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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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Last updated on 15-Apr-2025 at 5:18 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
10-Apr-2025dvaddbr 25688 The sum rule for derivatives at a point. For the (simpler but more limited) function version, see dvadd 25691. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Remove unnecessary hypotheses. (Revised by GG, 10-Apr-2025.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑌⟶ℂ)    &   (𝜑 → 𝑌 ⊆ 𝑆)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)    ⇒   (𝜑 → 𝐶(𝑆 D (𝐹 ∘f + 𝐺))(𝐾 + 𝐿))
 
9-Apr-2025issetft 3486 Closed theorem form of isset 3485 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3539. (Contributed by Wolf Lammen, 9-Apr-2025.)
(Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
 
4-Apr-2025iscsrg 41142 A commutative semiring is a semiring whose multiplication is a commutative monoid. (Contributed by metakunt, 4-Apr-2025.)
𝐺 = (mulGrp‘𝑅)    ⇒   (𝑅 ∈ CSRing ↔ (𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd))
 
4-Apr-2025df-csring 41141 Define the class of all commutative semirings. (Contributed by metakunt, 4-Apr-2025.)
CSRing = {𝑓 ∈ SRing ∣ (mulGrp‘𝑓) ∈ CMnd}
 
4-Apr-2025ghmf1 19160 Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g‘𝑅)    &    0 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁)))
 
3-Apr-2025df-suppos 34973 Define the function that, when given an 𝑛-ary operation 𝑓 and 𝑛 many 𝑚-ary operations (𝑔‘∅), ..., (𝑔‘∪ 𝑛), returns the superposition of 𝑓 with the (𝑔‘𝑖), itself another 𝑚-ary operation on 𝑎. Given 𝑥 (a sequence of 𝑚 arguments in 𝑎), the superposition effectively applies each of the (𝑔‘𝑖) to 𝑥, then applies 𝑓 to the resulting sequence of 𝑛 function values. This can be seen as a generalized version of function composition; see paragraph 3 of [Szendrei] p. 11. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑚 ∈ (ω ∖ 1o) ↦ (𝑓 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛)), 𝑔 ∈ ((𝑎 ↑m (𝑎 ↑m 𝑚)) ↑m 𝑛) ↦ (𝑥 ∈ (𝑎 ↑m 𝑚) ↦ (𝑓‘(𝑖 ∈ 𝑛 ↦ ((𝑔‘𝑖)‘𝑥)))))))
 
3-Apr-2025df-prj 34971 Define the function that, for a set 𝑎, arity 𝑛, and index 𝑖, returns the 𝑖-th 𝑛-ary projection on 𝑎. This is the 𝑛-ary operation on 𝑎 that, for any sequence of 𝑛 elements of 𝑎, returns the element having index 𝑖. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
 
3-Apr-2025df-cloneop 34969 Define the function that sends a set to the class of clone-theoretic operations on the set. For convenience, we take an operation on 𝑎 to be a function on finite sequences of elements of 𝑎 (rather than tuples) with values in 𝑎. Following line 6 of [Szendrei] p. 11, the arity 𝑛 of an operation (here, the length of the sequences at which the operation is defined) is always finite and non-zero, whence 𝑛 is taken to be a non-zero finite ordinal. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
 
3-Apr-2025quscrng 21029 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)    ⇒   ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
 
2-Apr-2025algextdeg 33070 The degree of an algebraic field extension is the degree of the minimal polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   (𝜑 → (𝐿[:]𝐾) = (𝐷‘(𝑀‘𝐴)))
 
2-Apr-2025algextdeglem8 33069 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   𝑅 = (rem1p‘𝐾)    &   𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    &   𝑇 = (◡( deg1 ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))    ⇒   (𝜑 → (dim‘(𝐻 “s 𝑃)) = (𝐷‘(𝑀‘𝐴)))
 
2-Apr-2025algextdeglem7 33068 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   𝑅 = (rem1p‘𝐾)    &   𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    &   𝑇 = (◡( deg1 ‘𝐾) “ (-∞[,)(𝐷‘(𝑀‘𝐴))))    &   (𝜑 → 𝑋 ∈ 𝑈)    ⇒   (𝜑 → (𝑋 ∈ 𝑇 ↔ (𝐻‘𝑋) = 𝑋))
 
2-Apr-2025algextdeglem6 33067 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    &   𝑅 = (rem1p‘𝐾)    &   𝐻 = (𝑝 ∈ 𝑈 ↦ (𝑝𝑅(𝑀‘𝐴)))    ⇒   (𝜑 → (dim‘𝑄) = (dim‘(𝐻 “s 𝑃)))
 
2-Apr-2025algextdeglem5 33066 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   (𝜑 → 𝑍 = ((RSpan‘𝑃)‘{(𝑀‘𝐴)}))
 
2-Apr-2025algextdeglem4 33065 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   (𝜑 → (dim‘𝑄) = (𝐿[:]𝐾))
 
2-Apr-2025algextdeglem3 33064 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   (𝜑 → 𝑄 ∈ LVec)
 
2-Apr-2025algextdeglem2 33063 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘𝐾)    &   𝑈 = (Base‘𝑃)    &   𝐺 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝐴))    &   𝑁 = (𝑥 ∈ 𝑈 ↦ [𝑥](𝑃 ~QG 𝑍))    &   𝑍 = (◡𝐺 “ {(0g‘𝐿)})    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝑍))    &   𝐽 = (𝑝 ∈ (Base‘𝑄) ↦ ∪ (𝐺 “ 𝑝))    ⇒   (𝜑 → 𝐺 ∈ (𝑃 LMHom ((subringAlg ‘𝐿)‘𝐹)))
 
2-Apr-2025algextdeglem1 33062 Lemma for algextdeg 33070. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝐴})))    &   𝐷 = ( deg1 ‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   (𝜑 → (𝐿 ↾s 𝐹) = 𝐾)
 
2-Apr-2025minplym1p 33061 A minimal polynomial is monic. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑍 = (0g‘(Poly1‘𝐸))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    &   𝑈 = (Monic1p‘(𝐸 ↾s 𝐹))    ⇒   (𝜑 → (𝑀‘𝐴) ∈ 𝑈)
 
2-Apr-2025evls1fldgencl 33033 Closure of the subring polynomial evaluation in the field extention. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝐸)    &   𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝑈)    ⇒   (𝜑 → ((𝑂‘𝐺)‘𝐴) ∈ (𝐸 fldGen (𝐹 ∪ {𝐴})))
 
2-Apr-2025lsssra 32963 A subring is a subspace of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑊 = ((subringAlg ‘𝑅)‘𝐶)    &   𝐴 = (Base‘𝑅)    &   𝑆 = (𝑅 ↾s 𝐵)    &   (𝜑 → 𝐵 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝐶 ∈ (SubRing‘𝑆))    ⇒   (𝜑 → 𝐵 ∈ (LSubSp‘𝑊))
 
2-Apr-2025resssra 32962 The subring algebra of a restricted structure is the restriction of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐴 = (Base‘𝑅)    &   𝑆 = (𝑅 ↾s 𝐵)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐶 ⊆ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝑉)    ⇒   (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
 
2-Apr-2025r1pquslmic 32956 The univariate polynomial remainder ring (𝐹 “s 𝑃) is module isomorphic with the quotient ring. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝐸 = (rem1p‘𝑅)    &   𝑁 = (Unic1p‘𝑅)    &   𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ 𝑁)    &    0 = (0g‘𝑃)    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝑃 /s (𝑃 ~QG 𝐾))    ⇒   (𝜑 → 𝑄 ≃𝑚 (𝐹 “s 𝑃))
 
2-Apr-2025r1plmhm 32955 The univariate polynomial remainder function 𝐹 is a module homomorphism. Its image (𝐹 “s 𝑃) is sometimes called the "ring of remainders" (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝐸 = (rem1p‘𝑅)    &   𝑁 = (Unic1p‘𝑅)    &   𝐹 = (𝑓 ∈ 𝑈 ↦ (𝑓𝐸𝑀))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ 𝑁)    ⇒   (𝜑 → 𝐹 ∈ (𝑃 LMHom (𝐹 “s 𝑃)))
 
2-Apr-2025r1pid2 32954 Identity law for polynomial remainder operation: it leaves a polynomial 𝐴 unchanged iff the degree of 𝐴 is less than the degree of the divisor 𝐵. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &   𝐸 = (rem1p‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   𝐷 = ( deg1 ‘𝑅)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑁)    ⇒   (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵)))
 
2-Apr-2025r1padd1 32953 Addition property of the polynomial remainder operation, similar to modadd1 13877. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &   𝐸 = (rem1p‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑁)    &   (𝜑 → (𝐴𝐸𝐷) = (𝐵𝐸𝐷))    &    + = (+g‘𝑃)    &   (𝜑 → 𝐵 ∈ 𝑈)    &   (𝜑 → 𝐶 ∈ 𝑈)    ⇒   (𝜑 → ((𝐴 + 𝐶)𝐸𝐷) = ((𝐵 + 𝐶)𝐸𝐷))
 
2-Apr-2025r1pcyc 32952 The polynomial remainder operation is periodic. See modcyc 13875. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &   𝐸 = (rem1p‘𝑅)    &    + = (+g‘𝑃)    &    · = (.r‘𝑃)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑁)    &   (𝜑 → 𝐶 ∈ 𝑈)    ⇒   (𝜑 → ((𝐴 + (𝐶 · 𝐵))𝐸𝐵) = (𝐴𝐸𝐵))
 
2-Apr-2025r1p0 32951 Polynomial remainder operation of a division of the zero polynomial. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &   𝐸 = (rem1p‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐷 ∈ 𝑁)    &    0 = (0g‘𝑃)    ⇒   (𝜑 → ( 0 𝐸𝐷) = 0 )
 
2-Apr-2025r1pvsca 32950 Scalar multiplication property of the polynomial remainder operation. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &   𝐸 = (rem1p‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑁)    &    × = ( ·𝑠 ‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐵 ∈ 𝐾)    ⇒   (𝜑 → ((𝐵 × 𝐴)𝐸𝐷) = (𝐵 × (𝐴𝐸𝐷)))
 
2-Apr-2025q1pvsca 32949 Scalar multiplication property of the polynomial division. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &    / = (quot1p‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐶 ∈ 𝑁)    &    × = ( ·𝑠 ‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐵 ∈ 𝐾)    ⇒   (𝜑 → ((𝐵 × 𝐴) / 𝐶) = (𝐵 × (𝐴 / 𝐶)))
 
2-Apr-2025q1pdir 32948 Distribution of univariate polynomial quotient over addition. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &    / = (quot1p‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐶 ∈ 𝑁)    &   (𝜑 → 𝐵 ∈ 𝑈)    &    + = (+g‘𝑃)    ⇒   (𝜑 → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))
 
2-Apr-2025deg1addlt 32945 If both factors have degree bounded by 𝐿, then the sum of the polynomials also has degree bounded by 𝐿. See also deg1addle 25854. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑌 = (Poly1‘𝑅)    &   𝐷 = ( deg1 ‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   𝐵 = (Base‘𝑌)    &    + = (+g‘𝑌)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    &   (𝜑 → 𝐿 ∈ ℝ*)    &   (𝜑 → (𝐷‘𝐹) < 𝐿)    &   (𝜑 → (𝐷‘𝐺) < 𝐿)    ⇒   (𝜑 → (𝐷‘(𝐹 + 𝐺)) < 𝐿)
 
2-Apr-2025ply1degleel 32941 Characterize elementhood in the set 𝑆 of polynomials of degree less than 𝑁. (Contributed by Thierry Arnoux, 2-Apr-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)))
 
2-Apr-2025imaslmhm 32742 Given a function 𝐹 with homomorphic properties, build the image of a left module. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &    + = (+g‘𝑊)    &   ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   ((𝜑 ∧ (𝑘 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 × 𝑎)) = (𝐹‘(𝑘 × 𝑏))))    &   (𝜑 → 𝑊 ∈ LMod)    &    × = ( ·𝑠 ‘𝑊)    ⇒   (𝜑 → ((𝐹 “s 𝑊) ∈ LMod ∧ 𝐹 ∈ (𝑊 LMHom (𝐹 “s 𝑊))))
 
2-Apr-2025imasrhm 32741 Given a function 𝐹 with homomorphic properties, build the image of a ring. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &    + = (+g‘𝑊)    &   ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &    · = (.r‘𝑊)    &   ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 → 𝑊 ∈ Ring)    ⇒   (𝜑 → ((𝐹 “s 𝑊) ∈ Ring ∧ 𝐹 ∈ (𝑊 RingHom (𝐹 “s 𝑊))))
 
2-Apr-2025imasghm 32740 Given a function 𝐹 with homomorphic properties, build the image of a group. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &    + = (+g‘𝑊)    &   ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑 → 𝑊 ∈ Grp)    ⇒   (𝜑 → ((𝐹 “s 𝑊) ∈ Grp ∧ 𝐹 ∈ (𝑊 GrpHom (𝐹 “s 𝑊))))
 
2-Apr-2025imasmhm 32739 Given a function 𝐹 with homomorphic properties, build the image of a monoid. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &    + = (+g‘𝑊)    &   ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑 → 𝑊 ∈ Mnd)    ⇒   (𝜑 → ((𝐹 “s 𝑊) ∈ Mnd ∧ 𝐹 ∈ (𝑊 MndHom (𝐹 “s 𝑊))))
 
2-Apr-2025lmhmimasvsca 32467 Value of the scalar product of the surjective image of a module. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑊 = (𝐹 “s 𝑉)    &   𝐵 = (Base‘𝑉)    &   𝐶 = (Base‘𝑊)    &   (𝜑 → 𝑋 ∈ 𝐾)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹:𝐵–onto→𝐶)    &   (𝜑 → 𝐹 ∈ (𝑉 LMHom 𝑊))    &    · = ( ·𝑠 ‘𝑉)    &    × = ( ·𝑠 ‘𝑊)    &   𝐾 = (Base‘(Scalar‘𝑉))    ⇒   (𝜑 → (𝑋 × (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
 
2-Apr-2025mhmimasplusg 32466 Value of the operation of the surjective image. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝑊 = (𝐹 “s 𝑉)    &   𝐵 = (Base‘𝑉)    &   𝐶 = (Base‘𝑊)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹:𝐵–onto→𝐶)    &   (𝜑 → 𝐹 ∈ (𝑉 MndHom 𝑊))    &    + = (+g‘𝑉)    &    ⨣ = (+g‘𝑊)    ⇒   (𝜑 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)) = (𝐹‘(𝑋 + 𝑌)))
 
2-Apr-2025lmhmghmd 32465 A module homomorphism is a group homomorphism. (Contributed by Thierry Arnoux, 2-Apr-2025.)
(𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    ⇒   (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
 
2-Apr-2025n0ons 27944 A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.)
(𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
 
2-Apr-2025n0scut 27943 A cut form for surreal naturals. (Contributed by Scott Fenton, 2-Apr-2025.)
(𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
 
1-Apr-2025aprilfools2025 41718 An abuse of notation. (Contributed by Prof. Loof Lirpa, 1-Apr-2025.) (New usage is discouraged.) (Proof modification is discouraged.)
{⟨“𝐴𝑝𝑟𝑖𝑙”⟩, ⟨“𝑓𝑜𝑜𝑙𝑠!”⟩} ∈ V
 
1-Apr-2025iddii 41708 Version of a1ii 2 with the hypotheses switched. The first hypothesis is redundant so this theorem should not normally appear in a proof. Inference associated with idd 24. (Contributed by SN, 1-Apr-2025.) (New usage is discouraged.)
𝜑    &   𝜓    ⇒   𝜓
 
1-Apr-2025nrt2irr 29993 The 𝑁-th root of 2 is irrational for 𝑁 greater than 2. For 𝑁 = 2, see sqrt2irr 16196. This short and rather elegant proof has the minor disadvantage that it refers to ax-flt 29992, which is still to be formalized. For a proof not requiring ax-flt 29992, see rtprmirr 41539. (Contributed by Prof. Loof Lirpa, 1-Apr-2025.) (Proof modification is discouraged.)
(𝑁 ∈ (ℤ≥‘3) → ¬ (2↑𝑐(1 / 𝑁)) ∈ ℚ)
 
1-Apr-2025ax-flt 29992 This factoid is e.g. useful for nrt2irr 29993. Andrew has a proof, I'll have a go at formalizing it after my coffee break. In the mean time let's add it as an axiom. (Contributed by Prof. Loof Lirpa, 1-Apr-2025.) (New usage is discouraged.)
((𝑁 ∈ (ℤ≥‘3) ∧ (𝑋 ∈ ℕ ∧ 𝑌 ∈ ℕ ∧ 𝑍 ∈ ℕ)) → ((𝑋↑𝑁) + (𝑌↑𝑁)) ≠ (𝑍↑𝑁))
 
1-Apr-2025recxpf1lem 26473 Complex exponentiation on positive real numbers is a one-to-one function. (Contributed by Thierry Arnoux, 1-Apr-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐵)    &   (𝜑 → 𝐶 ∈ ℝ+)    ⇒   (𝜑 → (𝐴 = 𝐵 ↔ (𝐴↑𝑐𝐶) = (𝐵↑𝑐𝐶)))
 
31-Mar-2025gg-cnfld1 35487 One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) Use mpocnfldmul 35477. (Revised by GG, 31-Mar-2025.)
1 = (1r‘ℂfld)
 
31-Mar-2025gg-cncrng 35486 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) Use mpocnfldmul 35477. (Revised by GG, 31-Mar-2025.)
ℂfld ∈ CRing
 
31-Mar-2025gg-cffldtocusgr 35485 The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (♯‘𝑥) = 2}    &   𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    ⇒   𝐺 ∈ ComplUSGraph
 
31-Mar-2025gg-cnfldfunALT 35484 The field of complex numbers is a function. Alternate proof of cnfldfun 21156 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.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
Fun ℂfld
 
31-Mar-2025gg-cnfldfun 35483 The field of complex numbers is a function. The proof is much shorter than the proof of gg-cnfldfunALT 35484 by using cnfldstr 21146 and structn0fun 17088: in addition, it must be shown that ∅ ∉ ℂfld. (Contributed by AV, 18-Nov-2021.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
Fun ℂfld
 
31-Mar-2025gg-cnfldunif 35482 The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
(metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
 
31-Mar-2025gg-cnfldds 35481 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
(abs ∘ − ) = (dist‘ℂfld)
 
31-Mar-2025gg-cnfldle 35480 The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
≤ = (le‘ℂfld)
 
31-Mar-2025gg-cnfldtset 35479 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
(MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
 
31-Mar-2025gg-cnfldcj 35478 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
∗ = (*𝑟‘ℂfld)
 
31-Mar-2025mpocnfldmul 35477 The multiplication operation of the field of complex numbers. Version of cnfldmul 21150 using maps-to notation. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
 
31-Mar-2025mpocnfldadd 35476 The addition operation of the field of complex numbers. Version of cnfldadd 21149 using maps-to notation. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
 
31-Mar-2025gg-cnfldbas 35475 The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
ℂ = (Base‘ℂfld)
 
31-Mar-2025gg-cnfldstr 35474 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21145. (Revised by GG, 31-Mar-2025.)
ℂfld Struct ⟨1, 13⟩
 
31-Mar-2025gg-dfcnfld 35473 Alternative definition of the field of complex numbers. Equivalent to df-cnfld 21145. (Contributed by GG, 31-Mar-2025.)
ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
 
31-Mar-2025mpoaddf 35471 Addition is an operation on complex numbers. Version of ax-addf 11191 using maps-to notation, proved from the axioms of set theory and ax-addcl 11172. (Contributed by GG, 31-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)):(ℂ × ℂ)⟶ℂ
 
