for the new proof steps generated, until the proof is done.
     The new tutorial, mmj2PATutorial.bat, explains this in detail. One way to reduce or avoid dummy $n's is to fill in the Hyp field with a comma-separated list of any known hypothesis matches to earlier proof steps, keeping a "?" in the list to indicate that the remaining hypotheses are unknown. Then "Unify+Get Hints" can be applied. The tutorial page \mmj2\data\mmp\PATutorial\Page405.mmp has an example.
     Don't forget that the eimm export/import program lets you go back and forth between the mmj2 and the metamath program proof assistants, without exiting from either one, to exploit the best features of each as required.

(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.

(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.

(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.

(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.

(28-Sep-2006) A blog entry, Metamath responds.

(26-Sep-2006) A blog entry, Metamath isn't hygienic.

(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.

(26-Jul-2006) A new open problem in predicate calculus was added.

(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.

(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C program used to create the MIDI files on the Metamath Music Page, so that it works on MacOS X. This is a nice accomplishment, since the original program was written before C was standardized by ANSI and will not compile on modern compilers.
Unfortunately, the original program source states no copyright terms. The main author, Tim Thompson, has kindly agreed to release his code to public domain, but two other authors have also contributed to the code, and so far I have been unable to contact them for copyright clearance. Therefore I cannot offer the MacOS X version for public download on this site until this is resolved. Update 10-May-2006: Another author, M. Czeiszperger, has released his contribution to public domain.
If you are interested in Luca's modified source code, please contact me directly.

(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.

(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.

(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java applet to Java 1.5: (1) QSort has been stripped out: its functionality is in the Collections class that Sun ships; (2) all Vectors have been replaced by ArrayLists; (3) generic types have been tossed in wherever they fit: this cuts back drastically on casting; and (4) any warnings Eclipse spouted out have been dealt with. I haven't yet updated it officially, because I don't know if it will work with Microsoft's JVM in older versions of Internet Explorer. The current official version is compiled with Java 1.3, because it won't work with Microsoft's JVM if it is compiled with Java 1.4. (As distasteful as that seems, I will get complaints from users if it doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new version runs on Microsoft's JVM, I would be grateful. Scott's new version is mm.java-1.5.gz; after uncompressing it, rename it to mm.java, use it to replace the existing mm.java file in the Metamath Solitaire download, and recompile according to instructions in the mm.java comments.
Scott has also created a second version, mmsol-0.2.tar.gz, that reads the axioms from ASCII files, instead of having the axioms hard-coded in the program. This can be very useful if you want to play with custom axioms, and you can also add a collection of starting theorems as "axioms" to work from. However, it must be run from the local directory with appletviewer, since the default Java security model doesn't allow reading files from a browser. It works with the JDK 5 Update 6 Java download.
To compile (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html

(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!

(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.

(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).

(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)

(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.

(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.

(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."

(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.

(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.

(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).

(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.

(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .

(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.

(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.

(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.

(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.

(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.

(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.

(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).

(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).

(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.

(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.

(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.

(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.

(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)

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