31-Mar-20252idlcpbl 21021 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
31-Mar-2025lidlmcl 20989 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof shortened by AV, 31-Mar-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
 
30-Mar-2025sum9cubes 41716 The sum of the first nine perfect cubes is 2025. (Contributed by SN, 30-Mar-2025.)
Σ𝑘 ∈ (1...9)(𝑘↑3) = 2025
 
30-Mar-2025sq45 41715 45 squared is 2025. (Contributed by SN, 30-Mar-2025.)
(45↑2) = 2025
 
30-Mar-2025sq9 41506 The square of 9 is 81. (Contributed by SN, 30-Mar-2025.)
(9↑2) = 81
 
30-Mar-20254t5e20 41505 4 times 5 equals 20. (Contributed by SN, 30-Mar-2025.)
(4 · 5) = 20
 
30-Mar-2025subrgmcl 20474 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 30-Mar-2025.)
· = (.r‘𝑅)    ⇒   ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
30-Mar-2025subrgsubrng 20468 A subring of a unital ring is a subring of a non-unital ring. (Contributed by AV, 30-Mar-2025.)
(𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubRng‘𝑅))
 
30-Mar-2025rimisrngim 20389 Each unital ring isomorphism is a non-unital ring isomorphism. (Contributed by AV, 30-Mar-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RngIso 𝑆))
 
30-Mar-2025opprring 20238 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) (Proof shortened by AV, 30-Mar-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝑂 ∈ Ring)
 
30-Mar-2025prdsmulrcl 20208 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.) (Proof shortened by AV, 30-Mar-2025.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = (.r‘𝑌)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑅:𝐼⟶Ring)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
 
30-Mar-2025ringm2neg 20194 Double negation of a product in a ring. (mul2neg 11657 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) (Proof shortened by AV, 30-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌))
 
30-Mar-2025ringrz 20182 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) (Proof shortened by AV, 30-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
 
30-Mar-2025ringlz 20181 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (Proof shortened by AV, 30-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 )
 
29-Mar-2025onscutleft 27928 A surreal ordinal is equal to the cut of its left set and the empty set. (Contributed by Scott Fenton, 29-Mar-2025.)
(𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
 
25-Mar-2025lmicdim 32977 Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Mar-2025.)
(𝜑 → 𝑆 ≃𝑚 𝑇)    &   (𝜑 → 𝑆 ∈ LVec)    ⇒   (𝜑 → (dim‘𝑆) = (dim‘𝑇))
 
25-Mar-2025pzriprng1 21267 The ring unity of the ring (ℤring ×s ℤring). Direct proof in contrast to pzriprng1ALT 21265. (Contributed by AV, 25-Mar-2025.)
(1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
 
25-Mar-2025pzriprng 21266 The non-unital ring (ℤring ×s ℤring) is unital. Direct proof in contrast to pzriprngALT 21264. (Contributed by AV, 25-Mar-2025.)
(ℤring ×s ℤring) ∈ Ring
 
25-Mar-2025ceqsal1t 3503 One direction of ceqsalt 3504 is based on fewer assumptions and fewer axioms. It is at the same time the reverse direction of vtoclgft 3539. Extracted from a proof of ceqsalt 3504. (Contributed by Wolf Lammen, 25-Mar-2025.)
((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
 
24-Mar-2025pzriprng1ALT 21265 The ring unity of the ring (ℤring ×s ℤring) constructed from the ring unity of the two-sided ideal (ℤ × {0}) and the ring unity of the quotient of the ring and the ideal (using ring2idlqus1 21078). (Contributed by AV, 24-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
 
24-Mar-2025zringsub 21226 The subtraction of elements in the ring of integers. (Contributed by AV, 24-Mar-2025.)
− = (-g‘ℤring)    ⇒   ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
 
24-Mar-2025rngqiprngimfo 21060 𝐹 is a function from (the base set of) a non-unital ring onto the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → 𝐹:𝐵–onto→(𝐶 × 𝐼))
 
24-Mar-2025rngqiprnglinlem1 21050 Lemma 1 for rngqiprnglin 21061. (Contributed by AV, 28-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · ( 1 · 𝐶)) = ( 1 · (𝐴 · 𝐶)))
 
24-Mar-2025rngqiprngimfolem 21049 Lemma for rngqiprngimfo 21060. (Contributed by AV, 5-Mar-2025.) (Proof shortened by AV, 24-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    ⇒   ((𝜑 ∧ 𝐴 ∈ 𝐼 ∧ 𝐶 ∈ 𝐵) → ( 1 · ((𝐶(-g‘𝑅)( 1 · 𝐶))(+g‘𝑅)𝐴)) = 𝐴)
 
24-Mar-2025rngqiprngghmlem3 21048 Lemma 3 for rngqiprngghm 21058. (Contributed by AV, 25-Feb-2025.) (Proof shortened by AV, 24-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · (𝐴(+g‘𝑅)𝐶)) = (( 1 · 𝐴)(+g‘𝐽)( 1 · 𝐶)))
 
24-Mar-2025xpsgrpsub 18980 Value of the subtraction operation in a binary structure product of groups. (Contributed by AV, 24-Mar-2025.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑆 ∈ Grp)    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &    · = (-g‘𝑅)    &    × = (-g‘𝑆)    &    − = (-g‘𝑇)    ⇒   (𝜑 → (⟨𝐴, 𝐵⟩ − ⟨𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
 
23-Mar-2025pzriprngALT 21264 The non-unital ring (ℤring ×s ℤring) is unital because it has the two-sided ideal (ℤ × {0}), which is unital, and the quotient of the ring and the ideal is also unital (using ring2idlqusb 21069). (Contributed by AV, 23-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(ℤring ×s ℤring) ∈ Ring
 
23-Mar-2025pzriprnglem14 21263 Lemma 14 for pzriprng 21266: The ring unity of the ring 𝑄. (Contributed by AV, 23-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   (1r‘𝑄) = (ℤ × {1})
 
23-Mar-2025pzriprnglem13 21262 Lemma 13 for pzriprng 21266: 𝑄 is a unital ring. (Contributed by AV, 23-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   𝑄 ∈ Ring
 
23-Mar-2025pzriprnglem12 21261 Lemma 12 for pzriprng 21266: 𝑄 has a ring unity. (Contributed by AV, 23-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   (𝑋 ∈ (Base‘𝑄) → (((ℤ × {1})(.r‘𝑄)𝑋) = 𝑋 ∧ (𝑋(.r‘𝑄)(ℤ × {1})) = 𝑋))
 
22-Mar-2025sumcubes 41513 The sum of the first 𝑁 perfect cubes is the sum of the first 𝑁 nonnegative integers, squared. This is the Proof by Nicomachus from https://proofwiki.org/wiki/Sum_of_Sequence_of_Cubes using induction and index shifting to collect all the odd numbers. (Contributed by SN, 22-Mar-2025.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
 
22-Mar-2025fz1sump1 41510 Add one more term to a sum. Special case of fsump1 15706 generalized to 𝑁 ∈ ℕ0. (Contributed by SN, 22-Mar-2025.)
(𝜑 → 𝑁 ∈ ℕ0)    &   ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   (𝜑 → Σ𝑘 ∈ (1...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (1...𝑁)𝐴 + 𝐵))
 
22-Mar-2025irngnminplynz 33060 Integral elements have nonzero minimal polynomials. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑍 = (0g‘(Poly1‘𝐸))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   𝑀 = (𝐸 minPoly 𝐹)    &   (𝜑 → 𝐴 ∈ (𝐸 IntgRing 𝐹))    ⇒   (𝜑 → (𝑀‘𝐴) ≠ 𝑍)
 
22-Mar-2025minplyirred 33059 A nonzero minimal polynomial is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   𝑀 = (𝐸 minPoly 𝐹)    &   𝑍 = (0g‘𝑃)    &   (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    ⇒   (𝜑 → (𝑀‘𝐴) ∈ (Irred‘𝑃))
 
22-Mar-2025minplyirredlem 33058 Lemma for minplyirred 33059. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   𝑀 = (𝐸 minPoly 𝐹)    &   𝑍 = (0g‘𝑃)    &   (𝜑 → (𝑀‘𝐴) ≠ 𝑍)    &   (𝜑 → 𝐺 ∈ (Base‘𝑃))    &   (𝜑 → 𝐻 ∈ (Base‘𝑃))    &   (𝜑 → (𝐺(.r‘𝑃)𝐻) = (𝑀‘𝐴))    &   (𝜑 → ((𝑂‘𝐺)‘𝐴) = (0g‘𝐸))    &   (𝜑 → 𝐺 ≠ 𝑍)    &   (𝜑 → 𝐻 ≠ 𝑍)    ⇒   (𝜑 → 𝐻 ∈ (Unit‘𝑃))
 
22-Mar-2025minplycl 33056 The minimal polynomial is a polynomial. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &    0 = (0g‘𝐸)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   𝐾 = (RSpan‘𝑃)    &   𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    &   𝑀 = (𝐸 minPoly 𝐹)    ⇒   (𝜑 → (𝑀‘𝐴) ∈ (Base‘𝑃))
 
22-Mar-2025ply1annnr 33053 The set 𝑄 of polynomials annihilating an element 𝐴 is not the whole polynomial ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝐴 ∈ 𝐵)    &    0 = (0g‘𝑅)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ NzRing)    ⇒   (𝜑 → 𝑄 ≠ 𝑈)
 
22-Mar-2025evls1fvcl 33047 Variant of fveval1fvcl 22072 for the subring evaluation function evalSub1 (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑀 ∈ 𝑈)    ⇒   (𝜑 → ((𝑂‘𝑀)‘𝑌) ∈ 𝐵)
 
22-Mar-2025rlmdim 32982 The left vector space induced by a ring over itself has dimension 1. (Contributed by Thierry Arnoux, 5-Aug-2023.) Generalize to division rings. (Revised by SN, 22-Mar-2025.)
𝑉 = (ringLMod‘𝐹)    ⇒   (𝐹 ∈ DivRing → (dim‘𝑉) = 1)
 
22-Mar-2025deg1le0eq0 32929 A polynomial with nonpositive degree is the zero polynomial iff its constant term is zero. Biconditional version of deg1scl 25866. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐷 = ( deg1 ‘𝑅)    &   𝑃 = (Poly1‘𝑅)    &    0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑂 = (0g‘𝑃)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → (𝐷‘𝐹) ≤ 0)    ⇒   (𝜑 → (𝐹 = 𝑂 ↔ ((coe1‘𝐹)‘0) = 0 ))
 
22-Mar-2025ply1asclunit 32928 A non-zero scalar polynomial is a unit. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Field)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑌 ≠ 0 )    ⇒   (𝜑 → (𝐴‘𝑌) ∈ (Unit‘𝑃))
 
22-Mar-2025evls1fvf 32916 The subring evaluation function for a univariate polynomial as a function, with domain and codomain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑃 = (Poly1‘(𝑅 ↾s 𝑆))    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑄 ∈ 𝑈)    ⇒   (𝜑 → (𝑂‘𝑄):𝐵⟶𝐵)
 
22-Mar-2025mxidlirred 32862 In a principal ideal domain, maximal ideals are exactly the ideals generated by irreducible elements. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    0 = (0g‘𝑅)    &   𝑀 = (𝐾‘{𝑋})    &   (𝜑 → 𝑅 ∈ PID)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    &   (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))    ⇒   (𝜑 → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑋 ∈ (Irred‘𝑅)))
 
22-Mar-2025mxidlirredi 32861 In an integral domain, the generator of a maximal ideal is irreducible. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    0 = (0g‘𝑅)    &   𝑀 = (𝐾‘{𝑋})    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    ⇒   (𝜑 → 𝑋 ∈ (Irred‘𝑅))
 
22-Mar-2025rhmquskerlem 32817 The mapping 𝐽 induced by a ring homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 22-Mar-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   (𝜑 → 𝐺 ∈ CRing)    ⇒   (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
 
22-Mar-2025unitpidl1 32816 The ideal 𝐼 generated by an element 𝑋 of an integral domain 𝑅 is the unit ideal 𝐵 iff 𝑋 is a ring unit. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝑈 = (Unit‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐼 = (𝐾‘{𝑋})    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → (𝐼 = 𝐵 ↔ 𝑋 ∈ 𝑈))
 
22-Mar-2025pidlnzb 32814 A principal ideal is nonzero iff it is generated by a nonzero elements (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐾 = (RSpan‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ≠ 0 ↔ (𝐾‘{𝑋}) ≠ { 0 }))
 
22-Mar-2025ghmquskerlem3 32805 The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 22-Mar-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
 
22-Mar-2025qusrn 32794 The natural map from elements to their cosets is surjective. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝐺)    &   𝑈 = (𝐵 / (𝐺 ~QG 𝑁))    &   𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (𝜑 → ran 𝐹 = 𝑈)
 
22-Mar-2025qusbas2 32791 Alternate definition of the group quotient set, as the set of all cosets of the form ({𝑥} ⊕ 𝑁). (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝐺)    &    ⊕ = (LSSum‘𝐺)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ (SubGrp‘𝐺))    ⇒   (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = ran (𝑥 ∈ 𝐵 ↦ ({𝑥} ⊕ 𝑁)))
 
22-Mar-2025rspsnasso 32766 Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff the ideals they generate are equal. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ (𝐾‘{𝑌}) = (𝐾‘{𝑋})))
 
22-Mar-2025dvdsrspss 32765 In a ring, an element 𝑋 divides 𝑌 iff the ideal generated by 𝑌 is a subset of the ideal generated by 𝑋 (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (𝑋 ∥ 𝑌 ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})))
 
22-Mar-2025dvdsruasso 32764 Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff they are unit multiples of each other. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ ∃𝑢 ∈ 𝑈 (𝑢 · 𝑋) = 𝑌))
 
22-Mar-2025dvdsruassoi 32763 If two elements 𝑋 and 𝑌 of a ring 𝑅 are unit multiples, then they are associates, i.e. each divides the other. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑉 ∈ 𝑈)    &   (𝜑 → (𝑉 · 𝑋) = 𝑌)    ⇒   (𝜑 → (𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋))
 
22-Mar-2025df-edom 32663 Define Euclidean Domains. (Contributed by Thierry Arnoux, 22-Mar-2025.)
EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0g‘𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ (𝑣 ∖ {(0g‘𝑑)})∃𝑞 ∈ 𝑣 ∃𝑟 ∈ 𝑣 (𝑎 = ((𝑏(.r‘𝑑)𝑞)(+g‘𝑑)𝑟) ∧ (𝑟 = (0g‘𝑑) ∨ (𝑒‘𝑟) < (𝑒‘𝑏))))}
 
22-Mar-2025eufid 32661 Utility theorem: index-independent form of df-euf 32659. (Contributed by Thierry Arnoux, 22-Mar-2025.)
EuclF = Slot (EuclF‘ndx)
 
22-Mar-2025eufndx 32660 Index value of the Euclidean function slot. Use ndxarg 17133. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.)
(EuclF‘ndx) = 21
 
22-Mar-2025df-euf 32659 Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 32661 instead. (New usage is discouraged.)
EuclF = Slot 21
 
22-Mar-2025idomrcan 32647 Right-cancellation law for integral domains. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → (𝑌 · 𝑋) = (𝑍 · 𝑋))    ⇒   (𝜑 → 𝑌 = 𝑍)
 
22-Mar-2025domnlcan 32646 Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍))    ⇒   (𝜑 → 𝑌 = 𝑍)
 
22-Mar-2025idomringd 32645 An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Ring)
 
22-Mar-2025idomdomd 32644 An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ Domn)
 
22-Mar-2025sltonex 27927 The class of ordinals less than any particular surreal is a set. Theorem 14 of [Conway] p. 27. (Contributed by Scott Fenton, 22-Mar-2025.)
(𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ V)
 
22-Mar-2025sltonold 27926 The class of ordinals less than any surreal is a subset of that surreal's old set. (Contributed by Scott Fenton, 22-Mar-2025.)
(𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ ( O ‘( bday ‘𝐴)))
 
22-Mar-2025pzriprnglem11 21260 Lemma 11 for pzriprng 21266: The base set of the quotient of 𝑅 and 𝐽. (Contributed by AV, 22-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
 
22-Mar-2025pzriprnglem10 21259 Lemma 10 for pzriprng 21266: The equivalence classes of 𝑅 modulo 𝐽. (Contributed by AV, 22-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    ⇒   ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = (ℤ × {𝑌}))
 
22-Mar-2025pzriprnglem9 21258 Lemma 9 for pzriprng 21266: The ring unity of the ring 𝐽. (Contributed by AV, 22-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    &    1 = (1r‘𝐽)    ⇒    1 = ⟨1, 0⟩
 
22-Mar-2025fsump1 15706 The addition of the next term in a finite sum of 𝐴(𝑘) is the current term plus 𝐵 i.e. 𝐴(𝑁 + 1). (Contributed by NM, 4-Nov-2005.) (Revised by Mario Carneiro, 21-Apr-2014.) (Proof shortened by SN, 22-Mar-2025.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   (𝜑 → Σ𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (𝑀...𝑁)𝐴 + 𝐵))
 
21-Mar-2025nicomachus 41512 Nicomachus's Theorem. The sum of the odd numbers from 𝑁↑2 − 𝑁 + 1 to 𝑁↑2 + 𝑁 − 1 is 𝑁↑3. Proof 2 from https://proofwiki.org/wiki/Nicomachus%27s_Theorem. (Contributed by SN, 21-Mar-2025.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(((𝑁↑2) − 𝑁) + ((2 · 𝑘) − 1)) = (𝑁↑3))
 
21-Mar-2025oddnumth 41511 The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15810. (Contributed by SN, 21-Mar-2025.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2))
 
21-Mar-2025fz1sumconst 41509 The sum of 𝑁 constant terms (𝑘 is not free in 𝐶). (Contributed by SN, 21-Mar-2025.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐶 ∈ ℂ)    ⇒   (𝜑 → Σ𝑘 ∈ (1...𝑁)𝐶 = (𝑁 · 𝐶))
 
21-Mar-2025slelss 27639 If two surreals 𝐴 and 𝐵 share a birthday, then 𝐴 ≤s 𝐵 if and only if the left set of 𝐴 is a non-strict subset of the left set of 𝐵. (Contributed by Scott Fenton, 21-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ ( L ‘𝐴) ⊆ ( L ‘𝐵)))
 
21-Mar-2025sraassa 21642 The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof shortened by SN, 21-Mar-2025.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)    ⇒   ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
 
21-Mar-2025sraassab 21641 A subring algebra is an associative algebra if and only if the subring is included in the ring's center. (Contributed by SN, 21-Mar-2025.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   𝑍 = (Cntr‘(mulGrp‘𝑊))    &   (𝜑 → 𝑊 ∈ Ring)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑊))    ⇒   (𝜑 → (𝐴 ∈ AssAlg ↔ 𝑆 ⊆ 𝑍))
 
21-Mar-2025pzriprnglem8 21257 Lemma 8 for pzriprng 21266: 𝐼 resp. 𝐽 is a two-sided ideal of the non-unital ring 𝑅. (Contributed by AV, 21-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    ⇒   𝐼 ∈ (2Ideal‘𝑅)
 
21-Mar-2025df2idl2rng 21037 Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼)))
 
21-Mar-2025isridlrng 21031 A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼))
 
21-Mar-2025dflidl2rng 21030 Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝐼 ∈ (SubGrp‘𝑅)) → (𝐼 ∈ 𝑈 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼))
 
21-Mar-2025crngbascntr 20149 The base set of a commutative ring is its center. (Contributed by SN, 21-Mar-2025.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntr‘(mulGrp‘𝐺))    ⇒   (𝐺 ∈ CRing → 𝐵 = 𝑍)
 
21-Mar-2025cmnbascntr 19714 The base set of a commutative monoid is its center. (Contributed by SN, 21-Mar-2025.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntr‘𝐺)    ⇒   (𝐺 ∈ CMnd → 𝐵 = 𝑍)
 
21-Mar-2025elcntr 19235 Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)
𝐵 = (Base‘𝑀)    &    + = (+g‘𝑀)    &   𝑍 = (Cntr‘𝑀)    ⇒   (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
 
21-Mar-2025cgsex4g 3519 An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) Avoid ax-10 2135, ax-11 2152. (Revised by Gino Giotto, 28-Jun-2024.) Avoid ax-9 2114, ax-ext 2701. (Revised by Wolf Lammen, 21-Mar-2025.)
(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → 𝜒)    &   (𝜒 → (𝜑 ↔ 𝜓))    ⇒   (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤(𝜒 ∧ 𝜑) ↔ 𝜓))
 
19-Mar-2025ig1pmindeg 32947 The polynomial ideal generator is of minimum degree. (Contributed by Thierry Arnoux, 19-Mar-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐺 = (idlGen1p‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑃))    &   𝐷 = ( deg1 ‘𝑅)    &    0 = (0g‘𝑃)    &   (𝜑 → 𝐹 ∈ 𝐼)    &   (𝜑 → 𝐹 ≠ 0 )    ⇒   (𝜑 → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹))
 
19-Mar-2025ig1pnunit 32946 The polynomial ideal generator is not a unit polynomial. (Contributed by Thierry Arnoux, 19-Mar-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐺 = (idlGen1p‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑃))    &   (𝜑 → 𝐼 ≠ 𝑈)    ⇒   (𝜑 → ¬ (𝐺‘𝐼) ∈ (Unit‘𝑃))
 
19-Mar-2025lidlunitel 32815 If an ideal 𝐼 contains a unit 𝐽, then it is the whole ring. (Contributed by Thierry Arnoux, 19-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑 → 𝐽 ∈ 𝑈)    &   (𝜑 → 𝐽 ∈ 𝐼)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    ⇒   (𝜑 → 𝐼 = 𝐵)
 
19-Mar-2025elons2d 27925 The cut of any set of surreals and the empty set is a surreal ordinal. (Contributed by Scott Fenton, 19-Mar-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐴 ⊆ No )    &   (𝜑 → 𝑋 = (𝐴 |s ∅))    ⇒   (𝜑 → 𝑋 ∈ Ons)
 
19-Mar-2025elons2 27924 A surreal is ordinal iff it is the cut of some set of surreals and the empty set. Definition from [Conway] p. 27. (Contributed by Scott Fenton, 19-Mar-2025.)
(𝐴 ∈ Ons ↔ ∃𝑎 ∈ 𝒫 No 𝐴 = (𝑎 |s ∅))
 
19-Mar-2025pzriprnglem7 21256 Lemma 7 for pzriprng 21266: 𝐽 is a unital ring. (Contributed by AV, 19-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    ⇒   𝐽 ∈ Ring
 
19-Mar-2025pzriprnglem6 21255 Lemma 6 for pzriprng 21266: 𝐽 has a ring unity. (Contributed by AV, 19-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    &   𝐽 = (𝑅 ↾s 𝐼)    ⇒   (𝑋 ∈ 𝐼 → ((⟨1, 0⟩(.r‘𝐽)𝑋) = 𝑋 ∧ (𝑋(.r‘𝐽)⟨1, 0⟩) = 𝑋))
 
18-Mar-2025currybi 34967 Biconditional version of Curry's paradox. If some proposition 𝜑 amounts to the self-referential statement "This very statement is equivalent to 𝜓", then 𝜓 is true. See bj-currypara 35739 in BJ's mathbox for the classical version. (Contributed by Adrian Ducourtial, 18-Mar-2025.)
((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)
 
18-Mar-20251ons 27923 Surreal one is a surreal ordinal. (Contributed by Scott Fenton, 18-Mar-2025.)
1s ∈ Ons
 
18-Mar-20250ons 27922 Surreal zero is a surreal ordinal. (Contributed by Scott Fenton, 18-Mar-2025.)
0s ∈ Ons
 
18-Mar-2025onsno 27921 A surreal ordinal is a surreal. (Contributed by Scott Fenton, 18-Mar-2025.)
(𝐴 ∈ Ons → 𝐴 ∈ No )
 
18-Mar-2025onssno 27920 The surreal ordinals are a subclass of the surreals. (Contributed by Scott Fenton, 18-Mar-2025.)
Ons ⊆ No
 
18-Mar-2025elons 27919 Membership in the class of surreal ordinals. (Contributed by Scott Fenton, 18-Mar-2025.)
(𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))
 
18-Mar-2025df-ons 27918 Define the surreal ordinals. These are the maximum members of each generation of surreals. Variant of definition from [Conway] p. 27. (Contributed by Scott Fenton, 18-Mar-2025.)
Ons = {𝑥 ∈ No ∣ ( R ‘𝑥) = ∅}
 
18-Mar-2025pzriprnglem5 21254 Lemma 5 for pzriprng 21266: 𝐼 is a subring of the non-unital ring 𝑅. (Contributed by AV, 18-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    ⇒   𝐼 ∈ (SubRng‘𝑅)
 
18-Mar-2025pzriprnglem4 21253 Lemma 4 for pzriprng 21266: 𝐼 is a subgroup of 𝑅. (Contributed by AV, 18-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    ⇒   𝐼 ∈ (SubGrp‘𝑅)
 
18-Mar-2025pzriprnglem3 21252 Lemma 3 for pzriprng 21266: An element of 𝐼 is an ordered pair. (Contributed by AV, 18-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    &   𝐼 = (ℤ × {0})    ⇒   (𝑋 ∈ 𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩)
 
18-Mar-2025xpsinv 18979 Value of the negation operation in a binary structure product. (Contributed by AV, 18-Mar-2025.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑆 ∈ Grp)    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   𝑀 = (invg‘𝑅)    &   𝑁 = (invg‘𝑆)    &   𝐼 = (invg‘𝑇)    ⇒   (𝜑 → (𝐼‘⟨𝐴, 𝐵⟩) = ⟨(𝑀‘𝐴), (𝑁‘𝐵)⟩)
 
17-Mar-2025n0sind 27942 Principle of Mathematical Induction (inference schema). Compare nnind 12234 and finds 7891. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝑥 = 0s → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   𝜓    &   (𝑦 ∈ ℕ0s → (𝜒 → 𝜃))    ⇒   (𝐴 ∈ ℕ0s → 𝜏)
 
17-Mar-2025dfn0s2 27941 Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
 
17-Mar-2025peano2n0s 27940 Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)
 
17-Mar-20250n0s 27939 Peano postulate: 0s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)
0s ∈ ℕ0s
 
17-Mar-2025nnssno 27938 The positive surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕs ⊆ No
 
17-Mar-2025nnssn0s 27937 The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕs ⊆ ℕ0s
 
17-Mar-2025n0ssno 27936 The non-negative surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕ0s ⊆ No
 
17-Mar-2025peano5n0s 27935 Peano's inductive postulate for non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
(( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s ⊆ 𝐴)
 
17-Mar-2025nnsex 27934 The set of all positive surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕs ∈ V
 
17-Mar-2025n0sex 27933 The set of all non-negative surreal integers exists. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕ0s ∈ V
 
17-Mar-2025df-nns 27932 Define the set of positive surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕs = (ℕ0s ∖ { 0s })
 
17-Mar-2025df-n0s 27931 Define the set of non-negative surreal integers. This set behaves similarly to ω and ℕ0, but it is a set of surreal numbers. Like those two sets, it satisfies the Peano axioms and is closed under (surreal) addition and multiplication. Compare df-nn 12217. (Contributed by Scott Fenton, 17-Mar-2025.)
ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
 
17-Mar-2025peano2no 27706 A theorem for surreals that is analagous to the second Peano postulate peano2 7883. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝐴 ∈ No → (𝐴 +s 1s ) ∈ No )
 
17-Mar-2025ssltsn 27530 Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    ⇒   (𝜑 → {𝐴} <<s {𝐵})
 
17-Mar-2025pzriprnglem2 21251 Lemma 2 for pzriprng 21266: The base set of 𝑅 is the cartesian product of the integers. (Contributed by AV, 17-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    ⇒   (Base‘𝑅) = (ℤ × ℤ)
 
17-Mar-2025pzriprnglem1 21250 Lemma 1 for pzriprng 21266: 𝑅 is a non-unital (actually a unital!) ring. (Contributed by AV, 17-Mar-2025.)
𝑅 = (ℤring ×s ℤring)    ⇒   𝑅 ∈ Rng
 
17-Mar-2025zringrng 21221 The ring of integers is a non-unital ring. (Contributed by AV, 17-Mar-2025.)
ℤring ∈ Rng
 
17-Mar-2025ring2idlqus1 21078 If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
· = (.r‘𝑅)    &    1 = (1r‘(𝑅 ↾s 𝐼))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ 𝐼 ∈ (2Ideal‘𝑅)) ∧ ((𝑅 ↾s 𝐼) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝐼)) ∈ Ring) ∧ 𝑈 ∈ (1r‘(𝑅 /s (𝑅 ~QG 𝐼)))) → (𝑅 ∈ Ring ∧ (1r‘𝑅) = ((𝑈 − ( 1 · 𝑈)) + 1 )))
 
17-Mar-2025rngqiprngu 21077 If a non-unital ring has a (two-sided) ideal which is unital, and the quotient of the ring and the ideal is also unital, then the ring is also unital with a ring unity which can be constructed from the ring unity of the ideal and a representative of the ring unity of the quotient. (Contributed by AV, 17-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &   𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   (𝜑 → (1r‘𝑅) = 𝑈)
 
16-Mar-2025gg-cxpcn 35470 Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.) Use mpomulcn 24605 instead of mulcn 24603 as direct dependency. (Revised by GG, 16-Mar-2025.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽 ↾t 𝐷)    ⇒   (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)
 
16-Mar-2025gg-dvfsumlem2 35469 Lemma for dvfsumrlim 25783. (Contributed by Mario Carneiro, 17-May-2016.) Use mpomulcn 24605 instead of mulcn 24603 as direct dependency. (Revised by GG, 16-Mar-2025.)
𝑆 = (𝑇(,)+∞)    &   𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝐷 ∈ ℝ)    &   (𝜑 → 𝑀 ≤ (𝐷 + 1))    &   (𝜑 → 𝑇 ∈ ℝ)    &   ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐵 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 𝐵))    &   (𝑥 = 𝑘 → 𝐵 = 𝐶)    &   (𝜑 → 𝑈 ∈ ℝ*)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆) ∧ (𝐷 ≤ 𝑥 ∧ 𝑥 ≤ 𝑘 ∧ 𝑘 ≤ 𝑈)) → 𝐶 ≤ 𝐵)    &   𝐻 = (𝑥 ∈ 𝑆 ↦ (((𝑥 − (⌊‘𝑥)) · 𝐵) + (Σ𝑘 ∈ (𝑀...(⌊‘𝑥))𝐶 − 𝐴)))    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    &   (𝜑 → 𝐷 ≤ 𝑋)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑌 ≤ 𝑈)    &   (𝜑 → 𝑌 ≤ ((⌊‘𝑋) + 1))    ⇒   (𝜑 → ((𝐻‘𝑌) ≤ (𝐻‘𝑋) ∧ ((𝐻‘𝑋) − ⦋𝑋 / 𝑥⦌𝐵) ≤ ((𝐻‘𝑌) − ⦋𝑌 / 𝑥⦌𝐵)))
 
16-Mar-2025gg-dvfsumle 35468 Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.) Use mpomulcn 24605 instead of mulcn 24603 as direct dependency. (Revised by GG, 16-Mar-2025.)
(𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))    &   ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)    &   (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))    &   (𝑥 = 𝑀 → 𝐴 = 𝐶)    &   (𝑥 = 𝑁 → 𝐴 = 𝐷)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)    &   ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)    ⇒   (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))
 
16-Mar-2025gg-cnfldex 35467 The field of complex numbers is a set. Alternative proof of cnfldex 21147. This version directly uses df-cnfld 21145, which is discouraged, however it saves all complex numbers axioms and ax-pow 5362. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 16-Mar-2025.)
ℂfld ∈ V
 
16-Mar-2025gg-cmvth 35466 Cauchy's Mean Value Theorem. If 𝐹, 𝐺 are real continuous functions on [𝐴, 𝐵] differentiable on (𝐴, 𝐵), then there is some 𝑥 ∈ (𝐴, 𝐵) such that 𝐹' (𝑥) / 𝐺' (𝑥) = (𝐹(𝐴) − 𝐹(𝐵)) / (𝐺(𝐴) − 𝐺(𝐵)). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 < 𝐵)    &   (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))    &   (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))    ⇒   (𝜑 → ∃𝑥 ∈ (𝐴(,)𝐵)(((𝐹‘𝐵) − (𝐹‘𝐴)) · ((ℝ D 𝐺)‘𝑥)) = (((𝐺‘𝐵) − (𝐺‘𝐴)) · ((ℝ D 𝐹)‘𝑥)))
 
16-Mar-2025gg-rmulccn 35465 Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = (topGen‘ran (,))    &   (𝜑 → 𝐶 ∈ ℝ)    ⇒   (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025gg-psercn2 35464 Since by pserulm 26170 the series converges uniformly, it is also continuous by ulmcn 26147. (Contributed by Mario Carneiro, 3-Mar-2015.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))    &   𝐹 = (𝑦 ∈ 𝑆 ↦ Σ𝑗 ∈ ℕ0 ((𝐺‘𝑦)‘𝑗))    &   (𝜑 → 𝐴:ℕ0⟶ℂ)    &   𝑅 = sup({𝑟 ∈ ℝ ∣ seq0( + , (𝐺‘𝑟)) ∈ dom ⇝ }, ℝ*, < )    &   𝐻 = (𝑖 ∈ ℕ0 ↦ (𝑦 ∈ 𝑆 ↦ (seq0( + , (𝐺‘𝑦))‘𝑖)))    &   (𝜑 → 𝑀 ∈ ℝ)    &   (𝜑 → 𝑀 < 𝑅)    &   (𝜑 → 𝑆 ⊆ (◡abs “ (0[,]𝑀)))    ⇒   (𝜑 → 𝐹 ∈ (𝑆–cn→ℂ))
 
16-Mar-2025drngmxidl 32867 The zero ideal is the only ideal of a division ring. (Contributed by Thierry Arnoux, 16-Mar-2025.)
0 = (0g‘𝑅)    ⇒   (𝑅 ∈ DivRing → (MaxIdeal‘𝑅) = {{ 0 }})
 
16-Mar-2025drng0mxidl 32866 In a division ring, the zero ideal is a maximal ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
0 = (0g‘𝑅)    ⇒   (𝑅 ∈ DivRing → { 0 } ∈ (MaxIdeal‘𝑅))
 
16-Mar-2025drnglidl1ne0 32865 In a nonzero ring, the zero ideal is different of the unit ideal. (Contributed by Thierry Arnoux, 16-Mar-2025.)
0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ NzRing → 𝐵 ≠ { 0 })
 
16-Mar-2025divsassd 27916 An associative law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) /su 𝐶) = (𝐴 ·s (𝐵 /su 𝐶)))
 
16-Mar-2025sltmuldiv2d 27915 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → ((𝐶 ·s 𝐴) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
 
16-Mar-2025sltmuldivd 27914 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → ((𝐴 ·s 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
 
16-Mar-2025sltdivmul2d 27913 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 ·s 𝐶)))
 
16-Mar-2025sltdivmuld 27912 Surreal less-than relationship between division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐶 ·s 𝐵)))
 
16-Mar-2025divscan1d 27911 A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    ⇒   (𝜑 → ((𝐴 /su 𝐵) ·s 𝐵) = 𝐴)
 
16-Mar-2025divscan2d 27910 A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    ⇒   (𝜑 → (𝐵 ·s (𝐴 /su 𝐵)) = 𝐴)
 
16-Mar-2025divscld 27909 Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    ⇒   (𝜑 → (𝐴 /su 𝐵) ∈ No )
 
16-Mar-2025divscl 27908 Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) ∈ No )
 
16-Mar-2025divsmuld 27907 Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    ⇒   (𝜑 → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
 
16-Mar-2025divsmul 27906 Relationship between surreal division and multiplication. (Contributed by Scott Fenton, 16-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
 
16-Mar-2025recsexd 27905 A non-zero surreal has a reciprocal. (Contributed by Scott Fenton, 16-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐴 ≠ 0s )    ⇒   (𝜑 → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
 
16-Mar-2025plycn 26010 A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
(𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
 
16-Mar-2025dvcobr 25697 The chain rule for derivatives at a point. For the (simpler but more limited) function version, see dvco 25699. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11192 and remove unnecessary hypotheses. (Revised by GG, 16-Mar-2025.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑌⟶𝑋)    &   (𝜑 → 𝑌 ⊆ 𝑇)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝑇 ⊆ ℂ)    &   (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑇 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)    ⇒   (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(𝐾 · 𝐿))
 
16-Mar-2025dvmulbr 25689 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmul 25692. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11192 and remove unnecessary hypotheses. (Revised by GG, 16-Mar-2025.)
(𝜑 → 𝐹:𝑋⟶ℂ)    &   (𝜑 → 𝑋 ⊆ 𝑆)    &   (𝜑 → 𝐺:𝑌⟶ℂ)    &   (𝜑 → 𝑌 ⊆ 𝑆)    &   (𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐶(𝑆 D 𝐹)𝐾)    &   (𝜑 → 𝐶(𝑆 D 𝐺)𝐿)    &   𝐽 = (TopOpen‘ℂfld)    ⇒   (𝜑 → 𝐶(𝑆 D (𝐹 ∘f · 𝐺))((𝐾 · (𝐺‘𝐶)) + (𝐿 · (𝐹‘𝐶))))
 
16-Mar-2025dvcnp2 25669 A function is continuous at each point for which it is differentiable. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = (𝐾 ↾t 𝐴)    &   𝐾 = (TopOpen‘ℂfld)    ⇒   (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐵))
 
16-Mar-2025mulcncf 25194 The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
(𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝑋–cn→ℂ))
 
16-Mar-2025reparphti 24743 Lemma for reparpht 24745. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
(𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   (𝜑 → 𝐺 ∈ (II Cn II))    &   (𝜑 → (𝐺‘0) = 0)    &   (𝜑 → (𝐺‘1) = 1)    &   𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))    ⇒   (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))
 
16-Mar-2025cnrehmeo 24698 The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12973 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘ℂfld)    ⇒   𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾)
 
16-Mar-2025icchmeo 24685 The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴)))    ⇒   ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾t (𝐴[,]𝐵))))
 
16-Mar-2025iimulcn 24681 Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
(𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II)
 
16-Mar-2025iihalf2cn 24676 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1))    ⇒   (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II)
 
16-Mar-2025iihalf1cn 24673 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2)))    ⇒   (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II)
 
16-Mar-2025negcncf 24662 The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)    ⇒   (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))
 
16-Mar-2025divccn 24611 Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025expcn 24610 The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
16-Mar-2025divcn 24606 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.) Avoid ax-mulf 11192. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽 ↾t (ℂ ∖ {0}))    ⇒    / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
 
16-Mar-2025mpomulcn 24605 Complex number multiplication is a continuous function. Version of mulcn 24603 using maps-to notation, which does not require ax-mulf 11192. (Contributed by GG, 16-Mar-2025.)
𝐽 = (TopOpen‘ℂfld)    ⇒   (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
16-Mar-2025rngqiprngfu 21076 The function value of 𝐹 at the constructed expected ring unity of 𝑅 is the ring unity of the product of the quotient with the two-sided ideal and the two-sided ideal. (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &   𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → (𝐹‘𝑈) = ⟨[𝐸] ∼ , 1 ⟩)
 
16-Mar-2025rngqipring1 21075 The ring unity of the product of the quotient with a two-sided ideal and the two-sided ideal, which both are rings. (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &   𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    &   𝑃 = (𝑄 ×s 𝐽)    ⇒   (𝜑 → (1r‘𝑃) = ⟨[𝐸] ∼ , 1 ⟩)
 
16-Mar-2025rngqiprngfulem5 21074 Lemma 5 for rngqiprngfu 21076. (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &   𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   (𝜑 → ( 1 · 𝑈) = 1 )
 
16-Mar-2025rngqiprngfulem4 21073 Lemma 4 for rngqiprngfu 21076. (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &   𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   (𝜑 → [𝑈] ∼ = [𝐸] ∼ )
 
16-Mar-2025rngqiprngfulem3 21072 Lemma 3 for rngqiprngfu 21076 (and lemma for rngqiprngu 21077). (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &   𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 )    ⇒   (𝜑 → 𝑈 ∈ 𝐵)
 
16-Mar-2025rngqiprngfulem2 21071 Lemma 2 for rngqiprngfu 21076 (and lemma for rngqiprngu 21077). (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    &   (𝜑 → 𝐸 ∈ (1r‘𝑄))    ⇒   (𝜑 → 𝐸 ∈ 𝐵)
 
16-Mar-2025rngqiprngfulem1 21070 Lemma 1 for rngqiprngfu 21076 (and lemma for rngqiprngu 21077). (Contributed by AV, 16-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   (𝜑 → 𝑄 ∈ Ring)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ )
 
16-Mar-2025rngisomring1 20359 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the ring unity of the second ring is the function value of the ring unity of the first ring for the isomorphism. (Contributed by AV, 16-Mar-2025.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))
 
16-Mar-2025xpsring1d 20221 The multiplicative identity element of a binary product of rings. (Contributed by AV, 16-Mar-2025.)
𝑌 = (𝑆 ×s 𝑅)    &   (𝜑 → 𝑆 ∈ Ring)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩)
 
16-Mar-2025ovmul 11207 Multiplication of complex numbers produces the same value as multiplication expressed in maps-to notation of the same complex numbers. (Contributed by GG, 16-Mar-2025.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))𝐵) = (𝐴 · 𝐵))
 
16-Mar-2025mpomulf 11206 Multiplication is an operation on complex numbers. Version of ax-mulf 11192 using maps-to notation, proved from the axioms of set theory and ax-mulcl 11174. (Contributed by GG, 16-Mar-2025.)
(𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)):(ℂ × ℂ)⟶ℂ
 
16-Mar-2025cbvsbv 2357 Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was extracted from a former cbvabv 2803 version. (Contributed by Wolf Lammen, 16-Mar-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
 
15-Mar-2025selvvvval 41459 Recover the original polynomial from a selectVars application. (Contributed by SN, 15-Mar-2025.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐷)    ⇒   (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑌))
 
15-Mar-2025mplmapghm 41430 The function 𝐻 mapping polynomials 𝑝 to their coefficient given a bag of variables 𝐹 is a group homomorphism. (Contributed by SN, 15-Mar-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝐹 ∈ 𝐷)    ⇒   (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅))
 
15-Mar-2025mxidlprmALT 32887 Every maximal ideal is prime - alternative proof. (Contributed by Thierry Arnoux, 15-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    ⇒   (𝜑 → 𝑀 ∈ (PrmIdeal‘𝑅))
 
15-Mar-2025recsex 27904 A non-zero surreal has a reciprocal. (Contributed by Scott Fenton, 15-Mar-2025.)
((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s )
 
15-Mar-2025precsex 27903 Every positive surreal has a reciprocal. Theorem 10(iv) of [Conway] p. 21. (Contributed by Scott Fenton, 15-Mar-2025.)
((𝐴 ∈ No ∧ 0s <s 𝐴) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 1s )
 
15-Mar-2025precsexlem11 27902 Lemma for surreal reciprocal. Show that the cut of the left and right sets is a multiplicative inverse for 𝐴. (Contributed by Scott Fenton, 15-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 0s <s 𝐴)    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    &   𝑌 = (∪ (𝐿 “ ω) |s ∪ (𝑅 “ ω))    ⇒   (𝜑 → (𝐴 ·s 𝑌) = 1s )
 
15-Mar-2025precsexlem10 27901 Lemma for surreal reciprocal. Show that the union of the left sets is less than the union of the right sets. Note that this is the first theorem in the surreal numbers to require the axiom of infinity. (Contributed by Scott Fenton, 15-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 0s <s 𝐴)    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    ⇒   (𝜑 → ∪ (𝐿 “ ω) <<s ∪ (𝑅 “ ω))
 
15-Mar-2025precsexlem7 27898 Lemma for surreal reciprocal. Show that 𝑅 is non-strictly increasing in its argument. (Contributed by Scott Fenton, 15-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝑅‘𝐼) ⊆ (𝑅‘𝐽))
 
15-Mar-2025precsexlem6 27897 Lemma for surreal reciprocal. Show that 𝐿 is non-strictly increasing in its argument. (Contributed by Scott Fenton, 15-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   ((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝐼 ⊆ 𝐽) → (𝐿‘𝐼) ⊆ (𝐿‘𝐽))
 
15-Mar-2025precsexlemcbv 27891 Lemma for surreal reciprocals. Change some bound variables. (Contributed by Scott Fenton, 15-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    ⇒   𝐹 = rec((𝑞 ∈ V ↦ ⦋(1st ‘𝑞) / 𝑚⦌⦋(2nd ‘𝑞) / 𝑠⦌⟨(𝑚 ∪ ({𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐴)∃𝑤 ∈ 𝑚 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝐴) ·s 𝑤)) /su 𝑧𝑅)} ∪ {𝑏 ∣ ∃𝑧𝐿 ∈ {𝑧 ∈ ( L ‘𝐴) ∣ 0s <s 𝑧}∃𝑡 ∈ 𝑠 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝐴) ·s 𝑡)) /su 𝑧𝐿)})), (𝑠 ∪ ({𝑏 ∣ ∃𝑧𝐿 ∈ {𝑧 ∈ ( L ‘𝐴) ∣ 0s <s 𝑧}∃𝑤 ∈ 𝑚 𝑏 = (( 1s +s ((𝑧𝐿 -s 𝐴) ·s 𝑤)) /su 𝑧𝐿)} ∪ {𝑏 ∣ ∃𝑧𝑅 ∈ ( R ‘𝐴)∃𝑡 ∈ 𝑠 𝑏 = (( 1s +s ((𝑧𝑅 -s 𝐴) ·s 𝑡)) /su 𝑧𝑅)}))⟩), ⟨{ 0s }, ∅⟩)
 
15-Mar-2025mul2negsd 27856 Surreal product of two negatives. (Contributed by Scott Fenton, 15-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (( -us ‘𝐴) ·s ( -us ‘𝐵)) = (𝐴 ·s 𝐵))
 
15-Mar-2025rngqiprng1elbas 21045 The ring unity of a two-sided ideal of a non-unital ring belongs to the base set of the ring. (Contributed by AV, 15-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    ⇒   (𝜑 → 1 ∈ 𝐵)
 
15-Mar-2025lmodvscld 20633 Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑 → 𝑊 ∈ LMod)    &   (𝜑 → 𝑅 ∈ 𝐾)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (𝑅 · 𝑋) ∈ 𝑉)
 
15-Mar-2025omun 7880 The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)
 
14-Mar-2025ovmpogad 41363 Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7564. (Contributed by SN, 14-Mar-2025.)
𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    &   ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝜑 → 𝐴 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → 𝑆 ∈ 𝑉)    ⇒   (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
14-Mar-2025fmpocos 41362 Composition of two functions. Variation of fmpoco 8083 with more context in the substitution hypothesis for 𝑇. (Contributed by SN, 14-Mar-2025.)
((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶)    &   (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅))    &   (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆))    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⦋𝑅 / 𝑧⦌𝑆 = 𝑇)    ⇒   (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇))
 
14-Mar-2025precsexlem9 27900 Lemma for surreal reciprocal. Show that the product of 𝐴 and a left element is less than one and the product of 𝐴 and a right element is greater than one. (Contributed by Scott Fenton, 14-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 0s <s 𝐴)    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    ⇒   ((𝜑 ∧ 𝐼 ∈ ω) → (∀𝑏 ∈ (𝐿‘𝐼)(𝐴 ·s 𝑏) <s 1s ∧ ∀𝑐 ∈ (𝑅‘𝐼) 1s <s (𝐴 ·s 𝑐)))
 
14-Mar-2025divsasswd 27889 An associative law for surreal division. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    &   (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) /su 𝐶) = (𝐴 ·s (𝐵 /su 𝐶)))
 
14-Mar-2025sltmuldiv2wd 27888 Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    &   (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐶 ·s 𝐴) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
 
14-Mar-2025sltmuldivwd 27887 Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    &   (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐴 ·s 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 /su 𝐶)))
 
14-Mar-2025sltdivmul2wd 27886 Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    &   (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐵 ·s 𝐶)))
 
14-Mar-2025sltdivmulwd 27885 Surreal less-than relationship between division and multiplication. Weak version. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    &   (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐴 /su 𝐶) <s 𝐵 ↔ 𝐴 <s (𝐶 ·s 𝐵)))
 
14-Mar-2025muls12d 27872 Commutative/associative law for surreal multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → (𝐴 ·s (𝐵 ·s 𝐶)) = (𝐵 ·s (𝐴 ·s 𝐶)))
 
14-Mar-2025sltmulneg2d 27868 Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 <s 0s )    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 ·s 𝐵) <s (𝐶 ·s 𝐴)))
 
14-Mar-2025sltmulneg1d 27867 Multiplication of both sides of surreal less-than by a negative number. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 <s 0s )    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ (𝐵 ·s 𝐶) <s (𝐴 ·s 𝐶)))
 
14-Mar-2025mulslidd 27838 Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → ( 1s ·s 𝐴) = 𝐴)
 
14-Mar-2025mulsridd 27809 Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → (𝐴 ·s 1s ) = 𝐴)
 
14-Mar-2025slenegd 27761 Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘𝐴)))
 
14-Mar-2025sltnegd 27760 Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 14-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
 
14-Mar-2025rabbida 3456 Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva 3437 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.) Avoid ax-10 2135, ax-11 2152. (Revised by Wolf Lammen, 14-Mar-2025.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
 
13-Mar-2025qsfld 32886 An ideal 𝑀 in the commutative ring 𝑅 is maximal if and only if the factor ring 𝑄 is a field. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑀 ∈ (LIdeal‘𝑅))    ⇒   (𝜑 → (𝑄 ∈ Field ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
 
13-Mar-2025qsdrng 32885 An ideal 𝑀 is both left and right maximal if and only if the factor ring 𝑄 is a division ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))    ⇒   (𝜑 → (𝑄 ∈ DivRing ↔ (𝑀 ∈ (MaxIdeal‘𝑅) ∧ 𝑀 ∈ (MaxIdeal‘𝑂))))
 
13-Mar-2025qsdrnglem2 32884 Lemma for qsdrng 32885. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑀 ∈ (2Ideal‘𝑅))    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑄 ∈ DivRing)    &   (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑀 ⊆ 𝐽)    &   (𝜑 → 𝑋 ∈ (𝐽 ∖ 𝑀))    ⇒   (𝜑 → 𝐽 = 𝐵)
 
13-Mar-2025opprqusdrng 32881 The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   (𝜑 → ((oppr‘𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))
 
13-Mar-2025opprqus0g 32878 The group identity element of the quotient of the opposite ring is the same as the group identity element of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    ⇒   (𝜑 → (0g‘(oppr‘𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
 
13-Mar-2025opprqusplusg 32877 The group operation of the quotient of the opposite ring is the same as the group operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐸 = (Base‘𝑄)    &   (𝜑 → 𝑋 ∈ 𝐸)    &   (𝜑 → 𝑌 ∈ 𝐸)    ⇒   (𝜑 → (𝑋(+g‘(oppr‘𝑄))𝑌) = (𝑋(+g‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
 
13-Mar-2025opprnsg 32872 Normal subgroups of the opposite ring are the same as the original normal subgroups. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (NrmSGrp‘𝑅) = (NrmSGrp‘𝑂)
 
13-Mar-2025crngmxidl 32859 In a commutative ring, maximal ideals of the opposite ring coincide with maximal ideals. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑀 = (MaxIdeal‘𝑅)    &   𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ CRing → 𝑀 = (MaxIdeal‘𝑂))
 
13-Mar-2025drngidlhash 32826 A ring is a division ring if and only if it admits exactly two ideals. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑈 = (LIdeal‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝑅 ∈ DivRing ↔ (♯‘𝑈) = 2))
 
13-Mar-2025drngidl 32825 A nonzero ring is a division ring if and only if its only left ideals are the zero ideal and the unit ideal. (Proposed by Gerard Lang, 13-Mar-2025.) (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    ⇒   (𝑅 ∈ NzRing → (𝑅 ∈ DivRing ↔ 𝑈 = {{ 0 }, 𝐵}))
 
13-Mar-2025qus0g 32792 The identity element of a quotient group. (Contributed by Thierry Arnoux, 13-Mar-2025.)
𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    ⇒   (𝑁 ∈ (NrmSGrp‘𝐺) → (0g‘𝑄) = 𝑁)
 
13-Mar-2025precsexlem8 27899 Lemma for surreal reciprocal. Show that the left and right functions give sets of surreals. (Contributed by Scott Fenton, 13-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 0s <s 𝐴)    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))( 0s <s 𝑥𝑂 → ∃𝑦 ∈ No (𝑥𝑂 ·s 𝑦) = 1s ))    ⇒   ((𝜑 ∧ 𝐼 ∈ ω) → ((𝐿‘𝐼) ⊆ No ∧ (𝑅‘𝐼) ⊆ No ))
 
13-Mar-2025precsexlem5 27896 Lemma for surreal reciprocals. Calculate the value of the recursive right function at a successor. (Contributed by Scott Fenton, 13-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   (𝐼 ∈ ω → (𝑅‘suc 𝐼) = ((𝑅‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)})))
 
13-Mar-2025precsexlem4 27895 Lemma for surreal reciprocals. Calculate the value of the recursive left function at a successor. (Contributed by Scott Fenton, 13-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   (𝐼 ∈ ω → (𝐿‘suc 𝐼) = ((𝐿‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})))
 
13-Mar-2025precsexlem2 27893 Lemma for surreal reciprocals. Calculate the value of the recursive right function at zero. (Contributed by Scott Fenton, 13-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   (𝑅‘∅) = ∅
 
13-Mar-2025precsexlem1 27892 Lemma for surreal reciprocals. Calculate the value of the recursive left function at zero. (Contributed by Scott Fenton, 13-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   (𝐿‘∅) = { 0s }
 
13-Mar-2025divs1 27890 A surreal divided by one is itself. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝐴 ∈ No → (𝐴 /su 1s ) = 𝐴)
 
13-Mar-2025divscan1wd 27884 A weak cancellation law for surreal division. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    &   (𝜑 → ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐴 /su 𝐵) ·s 𝐵) = 𝐴)
 
13-Mar-2025divscan2wd 27883 A weak cancellation law for surreal division. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    &   (𝜑 → ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s )    ⇒   (𝜑 → (𝐵 ·s (𝐴 /su 𝐵)) = 𝐴)
 
13-Mar-2025cutpos 27658 Reduce the elements of a cut for a positive number. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 0s <s 𝐴)    ⇒   (𝜑 → 𝐴 = (({ 0s } ∪ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}) |s ( R ‘𝐴)))
 
13-Mar-2025cutlt 27657 Eliminating all elements below a given element of a cut does not affect the cut. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐿 <<s 𝑅)    &   (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   (𝜑 → 𝑋 ∈ 𝐿)    ⇒   (𝜑 → 𝐴 = (({𝑋} ∪ {𝑦 ∈ 𝐿 ∣ 𝑋 <s 𝑦}) |s 𝑅))
 
13-Mar-2025coiniss 27656 Coinitiality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ⊆ No )    &   (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)
 
13-Mar-2025cofss 27655 Cofinality for a subset. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ⊆ No )    &   (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦)
 
13-Mar-20250elright 27642 Zero is in the right set of any negative number. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐴 <s 0s )    ⇒   (𝜑 → 0s ∈ ( R ‘𝐴))
 
13-Mar-20250elleft 27641 Zero is in the left set of any positive number. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 0s <s 𝐴)    ⇒   (𝜑 → 0s ∈ ( L ‘𝐴))
 
13-Mar-20250elold 27640 Zero is in the old set of any non-zero number. (Contributed by Scott Fenton, 13-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐴 ≠ 0s )    ⇒   (𝜑 → 0s ∈ ( O ‘( bday ‘𝐴)))
 
13-Mar-20252idlss 21017 A two-sided ideal is a subset of the base set. Formerly part of proof for 2idlcpbl 21021. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
𝐵 = (Base‘𝑊)    &   𝐼 = (2Ideal‘𝑊)    ⇒   (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵)
 
12-Mar-2025evlcl 41446 A polynomial over the ring 𝑅 evaluates to an element in 𝑅. (Contributed by SN, 12-Mar-2025.)
𝑄 = (𝐼 eval 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝐹)‘𝐴) ∈ 𝐾)
 
12-Mar-2025evlsmaprhm 41444 The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝑋 is a ring homomorphism. Compare evls1maprhm 33048. (Contributed by SN, 12-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑅)‘𝑆)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑅 ↾s 𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐹 = (𝑝 ∈ 𝐵 ↦ ((𝑄‘𝑝)‘𝐴))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))
 
12-Mar-2025evlscl 41432 A polynomial over the ring 𝑅 evaluates to an element in 𝑅. (Contributed by SN, 12-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑅)‘𝑆)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑅 ↾s 𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝐹)‘𝐴) ∈ 𝐾)
 
12-Mar-2025mplascl1 41429 The one scalar as a polynomial. (Contributed by SN, 12-Mar-2025.)
𝑊 = (𝐼 mPoly 𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝑂 = (1r‘𝑅)    &    1 = (1r‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (𝐴‘𝑂) = 1 )
 
12-Mar-2025mpllmodd 41418 The polynomial ring is a left module. (Contributed by SN, 12-Mar-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑃 ∈ LMod)
 
12-Mar-2025precsexlem3 27894 Lemma for surreal reciprocals. Calculate the value of the recursive function at a successor. (Contributed by Scott Fenton, 12-Mar-2025.)
𝐹 = rec((𝑝 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑙⦌⦋(2nd ‘𝑝) / 𝑟⦌⟨(𝑙 ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), (𝑟 ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ 𝑙 𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ 𝑟 𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩), ⟨{ 0s }, ∅⟩)    &   𝐿 = (1st ∘ 𝐹)    &   𝑅 = (2nd ∘ 𝐹)    ⇒   (𝐼 ∈ ω → (𝐹‘suc 𝐼) = ⟨((𝐿‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝑅)} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝐿)})), ((𝑅‘𝐼) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ {𝑥 ∈ ( L ‘𝐴) ∣ 0s <s 𝑥}∃𝑦𝐿 ∈ (𝐿‘𝐼)𝑎 = (( 1s +s ((𝑥𝐿 -s 𝐴) ·s 𝑦𝐿)) /su 𝑥𝐿)} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ (𝑅‘𝐼)𝑎 = (( 1s +s ((𝑥𝑅 -s 𝐴) ·s 𝑦𝑅)) /su 𝑥𝑅)}))⟩)
 
12-Mar-2025divsclwd 27882 Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    &   (𝜑 → ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s )    ⇒   (𝜑 → (𝐴 /su 𝐵) ∈ No )
 
12-Mar-2025divsclw 27881 Weak division closure law. (Contributed by Scott Fenton, 12-Mar-2025.)
(((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) ∧ ∃𝑥 ∈ No (𝐵 ·s 𝑥) = 1s ) → (𝐴 /su 𝐵) ∈ No )
 
12-Mar-2025divsmulwd 27880 Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. (Contributed by Scott Fenton, 12-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    &   (𝜑 → ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s )    ⇒   (𝜑 → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
 
12-Mar-2025divsmulw 27879 Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 27903, we can eliminate the existence hypothesis (see divsmul 27906). (Contributed by Scott Fenton, 12-Mar-2025.)
(((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))
 
12-Mar-2025noreceuw 27878 If a surreal has a reciprocal, then it has unique division. (Contributed by Scott Fenton, 12-Mar-2025.)
(((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)
 
12-Mar-2025norecdiv 27877 If a surreal has a reciprocal, then it has any division. (Contributed by Scott Fenton, 12-Mar-2025.)
(((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝐵 ∈ No ) ∧ ∃𝑥 ∈ No (𝐴 ·s 𝑥) = 1s ) → ∃𝑦 ∈ No (𝐴 ·s 𝑦) = 𝐵)
 
12-Mar-2025divsval 27876 The value of surreal division. (Contributed by Scott Fenton, 12-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
 
12-Mar-2025df-divs 27875 Define surreal division. This is not the definition used in the literature, but we use it here because it is technically easier to work with. (Contributed by Scott Fenton, 12-Mar-2025.)
/su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))
 
12-Mar-2025sgt0ne0d 27573 A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
(𝜑 → 0s <s 𝐴)    ⇒   (𝜑 → 𝐴 ≠ 0s )
 
12-Mar-2025sgt0ne0 27572 A positive surreal is not equal to zero. (Contributed by Scott Fenton, 12-Mar-2025.)
( 0s <s 𝐴 → 𝐴 ≠ 0s )
 
12-Mar-2025cuteq1 27571 Condition for a surreal cut to equal one. (Contributed by Scott Fenton, 12-Mar-2025.)
(𝜑 → 0s ∈ 𝐴)    &   (𝜑 → 𝐴 <<s { 1s })    &   (𝜑 → { 1s } <<s 𝐵)    ⇒   (𝜑 → (𝐴 |s 𝐵) = 1s )
 
12-Mar-2025sltne 27509 Surreal less-than implies not equal. (Contributed by Scott Fenton, 12-Mar-2025.)
((𝐴 ∈ No ∧ 𝐴 <s 𝐵) → 𝐵 ≠ 𝐴)
 
12-Mar-2025ply1scl1 22035 The one scalar is the unit polynomial. (Contributed by Stefan O'Rear, 1-Apr-2015.) (Proof shortened by SN, 12-Mar-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐴 = (algSc‘𝑃)    &    1 = (1r‘𝑅)    &   𝑁 = (1r‘𝑃)    ⇒   (𝑅 ∈ Ring → (𝐴‘ 1 ) = 𝑁)
 
12-Mar-2025reueq1 3413 Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2106. (Revised by Wolf Lammen, 12-Mar-2025.)
(𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
 
12-Mar-2025rmoeq1 3412 Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2106. (Revised by Wolf Lammen, 12-Mar-2025.)
(𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
 
11-Mar-2025lidlrng 46913 A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) (Proof shortened by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    ⇒   ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
 
11-Mar-2025evlvvvallem 41448 Lemma for theorems using evlvvval 41447. Version of evlsvvvallem2 41436 using df-evl 21855. (Contributed by SN, 11-Mar-2025.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    ↑ = (.g‘𝑀)    &    · = (.r‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0g‘𝑅))
 
11-Mar-2025mapcod 41373 Compose two mappings. (Contributed by SN, 11-Mar-2025.)
(𝜑 → 𝐹 ∈ (𝐴 ↑m 𝐵))    &   (𝜑 → 𝐺 ∈ (𝐵 ↑m 𝐶))    ⇒   (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶))
 
11-Mar-2025elmapssresd 41372 A restricted mapping is a mapping. EDITORIAL: Could be used to shorten elpm2r 8841 with some reordering involving mapsspm 8872. (Contributed by SN, 11-Mar-2025.)
(𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐶))    &   (𝜑 → 𝐷 ⊆ 𝐶)    ⇒   (𝜑 → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑m 𝐷))
 
11-Mar-2025f1o2d2 41361 Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by SN, 11-Mar-2025.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)    &   ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐼 ∈ 𝐴)    &   ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐽 ∈ 𝐵)    &   ((𝜑 ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐷)) → ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) ↔ 𝑧 = 𝐶))    ⇒   (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)
 
11-Mar-2025eqresfnbd 41360 Property of being the restriction of a function. Note that this is closer to funssres 6591 than fnssres 6672. (Contributed by SN, 11-Mar-2025.)
(𝜑 → 𝐹 Fn 𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → (𝑅 = (𝐹 ↾ 𝐴) ↔ (𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹)))
 
11-Mar-2025rng2idlsubgsubrng 21041 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
 
11-Mar-2025rng2idlsubrng 21038 A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → 𝐼 ∈ (SubRng‘𝑅))
 
11-Mar-2025rnglidlrng 21036 A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 𝑈 ∈ (SubGrp‘𝑅) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (SubGrp‘𝑅)) → 𝐼 ∈ Rng)
 
11-Mar-2025rnglidlmsgrp 21035 The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)
 
11-Mar-2025rnglidlmmgm 21034 The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅 ↾s 𝑈)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)
 
11-Mar-2025undifr 4481 Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.) (Proof shortened by SN, 11-Mar-2025.)
(𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵)
 
10-Mar-2025evlselv 41461 Evaluating a selection of variable assignments, then evaluating the rest of the variables, is the same as evaluating with all assignments. (Contributed by SN, 10-Mar-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐿 = (algSc‘𝑈)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))
 
10-Mar-2025evlselvlem 41460 Lemma for evlselv 41461. Used to re-index to and from bags of variables in 𝐼 and bags of variables in the subsets 𝐽 and 𝐼 ∖ 𝐽. (Contributed by SN, 10-Mar-2025.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}    &   𝐶 = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝐻 = (𝑐 ∈ 𝐶, 𝑒 ∈ 𝐸 ↦ (𝑐 ∪ 𝑒))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)
 
10-Mar-2025psrbagres 41417 Restrict a bag of variables in 𝐼 to a bag of variables in 𝐽 ⊆ 𝐼. (Contributed by SN, 10-Mar-2025.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐷)    ⇒   (𝜑 → (𝐹 ↾ 𝐽) ∈ 𝐸)
 
10-Mar-2025ricqusker 32819 The image 𝐻 of a ring homomorphism 𝐹 is isomorphic with the quotient ring 𝑄 over 𝐹's kernel 𝐾. This a part of what is sometimes called the first isomorphism theorem for rings. (Contributed by Thierry Arnoux, 10-Mar-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    &   (𝜑 → 𝐺 ∈ CRing)    ⇒   (𝜑 → 𝑄 ≃𝑟 𝐻)
 
10-Mar-2025lmicqusker 32809 The image 𝐻 of a module homomorphism 𝐹 is isomorphic with the quotient module 𝑄 over 𝐹's kernel 𝐾. This is part of what is sometimes called the first isomorphism theorem for modules. (Contributed by Thierry Arnoux, 10-Mar-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   (𝜑 → 𝑄 ≃𝑚 𝐻)
 
10-Mar-2025gicqusker 32807 The image 𝐻 of a group homomorphism 𝐹 is isomorphic with the quotient group 𝑄 over 𝐹's kernel 𝐾. Together with ghmker 19156 and ghmima 19151, this is sometimes called the first isomorphism theorem for groups. (Contributed by Thierry Arnoux, 10-Mar-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   (𝜑 → 𝑄 ≃𝑔 𝐻)
 
10-Mar-2025divsmo 27873 Uniqueness of surreal inversion. Given a non-zero surreal 𝐴, there is at most one surreal giving a particular product. (Contributed by Scott Fenton, 10-Mar-2025.)
((𝐴 ∈ No ∧ 𝐴 ≠ 0s ) → ∃*𝑥 ∈ No (𝐴 ·s 𝑥) = 𝐵)
 
10-Mar-2025mulscan1d 27871 Cancellation of surreal multiplication when the left term is non-zero. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    ⇒   (𝜑 → ((𝐶 ·s 𝐴) = (𝐶 ·s 𝐵) ↔ 𝐴 = 𝐵))
 
10-Mar-2025mulscan2d 27870 Cancellation of surreal multiplication when the right term is non-zero. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    ⇒   (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
 
10-Mar-2025mulscan2dlem 27869 Lemma for mulscan2d 27870. Cancellation of multiplication when the right term is positive. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → ((𝐴 ·s 𝐶) = (𝐵 ·s 𝐶) ↔ 𝐴 = 𝐵))
 
10-Mar-2025slemul1d 27866 Multiplication of both sides of surreal less-than or equal by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 ·s 𝐶) ≤s (𝐵 ·s 𝐶)))
 
10-Mar-2025slemul2d 27865 Multiplication of both sides of surreal less-than or equal by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐶 ·s 𝐴) ≤s (𝐶 ·s 𝐵)))
 
10-Mar-2025sltmul1d 27864 Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 ·s 𝐶) <s (𝐵 ·s 𝐶)))
 
10-Mar-2025sltmul2d 27863 Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 0s <s 𝐶)    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ (𝐶 ·s 𝐴) <s (𝐶 ·s 𝐵)))
 
10-Mar-2025sltmul2 27862 Multiplication of both sides of surreal less-than by a positive number. (Contributed by Scott Fenton, 10-Mar-2025.)
(((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐵 <s 𝐶 ↔ (𝐴 ·s 𝐵) <s (𝐴 ·s 𝐶)))
 
10-Mar-2025mulsassd 27861 Associative law for surreal multiplication. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))
 
10-Mar-2025mulsass 27860 Associative law for surreal multiplication. Part of theorem 7 of [Conway] p. 19. Much like the case for additive groups, this theorem together with mulscom 27834, addsdi 27849, mulsgt0 27839, and the addition theorems would make the surreals into an ordered ring except that they are a proper class. (Contributed by Scott Fenton, 10-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ·s 𝐵) ·s 𝐶) = (𝐴 ·s (𝐵 ·s 𝐶)))
 
10-Mar-2025mulsasslem3 27859 Lemma for mulsass 27860. Demonstrate the central equality. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   𝑃 ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))    &   𝑄 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵))    &   𝑅 ⊆ (( L ‘𝐶) ∪ ( R ‘𝐶))    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s 𝑦𝑂) ·s 𝑧𝑂) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝑧𝑂)))    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝑥𝑂 ·s 𝑦𝑂) ·s 𝐶) = (𝑥𝑂 ·s (𝑦𝑂 ·s 𝐶)))    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝑥𝑂 ·s 𝐵) ·s 𝑧𝑂) = (𝑥𝑂 ·s (𝐵 ·s 𝑧𝑂)))    &   (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝑦𝑂) ·s 𝑧𝑂) = (𝐴 ·s (𝑦𝑂 ·s 𝑧𝑂)))    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))((𝑥𝑂 ·s 𝐵) ·s 𝐶) = (𝑥𝑂 ·s (𝐵 ·s 𝐶)))    &   (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))((𝐴 ·s 𝑦𝑂) ·s 𝐶) = (𝐴 ·s (𝑦𝑂 ·s 𝐶)))    &   (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))((𝐴 ·s 𝐵) ·s 𝑧𝑂) = (𝐴 ·s (𝐵 ·s 𝑧𝑂)))    ⇒   (𝜑 → (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑄 ∃𝑧 ∈ 𝑅 𝑎 = ((((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧)) -s ((((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) -s (𝑥 ·s 𝑦)) ·s 𝑧)) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑄 ∃𝑧 ∈ 𝑅 𝑎 = (((𝑥 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧)))) -s (𝑥 ·s (((𝑦 ·s 𝐶) +s (𝐵 ·s 𝑧)) -s (𝑦 ·s 𝑧))))))
 
10-Mar-2025mulnegs2d 27855 Product with negative is negative of product. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 ·s ( -us ‘𝐵)) = ( -us ‘(𝐴 ·s 𝐵)))
 
10-Mar-2025mulnegs1d 27854 Product with negative is negative of product. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))
 
10-Mar-2025posdifsd 27800 Comparison of two surreals whose difference is positive. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 <s 𝐵 ↔ 0s <s (𝐵 -s 𝐴)))
 
10-Mar-2025slt0neg2d 27764 Comparison of a surreal and its negative to zero. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → ( 0s <s 𝐴 ↔ ( -us ‘𝐴) <s 0s ))
 
10-Mar-2025rspc3dv 3628 3-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2025.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   (𝑦 = 𝐵 → (𝜃 ↔ 𝜏))    &   (𝑧 = 𝐶 → (𝜏 ↔ 𝜒))    &   (𝜑 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 𝜓)    &   (𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐵 ∈ 𝐸)    &   (𝜑 → 𝐶 ∈ 𝐹)    ⇒   (𝜑 → 𝜒)
 
10-Mar-2025sbralie 3352 Implicit to explicit substitution that swaps variables in a rectrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004.) Avoid ax-ext 2701. (Revised by Wolf Lammen, 10-Mar-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
 
9-Mar-2025qsdrngi 32883 A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))    ⇒   (𝜑 → 𝑄 ∈ DivRing)
 
9-Mar-2025qsdrngilem 32882 Lemma for qsdrngi 32883. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑂))    &   (𝜑 → 𝑋 ∈ (Base‘𝑅))    &   (𝜑 → ¬ 𝑋 ∈ 𝑀)    ⇒   (𝜑 → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r‘𝑄)[𝑋](𝑅 ~QG 𝑀)) = (1r‘𝑄))
 
9-Mar-2025opprqus1r 32880 The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
 
9-Mar-2025opprqusmulr 32879 The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐸 = (Base‘𝑄)    &   (𝜑 → 𝑋 ∈ 𝐸)    &   (𝜑 → 𝑌 ∈ 𝐸)    ⇒   (𝜑 → (𝑋(.r‘(oppr‘𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
 
9-Mar-2025opprqusbas 32876 The base of the quotient of the opposite ring is the same as the base of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝐼 ⊆ 𝐵)    ⇒   (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
 
9-Mar-2025opprmxidlabs 32875 The maximal ideal of the opposite ring's opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    ⇒   (𝜑 → 𝑀 ∈ (MaxIdeal‘(oppr‘𝑂)))
 
9-Mar-2025oppr2idl 32874 Two sided ideal of the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
 
9-Mar-2025opprlidlabs 32873 The ideals of the opposite ring's opposite ring are the ideals of the original ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (LIdeal‘𝑅) = (LIdeal‘(oppr‘𝑂)))
 
9-Mar-2025oppreqg 32871 Group coset equivalence relation for the opposite ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   ((𝑅 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
 
9-Mar-2025opprabs 32870 The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑂 = (oppr‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → Fun 𝑅)    &   (𝜑 → (.r‘ndx) ∈ dom 𝑅)    &   (𝜑 → · Fn (𝐵 × 𝐵))    ⇒   (𝜑 → 𝑅 = (oppr‘𝑂))
 
9-Mar-2025mxidlmaxv 32858 An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑀 ⊆ 𝐼)    &   (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀))    ⇒   (𝜑 → 𝐼 = 𝐵)
 
9-Mar-2025qsnzr 32848 A quotient of a non-zero ring by a proper ideal is a non-zero ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ≠ 𝐵)    ⇒   (𝜑 → 𝑄 ∈ NzRing)
 
9-Mar-2025elrspunsn 32821 Membership to the span of an ideal 𝑅 and a single element 𝑋. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &    + = (+g‘𝑅)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝐼))    ⇒   (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
 
9-Mar-2025ringinveu 32664 If a ring unit element 𝑋 admits both a left inverse 𝑌 and a right inverse 𝑍, they are equal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → (𝑌 · 𝑋) = 1 )    &   (𝜑 → (𝑋 · 𝑍) = 1 )    ⇒   (𝜑 → 𝑍 = 𝑌)
 
9-Mar-2025urpropd 32648 Sufficient condition for ring unities to be equal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
𝐵 = (Base‘𝑆)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑇 ∈ 𝑊)    &   (𝜑 → 𝐵 = (Base‘𝑇))    &   (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘𝑇)𝑦))    ⇒   (𝜑 → (1r‘𝑆) = (1r‘𝑇))
 
9-Mar-2025mulsasslem2 27858 Lemma for mulsass 27860. Expand the right hand side of the formula. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → (𝐴 ·s (𝐵 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅)))) -s (𝑥𝐿 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝐿 ·s 𝑧𝑅))))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿)))) -s (𝑥𝐿 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝑅 ·s 𝑧𝐿))))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿)))) -s (𝑥𝑅 ·s (((𝑦𝐿 ·s 𝐶) +s (𝐵 ·s 𝑧𝐿)) -s (𝑦𝐿 ·s 𝑧𝐿))))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 ·s 𝐶)) +s (𝐴 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅)))) -s (𝑥𝑅 ·s (((𝑦𝑅 ·s 𝐶) +s (𝐵 ·s 𝑧𝑅)) -s (𝑦𝑅 ·s 𝑧𝑅))))}))))
 
9-Mar-2025mulsasslem1 27857 Lemma for mulsass 27860. Expand the left hand side of the formula. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) ·s 𝐶) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝐿))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝑅))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) ·s 𝑧𝑅))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝑅)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) ·s 𝑧𝑅))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) ·s 𝑧𝐿))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝐶) +s ((𝐴 ·s 𝐵) ·s 𝑧𝐿)) -s ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) ·s 𝑧𝐿))}))))
 
9-Mar-2025subsdird 27853 Distribution of surreal multiplication over subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐵) ·s 𝐶) = ((𝐴 ·s 𝐶) -s (𝐵 ·s 𝐶)))
 
9-Mar-2025subsdid 27852 Distribution of surreal multiplication over subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → (𝐴 ·s (𝐵 -s 𝐶)) = ((𝐴 ·s 𝐵) -s (𝐴 ·s 𝐶)))
 
9-Mar-2025addsdird 27851 Distributive law for surreal numbers. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 +s 𝐵) ·s 𝐶) = ((𝐴 ·s 𝐶) +s (𝐵 ·s 𝐶)))
 
9-Mar-2025addsdid 27850 Distributive law for surreal numbers. Commuted form of part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
 
9-Mar-2025addsdi 27849 Distributive law for surreal numbers. Commuted form of part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 9-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ·s (𝐵 +s 𝐶)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)))
 
9-Mar-2025addsdilem4 27848 Lemma for addsdi 27849. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))    &   (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))    &   (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))    &   (𝜓 → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))    ⇒   ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
 
9-Mar-2025addsdilem3 27847 Lemma for addsdi 27849. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))    &   (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝐴 ·s (𝑦𝑂 +s 𝐶)) = ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)))    &   (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)))    &   (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))    &   (𝜓 → 𝑌 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))    ⇒   ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑌 +s 𝐶))) -s (𝑋 ·s (𝑌 +s 𝐶))) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) +s (𝐴 ·s 𝐶)))
 
9-Mar-2025subsubs4d 27799 Law for double surreal subtraction. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐵) -s 𝐶) = (𝐴 -s (𝐵 +s 𝐶)))
 
9-Mar-2025adds12d 27730 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Scott Fenton, 9-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = (𝐵 +s (𝐴 +s 𝐶)))
 
9-Mar-20252idlridld 21016 A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝑂 = (oppr‘𝑅)    ⇒   (𝜑 → 𝐼 ∈ (LIdeal‘𝑂))
 
9-Mar-20252idllidld 21015 A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
 
9-Mar-2025quseccl 19102 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑆))    &   𝑉 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)    ⇒   ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝑋 ∈ 𝑉) → [𝑋](𝐺 ~QG 𝑆) ∈ 𝐵)
 
9-Mar-2025fovcl 7539 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.)
𝐹:(𝑅 × 𝑆)⟶𝐶    ⇒   ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
9-Mar-2025rexeqf 3348 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See rexeq 3319 for a version based on fewer axioms. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 9-Mar-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
 
9-Mar-2025raleleq 3335 All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.) Avoid ax-8 2106. (Revised by Wolf Lammen, 9-Mar-2025.)
(𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
 
9-Mar-2025cbvrexdva 3235 Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Avoid ax-9 2114, ax-ext 2701. (Revised by Wolf Lammen, 9-Mar-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒))
 
9-Mar-2025cbvraldva 3234 Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Avoid ax-9 2114, ax-ext 2701. (Revised by Wolf Lammen, 9-Mar-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))
 
8-Mar-2025mhphf 41471 A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄‘𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.) (Proof shortened by SN, 8-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &    · = (.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)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
8-Mar-2025evlsvvvallem2 41436 Lemma for theorems using evlsvvval 41437. (Contributed by SN, 8-Mar-2025.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝑀 = (mulGrp‘𝑆)    &    ↑ = (.g‘𝑀)    &    · = (.r‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0g‘𝑆))
 
8-Mar-2025crng12d 41392 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by SN, 8-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑌 · 𝑍)) = (𝑌 · (𝑋 · 𝑍)))
 
8-Mar-2025crngcomd 41391 Multiplication is commutative in a commutative ring. (Contributed by SN, 8-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 𝑌) = (𝑌 · 𝑋))
 
8-Mar-2025fsuppss 41371 A subset of a finitely supported function is a finitely supported function. (Contributed by SN, 8-Mar-2025.)
(𝜑 → 𝐹 ⊆ 𝐺)    &   (𝜑 → 𝐺 finSupp 𝑍)    ⇒   (𝜑 → 𝐹 finSupp 𝑍)
 
8-Mar-2025fsuppfund 41369 A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
(𝜑 → 𝐹 finSupp 𝑍)    ⇒   (𝜑 → Fun 𝐹)
 
8-Mar-2025addsdilem2 27846 Lemma for surreal distribution. Expand the right hand side of the main expression. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝐿 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝑅 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝐿 ·s 𝑧𝐿)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝑅 ·s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = ((((𝑥𝐿 ·s 𝐵) +s (𝐴 ·s 𝑦𝑅)) -s (𝑥𝐿 ·s 𝑦𝑅)) +s (𝐴 ·s 𝐶))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = ((((𝑥𝑅 ·s 𝐵) +s (𝐴 ·s 𝑦𝐿)) -s (𝑥𝑅 ·s 𝑦𝐿)) +s (𝐴 ·s 𝐶))}) ∪ ({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝐿 ·s 𝐶) +s (𝐴 ·s 𝑧𝑅)) -s (𝑥𝐿 ·s 𝑧𝑅)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = ((𝐴 ·s 𝐵) +s (((𝑥𝑅 ·s 𝐶) +s (𝐴 ·s 𝑧𝐿)) -s (𝑥𝑅 ·s 𝑧𝐿)))}))))
 
8-Mar-2025addsdilem1 27845 Lemma for surreal distribution. Expand the left hand side of the main expression. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → (𝐴 ·s (𝐵 +s 𝐶)) = ((({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝐿)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝑅)))})) |s (({𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑦𝑅 ∈ ( R ‘𝐵)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝑅 +s 𝐶))) -s (𝑥𝐿 ·s (𝑦𝑅 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝐿 ∈ ( L ‘𝐴)∃𝑧𝑅 ∈ ( R ‘𝐶)𝑎 = (((𝑥𝐿 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝑅))) -s (𝑥𝐿 ·s (𝐵 +s 𝑧𝑅)))}) ∪ ({𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑦𝐿 ∈ ( L ‘𝐵)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑦𝐿 +s 𝐶))) -s (𝑥𝑅 ·s (𝑦𝐿 +s 𝐶)))} ∪ {𝑎 ∣ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑧𝐿 ∈ ( L ‘𝐶)𝑎 = (((𝑥𝑅 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑧𝐿))) -s (𝑥𝑅 ·s (𝐵 +s 𝑧𝐿)))}))))
 
8-Mar-2025mulsuniflem 27843 Lemma for mulsunif 27844. State the theorem with some extra distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝜑 → 𝐿 <<s 𝑅)    &   (𝜑 → 𝑀 <<s 𝑆)    &   (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
 
8-Mar-2025mulscut 27827 Show the cut properties of surreal multiplication. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({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 {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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))})))
 
8-Mar-2025mulsval2 27806 The value of surreal multiplication, expressed with fewer distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-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))})))
 
8-Mar-2025mulsval2lem 27805 Lemma for mulsval2 27806. Change bound variables in one of the cases. (Contributed by Scott Fenton, 8-Mar-2025.)
{𝑎 ∣ ∃𝑝 ∈ 𝑋 ∃𝑞 ∈ 𝑌 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑏 ∣ ∃𝑟 ∈ 𝑋 ∃𝑠 ∈ 𝑌 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}
 
8-Mar-2025negsbday 27770 Negation of a surreal number preserves birthday. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) = ( bday ‘𝐴))
 
8-Mar-2025negsbdaylem 27769 Lemma for negsbday 27770. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)
(𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))
 
8-Mar-2025addscut2 27701 Show that the cut involved in surreal addition is legitimate. (Contributed by Scott Fenton, 8-Mar-2025.)
(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)}))
 
8-Mar-2025addscutlem 27699 Lemma for addscut 27700. Show the statement with some additional distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-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)})))
 
8-Mar-2025mhmima 18742 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.) (Proof shortened by AV, 8-Mar-2025.)
((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMnd‘𝑁))
 
8-Mar-2025raleq 3320 Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025.)
(𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))
 
8-Mar-2025rexeq 3319 Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) Remove usage of ax-10 2135, ax-11 2152, and ax-12 2169. (Revised by Steven Nguyen, 30-Apr-2023.) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025.)
(𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
 
8-Mar-2025cbvrexsvw 3313 Change bound variable by using a substitution. Version of cbvrexsv 3361 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 2-Mar-2008.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.)
(∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
 
8-Mar-2025cbvralsvw 3312 Change bound variable by using a substitution. Version of cbvralsv 3360 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2369. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.)
(∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
 
7-Mar-2025mulsunif 27844 Surreal multiplication has the uniformity property. That is, any cuts that define 𝐴 and 𝐵 can be used in the definition of (𝐴 ·s 𝐵). Theorem 3.5 of [Gonshor] p. 18. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → 𝐿 <<s 𝑅)    &   (𝜑 → 𝑀 <<s 𝑆)    &   (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
 
7-Mar-2025ssltmul2 27842 One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → 𝐿 <<s 𝑅)    &   (𝜑 → 𝑀 <<s 𝑆)    &   (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   (𝜑 → {(𝐴 ·s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
 
7-Mar-2025ssltmul1 27841 One surreal set less-than relationship for cuts of 𝐴 and 𝐵. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → 𝐿 <<s 𝑅)    &   (𝜑 → 𝑀 <<s 𝑆)    &   (𝜑 → 𝐴 = (𝐿 |s 𝑅))    &   (𝜑 → 𝐵 = (𝑀 |s 𝑆))    ⇒   (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)})
 
7-Mar-2025slemuld 27833 An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → 𝐴 ≤s 𝐵)    &   (𝜑 → 𝐶 ≤s 𝐷)    ⇒   (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) ≤s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
 
7-Mar-2025mulscut2 27828 Show that the cut involved in surreal multiplication is actually a cut. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ({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))}))
 
7-Mar-2025mulscutlem 27826 Lemma for mulscut 27827. State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({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 {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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))})))
 
7-Mar-2025slesubsub3bd 27794 Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-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)))
 
7-Mar-2025slesubsub2bd 27793 Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-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)))
 
7-Mar-2025slesubsubbd 27792 Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-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)))
 
7-Mar-2025rngqiprngimf1 21059 𝐹 is a one-to-one function from (the base set of) a non-unital ring to the product of the (base set of the) quotient with a two-sided ideal and the (base set of the) two-sided ideal. (Contributed by AV, 7-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → 𝐹:𝐵–1-1→(𝐶 × 𝐼))
 
7-Mar-2025rngqiprngimf1lem 21053 Lemma for rngqiprngimf1 21059. (Contributed by AV, 7-Mar-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ 𝐴 ∈ 𝐵) → (([𝐴] ∼ = (0g‘𝑄) ∧ ( 1 · 𝐴) = (0g‘𝐽)) → 𝐴 = (0g‘𝑅)))
 
7-Mar-2025ring1nzdiv 20290 In a unitary ring, the ring unity is not a zero divisor. (Contributed by AV, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    1 = (1r‘𝑅)    ⇒   (𝜑 → (( 1 · 𝑌) = 0 ↔ 𝑌 = 0 ))
 
7-Mar-2025ringunitnzdiv 20289 In a unitary ring, a unit is not a zero divisor. (Contributed by AV, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ (Unit‘𝑅))    ⇒   (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 ))
 
7-Mar-2025gsummulc2 20205 A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) Remove unused hypothesis. (Revised by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    ⇒   (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))))
 
7-Mar-2025gsummulc1 20204 A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) Remove unused hypothesis. (Revised by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    ⇒   (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌))
 
7-Mar-2025ringrzd 20184 The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 0 ) = 0 )
 
7-Mar-2025ringlzd 20183 The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 0 · 𝑋) = 0 )
 
7-Mar-2025qusecsub 19744 Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
𝐵 = (Base‘𝐺)    &    − = (-g‘𝐺)    &    ∼ = (𝐺 ~QG 𝑆)    ⇒   (((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ = [𝑌] ∼ ↔ (𝑌 − 𝑋) ∈ 𝑆))
 
6-Mar-2025n0nsn2el 46033 If a class with one element is not a singleton, there is at least another element in this class. (Contributed by AV, 6-Mar-2025.)
((𝐴 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
 
6-Mar-2025evlsvvvallem 41435 Lemma for evlsvvval 41437 akin to psrbagev2 21859. (Contributed by SN, 6-Mar-2025.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐾 = (Base‘𝑆)    &   𝑀 = (mulGrp‘𝑆)    &    ↑ = (.g‘𝑀)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (𝜑 → 𝐵 ∈ 𝐷)    ⇒   (𝜑 → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)
 
6-Mar-2025fsuppsssuppgd 41370 If the support of a function is a subset of a finite support, it is finite. Deduction associated with fsuppsssupp 9381. (Contributed by SN, 6-Mar-2025.)
(𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝑍 ∈ 𝑊)    &   (𝜑 → Fun 𝐺)    &   (𝜑 → 𝐹 finSupp 𝑂)    &   (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑂))    ⇒   (𝜑 → 𝐺 finSupp 𝑍)
 
6-Mar-2025mulsgt0d 27840 The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 0s <s 𝐴)    &   (𝜑 → 0s <s 𝐵)    ⇒   (𝜑 → 0s <s (𝐴 ·s 𝐵))
 
6-Mar-2025mulsgt0 27839 The product of two positive surreals is positive. Theorem 9 of [Conway] p. 20. (Contributed by Scott Fenton, 6-Mar-2025.)
(((𝐴 ∈ No ∧ 0s <s 𝐴) ∧ (𝐵 ∈ No ∧ 0s <s 𝐵)) → 0s <s (𝐴 ·s 𝐵))
 
6-Mar-2025mulscomd 27835 Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴))
 
6-Mar-2025mulscom 27834 Surreal multiplication commutes. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) = (𝐵 ·s 𝐴))
 
6-Mar-2025sltmuld 27832 An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    &   (𝜑 → 𝐶 <s 𝐷)    ⇒   (𝜑 → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) <s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶)))
 
6-Mar-2025mulscld 27830 The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 ·s 𝐵) ∈ No )
 
6-Mar-2025rspc2dv 3625 2-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 6-Mar-2025.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))    &   (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓)    &   (𝜑 → 𝐴 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝐷)    ⇒   (𝜑 → 𝜒)
 
5-Mar-2025evlsmhpvvval 41469 Give a formula for the evaluation of a homogeneous polynomial given assignments from variables to values. The difference between this and evlsvvval 41437 is that 𝑏 ∈ 𝐷 is restricted to 𝑏 ∈ 𝐺, that is, we can evaluate an 𝑁-th degree homogeneous polynomial over just the terms where the sum of all variable degrees is 𝑁. (Contributed by SN, 5-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   𝐾 = (Base‘𝑆)    &   𝑀 = (mulGrp‘𝑆)    &    ↑ = (.g‘𝑀)    &    · = (.r‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐹 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
 
5-Mar-2025evlvvval 41447 Give a formula for the evaluation of a polynomial given assignments from variables to values. (Contributed by SN, 5-Mar-2025.)
𝑄 = (𝐼 eval 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐾 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    ↑ = (.g‘𝑀)    &    · = (.r‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑅 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
 
5-Mar-2025evlsvvval 41437 Give a formula for the evaluation of a polynomial given assignments from variables to values. This is the sum of the evaluations for each term (corresponding to a bag of variables), that is, the coefficient times the product of each variable raised to the corresponding power. (Contributed by SN, 5-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝐵 = (Base‘𝑃)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐾 = (Base‘𝑆)    &   𝑀 = (mulGrp‘𝑆)    &    ↑ = (.g‘𝑀)    &    · = (.r‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
 
5-Mar-2025sltmul 27831 An ordering relationship for surreal multiplication. Compare theorem 8(iii) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.)
(((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No )) → ((𝐴 <s 𝐵 ∧ 𝐶 <s 𝐷) → ((𝐴 ·s 𝐷) -s (𝐴 ·s 𝐶)) <s ((𝐵 ·s 𝐷) -s (𝐵 ·s 𝐶))))
 
5-Mar-2025mulscl 27829 The surreals are closed under multiplication. Theorem 8(i) of [Conway] p. 19. (Contributed by Scott Fenton, 5-Mar-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·s 𝐵) ∈ No )
 
5-Mar-2025mulsprop 27825 Surreals are closed under multiplication and obey a particular ordering law. Theorem 3.4 of [Gonshor] p. 17. (Contributed by Scott Fenton, 5-Mar-2025.)
(((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))))
 
5-Mar-2025mulsproplem14 27824 Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 27822 and mulsproplem13 27823. This completes the induction on surreal multiplication. mulsprop 27825 brings all this together technically. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → 𝐸 ∈ No )    &   (𝜑 → 𝐹 ∈ No )    &   (𝜑 → 𝐶 <s 𝐷)    &   (𝜑 → 𝐸 <s 𝐹)    ⇒   (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
 
5-Mar-2025mulsproplem13 27823 Lemma for surreal multiplication. Remove the restriction on 𝐶 and 𝐷 from mulsproplem12 27822. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → 𝐸 ∈ No )    &   (𝜑 → 𝐹 ∈ No )    &   (𝜑 → 𝐶 <s 𝐷)    &   (𝜑 → 𝐸 <s 𝐹)    &   (𝜑 → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))    ⇒   (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
 
5-Mar-2025mulsproplem12 27822 Lemma for surreal multiplication. Demonstrate the second half of the inductive statement assuming 𝐶 and 𝐷 are not the same age and 𝐸 and 𝐹 are not the same age. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐷 ∈ No )    &   (𝜑 → 𝐸 ∈ No )    &   (𝜑 → 𝐹 ∈ No )    &   (𝜑 → 𝐶 <s 𝐷)    &   (𝜑 → 𝐸 <s 𝐹)    &   (𝜑 → (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)))    &   (𝜑 → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))    ⇒   (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
 
5-Mar-2025mulsproplem11 27821 Lemma for surreal multiplication. Under the inductive hypothesis, demonstrate closure of surreal multiplication. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 ·s 𝐵) ∈ No )
 
5-Mar-2025mulsproplem10 27820 Lemma for surreal multiplication. State the cut properties of surreal multiplication. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<s ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
 
5-Mar-2025mulsproplem9 27819 Lemma for surreal multiplication. Show that the cut involved in surreal multiplication makes sense. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
 
5-Mar-2025mulsproplem8 27818 Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (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)))
 
5-Mar-2025mulsproplem7 27817 Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (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)))
 
5-Mar-2025mulsproplem6 27816 Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (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)))
 
5-Mar-2025mulsproplemcbv 27810 Lemma for surreal multiplication. Change some bound variables for later use. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    ⇒   (𝜑 → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
 
5-Mar-2025cbvral6vw 3240 Change bound variables of sextuple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝑥 = 𝑎 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑏 → (𝜒 ↔ 𝜃))    &   (𝑧 = 𝑐 → (𝜃 ↔ 𝜏))    &   (𝑤 = 𝑑 → (𝜏 ↔ 𝜂))    &   (𝑝 = 𝑒 → (𝜂 ↔ 𝜁))    &   (𝑞 = 𝑓 → (𝜁 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 𝜓)
 
5-Mar-20256ralbidv 3221 Formula-building rule for restricted universal quantifiers (deduction form.) (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜒))
 
5-Mar-2025ralimd6v 3206 Deduction sextupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 𝜒))
 
5-Mar-20256ralimi 3125 Inference quantifying both antecedent and consequent six times, with strong hypothesis. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → 𝜓)    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 ∀𝑢 ∈ 𝐹 𝜓)
 
5-Mar-20255ralimi 3124 Inference quantifying both antecedent and consequent five times, with strong hypothesis. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → 𝜓)    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑡 ∈ 𝐸 𝜓)
 
5-Mar-20254ralimi 3123 Inference quantifying both antecedent and consequent four times, with strong hypothesis. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → 𝜓)    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓)
 
5-Mar-20253ralimi 3122 Inference quantifying both antecedent and consequent three times, with strong hypothesis. (Contributed by Scott Fenton, 5-Mar-2025.)
(𝜑 → 𝜓)    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
 
5-Mar-2025nabbib 3043 Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.) Definitial form. (Revised by Wolf Lammen, 5-Mar-2025.)
({𝑥 ∣ 𝜑} ≠ {𝑥 ∣ 𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
 
4-Mar-2025mulsproplem5 27815 Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (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)))
 
4-Mar-2025mulsproplem4 27814 Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))    &   (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))    ⇒   (𝜑 → (𝑋 ·s 𝑌) ∈ No )
 
4-Mar-2025mulsproplem3 27813 Lemma for surreal multiplication. Under the inductive hypothesis, the product of 𝐴 itself and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))    ⇒   (𝜑 → (𝐴 ·s 𝑌) ∈ No )
 
4-Mar-2025mulsproplem2 27812 Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and 𝐵 itself is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝑋 ·s 𝐵) ∈ No )
 
4-Mar-2025mulsproplem1 27811 Lemma for surreal multiplication. Instantiate some variables. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → 𝑍 ∈ No )    &   (𝜑 → 𝑊 ∈ No )    &   (𝜑 → 𝑇 ∈ No )    &   (𝜑 → 𝑈 ∈ No )    &   (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘𝑍) +no ( bday ‘𝑇)) ∪ (( bday ‘𝑊) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑍) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑊) +no ( bday ‘𝑇))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))    ⇒   (𝜑 → ((𝑋 ·s 𝑌) ∈ No ∧ ((𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈) → ((𝑍 ·s 𝑈) -s (𝑍 ·s 𝑇)) <s ((𝑊 ·s 𝑈) -s (𝑊 ·s 𝑇)))))
 
4-Mar-2025addsubsd 27788 Law for surreal addition and subtraction. (Contributed by Scott Fenton, 4-Mar-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 +s 𝐵) -s 𝐶) = ((𝐴 -s 𝐶) +s 𝐵))
 
3-Mar-2025ablsubaddsub 19723 Double subtraction and addition in abelian groups. (cnambpcma 46300 analog.) (Contributed by AV, 3-Mar-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    − = (-g‘𝐺)    ⇒   ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((𝑋 − 𝑌) + 𝑍) − 𝑋) = (𝑍 − 𝑌))
 
2-Mar-2025tfsconcatrnsson 42403 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-2025tfsconcatrnss 42402 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-2025tfsconcatrnss12 42401 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-2025tfsconcatrev 42400 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-2025isdrng4 32665 A division ring is a ring in which 1 ≠ 0 and every nonzero element has a left and right inverse. (Contributed by Thierry Arnoux, 2-Mar-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (𝑅 ∈ DivRing ↔ ( 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 1 ∧ (𝑦 · 𝑥) = 1 ))))
 
2-Mar-2025isassad 21638 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-2025assasca 21636 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-2025isassa 21630 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-2025df-assa 21627 Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a 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-2025ablsubadd23 19722 Commutative/associative law for addition and subtraction in abelian groups. (subadd23d 11597 analog.) (Contributed by AV, 2-Mar-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    − = (-g‘𝐺)    ⇒   ((𝐺 ∈ Abel ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 − 𝑌) + 𝑍) = (𝑋 + (𝑍 − 𝑌)))
 
2-Mar-2025sbcie2s 17098 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))
 
2-Mar-2025cbvral8vw 3241 Change bound variables of octuple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 2-Mar-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))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 ∀𝑝 ∈ 𝐸 ∀𝑞 ∈ 𝐹 ∀𝑟 ∈ 𝐺 ∀𝑠 ∈ 𝐻 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 ∀𝑒 ∈ 𝐸 ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐺 ∀ℎ ∈ 𝐻 𝜓)
 
2-Mar-2025cbvral4vw 3239 Change bound variables of quadruple restricted universal quantification, using implicit substitution. (Contributed by Scott Fenton, 2-Mar-2025.)
(𝑥 = 𝑎 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑏 → (𝜒 ↔ 𝜃))    &   (𝑧 = 𝑐 → (𝜃 ↔ 𝜏))    &   (𝑤 = 𝑑 → (𝜏 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜑 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ 𝐷 𝜓)
 
2-Mar-2025ralimd4v 3205 Deduction quadrupally quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜒))
 
2-Mar-2025ralimdvv 3204 Deduction doubly quantifying both antecedent and consequent. (Contributed by Scott Fenton, 2-Mar-2025.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
 
1-Mar-2025rp-tfslim 42405 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-2025tfsnfin 42404 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-2025tfsconcat0b 42398 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))
 
1-Mar-2025quselbas 19099 Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)
∼ = (𝐺 ~QG 𝑆)    &   𝑈 = (𝐺 /s ∼ )    &   𝐵 = (Base‘𝐺)    ⇒   ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))
 
28-Feb-2025tfsconcat0i 42397 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))
 
28-Feb-2025sltaddsub2d 27798 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 +s 𝐵) <s 𝐶 ↔ 𝐵 <s (𝐶 -s 𝐴)))
 
28-Feb-2025sltaddsubd 27797 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 +s 𝐵) <s 𝐶 ↔ 𝐴 <s (𝐶 -s 𝐵)))
 
28-Feb-2025rngqiprnglin 21061 𝐹 is linear with respect to the multiplication. (Contributed by AV, 28-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝐹‘(𝑎 · 𝑏)) = ((𝐹‘𝑎)(.r‘𝑃)(𝐹‘𝑏)))
 
28-Feb-2025rngqiprnglinlem3 21052 Lemma 3 for rngqiprnglin 21061. (Contributed by AV, 28-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (.r‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
 
28-Feb-2025rngqiprnglinlem2 21051 Lemma 2 for rngqiprnglin 21061. (Contributed by AV, 28-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴 · 𝐶)] ∼ = ([𝐴] ∼ (.r‘𝑄)[𝐶] ∼ ))
 
28-Feb-2025qusmulrng 21028 Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 21029. Similar to qusmul2 21025. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
∼ = (𝑅 ~QG 𝑆)    &   𝐻 = (𝑅 /s ∼ )    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    ∙ = (.r‘𝐻)    ⇒   (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
 
28-Feb-2025xpsringd 20220 A product of two rings is a ring (xpsmnd 18699 analog). (Contributed by AV, 28-Feb-2025.)
𝑌 = (𝑆 ×s 𝑅)    &   (𝜑 → 𝑆 ∈ Ring)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝑌 ∈ Ring)
 
28-Feb-20254ralbii 3129 Inference adding four restricted universal quantifiers to both sides of an equivalence. (Contributed by Scott Fenton, 28-Feb-2025.)
(𝜑 ↔ 𝜓)    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ∀𝑤 ∈ 𝐷 𝜓)
 
27-Feb-2025sltsubadd2d 27796 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐵) <s 𝐶 ↔ 𝐴 <s (𝐵 +s 𝐶)))
 
27-Feb-2025sltsubaddd 27795 Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐵) <s 𝐶 ↔ 𝐴 <s (𝐶 +s 𝐵)))
 
27-Feb-2025rngisomring 20358 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then both rings are unital. (Contributed by AV, 27-Feb-2025.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑆 ∈ Ring)
 
27-Feb-2025rngisom1 20357 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
1 = (1r‘𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r‘𝑆)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ 𝐵 (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))
 
27-Feb-2025rngisomfv1 20356 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is an element of the base set of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
1 = (1r‘𝑅)    &   𝐵 = (Base‘𝑆)    ⇒   ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
 
27-Feb-2025rngimcnv 20347 The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025.)
(𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))
 
27-Feb-2025imasringf1 20219 The image of a ring under an injection is a ring (imasmndf1 18698 analog). (Contributed by AV, 27-Feb-2025.)
𝑈 = (𝐹 “s 𝑅)    &   𝑉 = (Base‘𝑅)    ⇒   ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring)
 
26-Feb-2025qus0subgadd 19114 The addition in a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.)
0 = (0g‘𝐺)    &   𝑆 = { 0 }    &    ∼ = (𝐺 ~QG 𝑆)    &   𝑈 = (𝐺 /s ∼ )    &   𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ({𝑎} (+g‘𝑈){𝑏}) = {(𝑎(+g‘𝐺)𝑏)})
 
26-Feb-2025qus0subgbas 19113 The base set of a quotient of a group by the trivial (zero) subgroup. (Contributed by AV, 26-Feb-2025.)
0 = (0g‘𝐺)    &   𝑆 = { 0 }    &    ∼ = (𝐺 ~QG 𝑆)    &   𝑈 = (𝐺 /s ∼ )    &   𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → (Base‘𝑈) = {𝑢 ∣ ∃𝑥 ∈ 𝐵 𝑢 = {𝑥}})
 
26-Feb-2025eqg0subgecsn 19112 The equivalence classes modulo the coset equivalence relation for the trivial (zero) subgroup of a group are singletons. (Contributed by AV, 26-Feb-2025.)
0 = (0g‘𝐺)    &   𝑆 = { 0 }    &   𝐵 = (Base‘𝐺)    &   𝑅 = (𝐺 ~QG 𝑆)    ⇒   ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → [𝑋]𝑅 = {𝑋})
 
26-Feb-2025abid2f 2934 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-2025tfsconcat00 42399 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-2025tfsconcatb0 42396 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-2025evls1maprnss 33050 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-2025evls1maplmhm 33049 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-2025lmimdim 32976 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-2025evls1vsca 32924 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-2025rhmqusker 32818 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 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    &   (𝜑 → 𝐺 ∈ CRing)    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   (𝜑 → 𝐽 ∈ (𝑄 RingIso 𝐻))
 
25-Feb-2025lmhmqusker 32808 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 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 LMHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   (𝜑 → ran 𝐹 = (Base‘𝐻))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   (𝜑 → 𝐽 ∈ (𝑄 LMIso 𝐻))
 
25-Feb-2025fldgenssp 32678 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-2025fldgenssv 32675 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-2025rndrhmcl 32666 The image of a division ring by a ring homomorphism is a division ring. (Contributed by Thierry Arnoux, 25-Feb-2025.)
𝑅 = (𝑁 ↾s ran 𝐹)    &    0 = (0g‘𝑁)    &   (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   (𝜑 → ran 𝐹 ≠ { 0 })    &   (𝜑 → 𝑀 ∈ DivRing)    ⇒   (𝜑 → 𝑅 ∈ DivRing)
 
25-Feb-2025rngqiprngghmlem2 21047 Lemma 2 for rngqiprngghm 21058. (Contributed by AV, 25-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴)(+g‘𝐽)( 1 · 𝐶)) ∈ (Base‘𝐽))
 
25-Feb-2025rngqiprngghmlem1 21046 Lemma 1 for rngqiprngghm 21058. (Contributed by AV, 25-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    ⇒   ((𝜑 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ (Base‘𝐽))
 
25-Feb-2025subrngringnsg 20441 A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
 
25-Feb-2025rngansg 20064 Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
(𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
 
25-Feb-2025eqg0subg 19111 The coset equivalence relation for the trivial (zero) subgroup of a group is the identity relation restricted to the base set of the group. (Contributed by AV, 25-Feb-2025.)
0 = (0g‘𝐺)    &   𝑆 = { 0 }    &   𝐵 = (Base‘𝐺)    &   𝑅 = (𝐺 ~QG 𝑆)    ⇒   (𝐺 ∈ Grp → 𝑅 = ( I ↾ 𝐵))
 
25-Feb-2025ecqusaddd 19107 Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)
(𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → [(𝐴(+g‘𝑅)𝐶)] ∼ = ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ))
 
25-Feb-2025xpsmnd0 18700 The identity element of a binary product of monoids. (Contributed by AV, 25-Feb-2025.)
𝑇 = (𝑅 ×s 𝑆)    ⇒   ((𝑅 ∈ Mnd ∧ 𝑆 ∈ Mnd) → (0g‘𝑇) = ⟨(0g‘𝑅), (0g‘𝑆)⟩)
 
25-Feb-2025ressbasss 17187 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-2025tfsconcatfo 42395 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-2025tfsconcatrn 42394 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-2025tfsconcatfv 42393 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))))
 
24-Feb-2025rngqiprngghm 21058 𝐹 is a homomorphism of the additive groups of non-unital rings. (Contributed by AV, 24-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑃))
 
24-Feb-2025rngqiprngimfv 21057 The value of the function 𝐹 at an element of (the base set of) a non-unital ring. (Contributed by AV, 24-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   ((𝜑 ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = ⟨[𝐴] ∼ , ( 1 · 𝐴)⟩)
 
24-Feb-2025ecqusaddcl 19108 Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.)
(𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    ⇒   ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ([𝐴] ∼ (+g‘𝑄)[𝐶] ∼ ) ∈ (Base‘𝑄))
 
24-Feb-2025quseccl0 19100 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 19102 for arbitrary sets 𝐺. (Revised by AV, 24-Feb-2025.)
∼ = (𝐺 ~QG 𝑆)    &   𝐻 = (𝐺 /s ∼ )    &   𝐶 = (Base‘𝐺)    &   𝐵 = (Base‘𝐻)    ⇒   ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → [𝑋] ∼ ∈ 𝐵)
 
23-Feb-2025tfsconcatfv2 42392 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-2025tfsconcatfv1 42391 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-2025tfsconcatun 42389 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-2025tfsconcatlem 42388 Lemma for tfsconcatun 42389. (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-2025rngqiprng 21055 The product of the quotient with a two-sided ideal and the two-sided ideal is a non-unital ring. (Contributed by AV, 23-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    ⇒   (𝜑 → 𝑃 ∈ Rng)
 
23-Feb-2025qus2idrng 21044 The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 21023 analog). (Contributed by AV, 23-Feb-2025.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
 
23-Feb-20252idlcpblrng 21020 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
23-Feb-2025lringuplu 20432 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-2025lringnz 20431 A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
1 = (1r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ LRing → 1 ≠ 0 )
 
23-Feb-2025lringring 20430 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-2025lringnzr 20429 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
(𝑅 ∈ LRing → 𝑅 ∈ NzRing)
 
23-Feb-2025islring 20428 The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    1 = (1r‘𝑅)    &   𝑈 = (Unit‘𝑅)    ⇒   (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
 
23-Feb-2025df-lring 20427 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) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
 
23-Feb-202501eq0ring 20419 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 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
 
23-Feb-2025nzrring 20407 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)
 
23-Feb-2025qusrng 20074 The quotient structure of a non-unital ring is a non-unital ring (qusring2 20222 analog). (Contributed by AV, 23-Feb-2025.)
(𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → ∼ Er 𝑉)    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞)))    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑈 ∈ Rng)
 
23-Feb-2025rngsubdir 20066 Ring multiplication distributes over subtraction. (subdir 11652 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 20196. (Revised by AV, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍)))
 
23-Feb-2025rngsubdi 20065 Ring multiplication distributes over subtraction. (subdi 11651 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 20195. (Revised by AV, 23-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍)))
 
23-Feb-2025abbib 2802 Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2106 and df-clel 2808 (by avoiding use of cleqh 2861). (Revised by BJ, 23-Jun-2019.) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025.)
({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
 
22-Feb-2025tfsconcatfn 42390 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-2025ordsssucb 42387 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 42455, limsssuc 7841. (Contributed by RP, 22-Feb-2025.)
((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
 
22-Feb-2025xpsrngd 20073 A product of two non-unital rings is a non-unital ring (xpsmnd 18699 analog). (Contributed by AV, 22-Feb-2025.)
𝑌 = (𝑆 ×s 𝑅)    &   (𝜑 → 𝑆 ∈ Rng)    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑌 ∈ Rng)
 
22-Feb-2025imasrngf1 20072 The image of a non-unital ring under an injection is a non-unital ring (imasmndf1 18698 analog). (Contributed by AV, 22-Feb-2025.)
𝑈 = (𝐹 “s 𝑅)    &   𝑉 = (Base‘𝑅)    ⇒   ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Rng) → 𝑈 ∈ Rng)
 
22-Feb-2025imasrng 20071 The image structure of a non-unital ring is a non-unital ring (imasring 20218 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑 → 𝑅 ∈ Rng)    ⇒   (𝜑 → 𝑈 ∈ Rng)
 
22-Feb-2025prdsrngd 20070 A product of non-unital rings is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅:𝐼⟶Rng)    ⇒   (𝜑 → 𝑌 ∈ Rng)
 
22-Feb-2025rngmgpf 20051 Restricted functionality of the multiplicative group on non-unital rings (mgpf 20142 analog). (Contributed by AV, 22-Feb-2025.)
(mulGrp ↾ Rng):Rng⟶Smgrp
 
22-Feb-2025imasabl 19785 The image structure of an abelian group is an abelian group (imasgrp 18975 analog). (Contributed by AV, 22-Feb-2025.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → 𝐹:𝑉–onto→𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑 → 𝑅 ∈ Abel)    &    0 = (0g‘𝑅)    ⇒   (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
 
21-Feb-2025sltsubsub3bd 27791 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-2025sltsubsub2bd 27790 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-2025lltropt 27604 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)
 
21-Feb-2025rngqiprngim 21063 𝐹 is an isomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → 𝐹 ∈ (𝑅 RngIso 𝑃))
 
21-Feb-2025rngqiprngho 21062 𝐹 is a homomorphism of non-unital rings. (Contributed by AV, 21-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑃))
 
21-Feb-2025rngqiprngimf 21056 𝐹 is a function from (the base set of) a non-unital ring to the product of the (base set 𝐶 of the) quotient with a two-sided ideal and the (base set 𝐼 of the) two-sided ideal (because of 2idlbas 21018, (Base‘𝐽) = 𝐼!) (Contributed by AV, 21-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ ⟨[𝑥] ∼ , ( 1 · 𝑥)⟩)    ⇒   (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼))
 
21-Feb-2025rngqipbas 21054 The base set of the product of the quotient with a two-sided ideal and the two-sided ideal is the cartesian product of the base set of the quotient and the base set of the two-sided ideal. (Contributed by AV, 21-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝐽)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑄 = (𝑅 /s ∼ )    &   𝐶 = (Base‘𝑄)    &   𝑃 = (𝑄 ×s 𝐽)    ⇒   (𝜑 → (Base‘𝑃) = (𝐶 × 𝐼))
 
21-Feb-2025prdsmulrngcl 20069 Closure of the multiplication in a structure product of non-unital rings. (Contributed by Mario Carneiro, 11-Mar-2015.) Generalization of prdsmulrcl 20208. (Revised by AV, 21-Feb-2025.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = (.r‘𝑌)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑅:𝐼⟶Rng)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
 
21-Feb-2025prdssgrpd 18658 The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅:𝐼⟶Smgrp)    ⇒   (𝜑 → 𝑌 ∈ Smgrp)
 
21-Feb-2025prdsplusgsgrpcl 18657 Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    + = (+g‘𝑌)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑅:𝐼⟶Smgrp)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (𝐹 + 𝐺) ∈ 𝐵)
 
20-Feb-2025ply1degltdim 32996 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-2025ply1degltdimlem 32995 Lemma for ply1degltdim 32996. (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-2025ply1gsumz 32944 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 = (0g‘𝑅)    &   𝑍 = (0g‘𝑃)    &   (𝜑 → 𝐴:(0..^𝑁)⟶𝐵)    &   (𝜑 → (𝑃 Σg (𝐴 ∘f ( ·𝑠 ‘𝑃)𝐹)) = 𝑍)    ⇒   (𝜑 → 𝐴 = ((0..^𝑁) × { 0 }))
 
20-Feb-2025gsummoncoe1fzo 32943 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 = (0g‘𝑅)    &   (𝜑 → ∀𝑘 ∈ (0..^𝑁)𝐴 ∈ 𝐾)    &   (𝜑 → 𝐿 ∈ (0..^𝑁))    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝑘 = 𝐿 → 𝐴 = 𝐶)    ⇒   (𝜑 → ((coe1‘(𝑃 Σg (𝑘 ∈ (0..^𝑁) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = 𝐶)
 
20-Feb-2025ply1degltlss 32942 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-2025ply1degltel 32940 Characterize elementhood in 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-2025ply1moneq 32939 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-2025coe1mon 32938 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 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝜑 → (coe1‘(𝑁 ↑ 𝑋)) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝑁, 1 , 0 )))
 
20-Feb-2025islbs5 32770 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 = (0g‘𝑊)    &    0 = (0g‘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 Σg (𝑎 ∘f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵)))
 
20-Feb-2025suppssnn0 32284 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-2025nn0difffzod 32283 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-2025mptiffisupp 32182 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-2025ifnebib 32048 The converse of ifbi 4549 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-2025ifnefals 32047 Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
 
20-Feb-2025ifnetrue 32046 Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
 
20-Feb-2025eqelbid 31982 A variable elimination law for equality within a given set 𝐴. See equvel 2453. (Contributed by Thierry Arnoux, 20-Feb-2025.)
(𝜑 → 𝐵 ∈ 𝐴)    &   (𝜑 → 𝐶 ∈ 𝐴)    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶))
 
20-Feb-2025ring2idlqusb 21069 A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
(𝑅 ∈ Rng → (𝑅 ∈ Ring ↔ ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring)))
 
20-Feb-2025rngringbd 21067 A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   (𝜑 → (𝑅 ∈ Ring ↔ 𝑄 ∈ Ring))
 
20-Feb-2025rngringbdlem1 21065 In a unital ring, the quotient of the ring and a two-sided ideal is unital. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ((𝜑 ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
 
20-Feb-2025rng2idlsubg0 21043 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → (0g‘𝑅) ∈ 𝐼)
 
20-Feb-2025rng2idlsubgnsg 21042 A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → 𝐼 ∈ (SubGrp‘𝑅))    ⇒   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
 
20-Feb-2025rng2idl0 21040 The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → (0g‘𝑅) ∈ 𝐼)
 
20-Feb-2025rng2idlnsg 21039 A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng)    ⇒   (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅))
 
20-Feb-20252idlelbas 21019 The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   𝐵 = (Base‘𝐽)    ⇒   (𝜑 → (𝐵 ∈ (LIdeal‘𝑅) ∧ 𝐵 ∈ (LIdeal‘(oppr‘𝑅))))
 
20-Feb-20252idlbas 21018 The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
(𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   𝐵 = (Base‘𝐽)    ⇒   (𝜑 → 𝐵 = 𝐼)
 
20-Feb-20252idlelb 21014 Membership in a two-sided ideal. Formerly part of proof for 2idlcpbl 21021. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr‘𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)    ⇒   (𝑈 ∈ 𝑇 ↔ (𝑈 ∈ 𝐼 ∧ 𝑈 ∈ 𝐽))
 
20-Feb-2025naddlid 8685 Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 20-Feb-2025.)
(𝐴 ∈ On → (∅ +no 𝐴) = 𝐴)
 
20-Feb-2025rspc8v 3631 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-2025rspc6v 3630 6-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 ↔ 𝜓))    ⇒   (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑈) ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑇 ∀𝑤 ∈ 𝑈 ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑊 𝜑 → 𝜓))
 
20-Feb-20254ralbidv 3220 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-20253ralbidv 3219 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-2025onmcl 42383 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-2025oaabsb 42346 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-2025rimrcl2 41395 Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)
 
19-Feb-2025rimrcl1 41394 Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)
 
19-Feb-2025ply1lvec 32912 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-2025rnglidl1 21033 The base set of every non-unital ring is an ideal. For unital rings, such ideals are called "unit ideals", see lidl1 20994. (Contributed by AV, 19-Feb-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝐵 ∈ 𝑈)
 
19-Feb-2025rnglidl0 21032 Every non-unital ring contains a zero ideal. (Contributed by AV, 19-Feb-2025.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Rng → { 0 } ∈ 𝑈)
 
19-Feb-2025rngridlmcl 20983 A right ideal (which is a left ideal over the opposite ring) containing the zero element is closed under right-multiplication by elements of the full non-unital ring. (Contributed by AV, 19-Feb-2025.)
0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (LIdeal‘(oppr‘𝑅))    ⇒   (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑌 · 𝑋) ∈ 𝐼)
 
19-Feb-2025sdrgunit 20555 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 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑆)    ⇒   (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 )))
 
19-Feb-2025sdrgbas 20553 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-2025sdrgsubrg 20550 A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
(𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))
 
19-Feb-2025sdrgdrng 20549 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-2025sdrgrcl 20548 Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
(𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
 
19-Feb-2025isdrngrd 20534 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 20533 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 = (0g‘𝑅))    &   (𝜑 → 1 = (1r‘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-2025isdrngd 20533 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 20534 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 = (0g‘𝑅))    &   (𝜑 → 1 = (1r‘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-2025selvmul 41463 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-2025evlmulval 41450 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-2025ricfld 41408 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-2025ricdrng 41407 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-2025ricdrng1 41406 A ring isomorphism maps a division ring to a division ring. (Contributed by SN, 18-Feb-2025.)
((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ DivRing) → 𝑆 ∈ DivRing)
 
18-Feb-2025rnglidlmcl 20982 A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven as in lidlmcl 20989. (Contributed by AV, 18-Feb-2025.)
0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ 𝐼 ∈ 𝑈 ∧ 0 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐼)) → (𝑋 · 𝑌) ∈ 𝐼)
 
17-Feb-2025imadrhmcl 20556 The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)
𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))    &    0 = (0g‘𝑁)    &   (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   (𝜑 → 𝑆 ∈ (SubDRing‘𝑀))    &   (𝜑 → ran 𝐹 ≠ { 0 })    ⇒   (𝜑 → 𝑅 ∈ DivRing)
 
17-Feb-2025subrngpropd 20456 If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))
 
17-Feb-2025cntzsubrng 20455 Centralizers in a non-unital ring are subrings. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (Cntz‘𝑀)    ⇒   ((𝑅 ∈ Rng ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRng‘𝑅))
 
17-Feb-2025rngm2neg 20063 Double negation of a product in a non-unital ring (mul2neg 11657 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 20194. (Revised by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌))
 
17-Feb-2025rngmneg2 20062 Negation of a product in a non-unital ring (mulneg2 11655 analog). In contrast to ringmneg2 20193, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌)))
 
17-Feb-2025rngmneg1 20061 Negation of a product in a non-unital ring (mulneg1 11654 analog). In contrast to ringmneg1 20192, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑁 = (invg‘𝑅)    &   (𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))
 
17-Feb-2025cntzsgrpcl 19239 Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)    &   𝐶 = (𝑍‘𝑆)    ⇒   ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
 
16-Feb-2025sltled 27508 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)
 
16-Feb-2025rhmimasubrng 20454 The homomorphic image of a subring is a subring. (Contributed by AV, 16-Feb-2025.)
((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))
 
16-Feb-2025rhmimasubrnglem 20453 Lemma for rhmimasubrng 20454: Modified part of mhmima 18742. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)
𝑀 = (mulGrp‘𝑅)    ⇒   ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
 
16-Feb-2025rngrz 20060 The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 20182. (Revised by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
 
16-Feb-2025rng0cl 20057 The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Rng → 0 ∈ 𝐵)
 
16-Feb-2025rngacl 20056 Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
16-Feb-2025rnggrp 20052 A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
(𝑅 ∈ Rng → 𝑅 ∈ Grp)
 
16-Feb-2025mhmimalem 18741 Lemma for mhmima 18742 and similar theorems, formerly part of proof for mhmima 18742. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)
(𝜑 → 𝐹 ∈ (𝑀 MndHom 𝑁))    &   (𝜑 → 𝑋 ⊆ (Base‘𝑀))    &   (𝜑 → ⊕ = (+g‘𝑀))    &   (𝜑 → + = (+g‘𝑁))    &   ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧 ⊕ 𝑥) ∈ 𝑋)    ⇒   (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋))
 
15-Feb-2025rexanuz2nf 44501 A simple counterexample related to theorem rexanuz2 15300, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15300 and similar theorems would not hold (see rexanre 15297 and rexanuz 15296). (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-2025cvgcaule 44500 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-2025cvgcau 44499 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-2025caucvgbf 44498 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-2025fvmpt4d 44279 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-2025rspced 44163 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-2025rexeqif 44162 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-2025notbicom 44161 Commutative law for the negation of a biconditional. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜑 ↔ 𝜓)    ⇒    ¬ (𝜓 ↔ 𝜑)
 
15-Feb-2025nimnbi2 44160 If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜓 → 𝜑)    ⇒    ¬ (𝜑 ↔ 𝜓)
 
15-Feb-2025nimnbi 44159 If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜑 → 𝜓)    ⇒    ¬ (𝜑 ↔ 𝜓)
 
15-Feb-2025eliund 44158 Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
(𝜑 → ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)    ⇒   (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)
 
15-Feb-2025naddwordnexlem4 42454 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 ∖ 1o)((𝐶 +o 𝑥) = 𝐷 ∧ (𝐴 +o (ω ·o 𝑥)) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +no (ω ·o 𝑥))))
 
15-Feb-2025ghmqusker 32806 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 = (0g‘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-2025ghmquskerco 32803 In the case of theorem ghmqusker 32806, 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 = (0g‘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-2025subsubrng2 20452 The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))
 
15-Feb-2025subsubrng 20451 A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
 
15-Feb-2025subrngmre 20450 The subrings of a non-unital ring are a Moore system. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Rng → (SubRng‘𝑅) ∈ (Moore‘𝐵))
 
15-Feb-2025subrngin 20449 The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.)
((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRng‘𝑅))
 
15-Feb-2025subrngint 20448 The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
((𝑆 ⊆ (SubRng‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRng‘𝑅))
 
15-Feb-2025opprsubrng 20447 Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (SubRng‘𝑅) = (SubRng‘𝑂)
 
15-Feb-2025issubrng2 20446 Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
 
15-Feb-2025opprrngb 20237 A class is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 20236. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng)
 
15-Feb-2025opprrng 20236 An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝑂 ∈ Rng)
 
15-Feb-2025rngpropd 20068 If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a non-unital ring iff the other one is. (Contributed by AV, 15-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ Rng ↔ 𝐿 ∈ Rng))
 
15-Feb-2025sgrppropd 18656 If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)
(𝜑 → 𝐾 ∈ 𝑉)    &   (𝜑 → 𝐿 ∈ 𝑊)    &   (𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ Smgrp ↔ 𝐿 ∈ Smgrp))
 
15-Feb-2025sgrpcl 18651 Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
𝐵 = (Base‘𝐺)    &    ⚬ = (+g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
 
15-Feb-2025naddword2 8693 Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 15-Feb-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +no 𝐴))
 
15-Feb-2025eqimsscd 4038 Equality implies inclusion, deduction version. (Contributed by SN, 15-Feb-2025.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → 𝐵 ⊆ 𝐴)
 
14-Feb-2025oawordex3 42453 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 8559. (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-2025naddwordnexlem3 42452 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-2025naddwordnexlem2 42451 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-2025naddwordnexlem1 42450 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-2025naddwordnexlem0 42449 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-2025ghmquskerlem2 32804 Lemma for ghmqusker 32806. (Contributed by Thierry Arnoux, 14-Feb-2025.)
0 = (0g‘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-2025ghmquskerlem1 32802 Lemma for ghmqusker 32806 (Contributed by Thierry Arnoux, 14-Feb-2025.)
0 = (0g‘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-2025ecref 32200 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-2025imaexd 32171 The image of a set is a set. Deduction version of imaexg 7908. (Contributed by Thierry Arnoux, 14-Feb-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 “ 𝐵) ∈ V)
 
14-Feb-2025rnexd 32170 The range of a set is a set. Deduction version of rnexd 32170. (Contributed by Thierry Arnoux, 14-Feb-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → ran 𝐴 ∈ V)
 
14-Feb-2025negsunif 27768 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 → ( -us ‘𝐴) = (( -us “ 𝑅) |s ( -us “ 𝐿)))
 
14-Feb-2025mins2 27507 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-2025mins1 27506 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-2025maxs2 27505 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-2025maxs1 27504 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-2025sletric 27503 Surreal trichotomy law. (Contributed by Scott Fenton, 14-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ∨ 𝐵 ≤s 𝐴))
 
14-Feb-2025rngringbdlem2 21066 A non-unital ring is unital if and only if there is a (two-sided) ideal of the ring which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 14-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &   𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ((𝜑 ∧ 𝑄 ∈ Ring) → 𝑅 ∈ Ring)
 
14-Feb-2025subrngmcl 20445 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 20474. (Revised by AV, 14-Feb-2025.)
· = (.r‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
14-Feb-2025subrngacl 20444 A subring is closed under addition. (Contributed by AV, 14-Feb-2025.)
+ = (+g‘𝑅)    ⇒   ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
 
14-Feb-2025subrng0 20443 A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    &    0 = (0g‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 0 = (0g‘𝑆))
 
14-Feb-2025subrngbas 20442 Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
 
14-Feb-2025subrngsubg 20440 A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
14-Feb-2025subrngrcl 20439 Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
(𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
 
14-Feb-2025subrngrng 20438 A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
𝑆 = (𝑅 ↾s 𝐴)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
 
14-Feb-2025subrngid 20437 Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))
 
14-Feb-2025subrngss 20436 A subring is a subset. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
 
14-Feb-2025issubrng 20435 The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    ⇒   (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
 
14-Feb-2025df-subrng 20434 Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
 
14-Feb-2025isrngd 20067 Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝑅))    &   (𝜑 → + = (+g‘𝑅))    &   (𝜑 → · = (.r‘𝑅))    &   (𝜑 → 𝑅 ∈ Abel)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    ⇒   (𝜑 → 𝑅 ∈ Rng)
 
14-Feb-2025rngdi 20054 Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
13-Feb-2025oaun3 42434 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-2025oaun2 42433 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-2025oadif1 42432 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-2025oadif1lem 42431 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-2025rp-abid 42430 Two ways to express a class. (Contributed by RP, 13-Feb-2025.)
𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}
 
13-Feb-2025oaun3lem3 42428 The class of all ordinal sums of elements from two ordinals is an ordinal. Lemma for oaun3 42434. (Contributed by RP, 13-Feb-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ∈ On)
 
13-Feb-2025oaun3lem2 42427 The class of all ordinal sums of elements from two ordinals is bounded by the sum. Lemma for oaun3 42434. (Contributed by RP, 13-Feb-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)} ⊆ (𝐴 +o 𝐵))
 
13-Feb-2025oaun3lem1 42426 The class of all ordinal sums of elements from two ordinals is ordinal. Lemma for oaun3 42434. (Contributed by RP, 13-Feb-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord {𝑥 ∣ ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑥 = (𝑎 +o 𝑏)})
 
13-Feb-2025ring2idlqus 21068 For every unital ring there is a (two-sided) ideal of the ring (in fact the base set of the ring itself) which is unital, and the quotient of the ring and the ideal is unital. (Proposed by GL, 12-Feb-2025.) (Contributed by AV, 13-Feb-2025.)
(𝑅 ∈ Ring → ∃𝑖 ∈ (2Ideal‘𝑅)((𝑅 ↾s 𝑖) ∈ Ring ∧ (𝑅 /s (𝑅 ~QG 𝑖)) ∈ Ring))
 
13-Feb-2025rng2idl1cntr 21064 The unity of a two-sided ideal of a non-unital ring is central, i.e., an element of the center of the multiplicative semigroup of the non-unital ring. This is part of the proof given in MathOverflow, which seems to be sufficient to show that 𝐹 given below (see rngqiprngimf 21056) is an isomorphism. In our proof, however we show that 𝐹 is linear regarding the multiplication (rngqiprnglin 21061) via rngqiprnglinlem1 21050 instead. (Contributed by AV, 13-Feb-2025.)
(𝜑 → 𝑅 ∈ Rng)    &   (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   𝐽 = (𝑅 ↾s 𝐼)    &   (𝜑 → 𝐽 ∈ Ring)    &    1 = (1r‘𝐽)    &   𝑀 = (mulGrp‘𝑅)    ⇒   (𝜑 → 1 ∈ (Cntr‘𝑀))
 
13-Feb-20252idl1 21013 Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝐼)
 
13-Feb-20252idl0 21012 Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
𝐼 = (2Ideal‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Ring → { 0 } ∈ 𝐼)
 
13-Feb-2025ridl1 21011 Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)
 
13-Feb-2025ridl0 21010 Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Ring → { 0 } ∈ 𝑈)
 
13-Feb-2025df2idl2 21009 Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
 
13-Feb-2025isridl 21008 A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘(oppr‘𝑅))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑦 · 𝑥) ∈ 𝐼)))
 
13-Feb-2025dflidl2 20992 Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
 
13-Feb-2025dflidl2lem 20991 Lemma for dflidl2 20992: a sufficient condition for a set to be a left ideal. (Contributed by AV, 13-Feb-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼) → 𝐼 ∈ 𝑈)
 
13-Feb-2025rngass 20053 Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
13-Feb-2025issgrpd 18655 Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
(𝜑 → 𝐵 = (Base‘𝐺))    &   (𝜑 → + = (+g‘𝐺))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → 𝐺 ∈ Smgrp)
 
13-Feb-2025eqab 2870 One direction of eqabb 2871 is provable from fewer axioms. (Contributed by Wolf Lammen, 13-Feb-2025.)
(∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → 𝐴 = {𝑥 ∣ 𝜑})
 
12-Feb-2025oaun3lem4 42429 The class of all ordinal sums of elements from two ordinals is less than the successor to the sum. Lemma for oaun3 42434. (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-2025onsucunitp 42425 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-2025onsucunipr 42424 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-2025onsucunifi 42422 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-2025eqabb 2871 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 2802 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 5300 to inex1 5316 (look at the instance of zfauscl 5300 that occurs in the proof of inex1 5316). 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 4249 and cp 9888; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9887. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.

Usage of eqabbw 2807 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-2025sucunisn 42423 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-2025cantnfub2 42374 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-2025selvval2 41458 Value of the "variable selection" function. Convert selvval 21900 into a simpler form by using evlsevl 41445. (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-2025evladdval 41449 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-2025evlsevl 41445 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-2025evlsvval 41434 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 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝐴‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))))
 
9-Feb-2025mplsubrgcl 41421 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-2025ply1annprmidl 33057 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 = (0g‘𝐸)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    ⇒   (𝜑 → 𝑄 ∈ (PrmIdeal‘𝑃))
 
9-Feb-2025minplyval 33055 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 33054, 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 = (0g‘𝐸)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   𝐾 = (RSpan‘𝑃)    &   𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    &   𝑀 = (𝐸 minPoly 𝐹)    ⇒   (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))
 
9-Feb-2025ply1annig1p 33054 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 = (0g‘𝐸)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   𝐾 = (RSpan‘𝑃)    &   𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))    ⇒   (𝜑 → 𝑄 = (𝐾‘{(𝐺‘𝑄)}))
 
9-Feb-2025ply1annidl 33052 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 = (0g‘𝑅)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    ⇒   (𝜑 → 𝑄 ∈ (LIdeal‘𝑃))
 
9-Feb-2025ply1annidllem 33051 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 = (0g‘𝑅)    &   𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }    &   𝐹 = (𝑝 ∈ (Base‘𝑃) ↦ ((𝑂‘𝑝)‘𝐴))    ⇒   (𝜑 → 𝑄 = (◡𝐹 “ { 0 }))
 
9-Feb-2025evls1dm 32915 The domain of the subring polynomial evaluation function. (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 → dom 𝑂 = 𝑈)
 
9-Feb-2025evls1fn 32914 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-2025rhmmpl 41427 Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20394. TODO: Currently mhmvlin 22119 would have to be moved up. Investigate the usefulness of surrounding theorems like mndvcl 22113 and the difference between mhmvlin 22119, ofco 7695, and ofco2 22173. (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-2025rhmcomulmpl 41426 Show that the ring homomorphism in rhmmpl 41427 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-2025mhmcoaddmpl 41425 Show that the ring homomorphism in rhmmpl 41427 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-2025rhmmpllem2 41424 Lemma for rhmmpl 41427. A subproof of psrmulcllem 21725. (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 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑋‘𝑥)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑥))))) ∈ (Base‘𝑅))
 
8-Feb-2025rhmmpllem1 41423 Lemma for rhmmpl 41427. A subproof of psrmulcllem 21725. (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 (0g‘𝑅))
 
8-Feb-2025evls1maprhm 33048 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-2025evls1addd 32922 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-2025selvadd 41462 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-2025mhmcompl 41422 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-2025mplcrngd 41420 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-2025mplringd 41419 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-2025coexd 41354 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-2025psrgrp 21736 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-2025resrhm2b 20492 Restriction of the codomain of a (ring) homomorphism. resghm2b 19148 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-2025eluzsubi 12859 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-2025eluzaddi 12857 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-2025eluzsub 12856 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-2025eluzadd 12855 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-2025naddel12 8701 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-2025rspc4v 3629 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-2025oalim2cl 42341 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-2025sltsubsubbd 27789 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-2025addsubsassd 27787 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-2025zzlesq 14174 An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
(𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
 
5-Feb-2025irngnzply1 33044 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 = (0g‘(Poly1‘𝐸))    &    0 = (0g‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    ⇒   (𝜑 → (𝐸 IntgRing 𝐹) = ∪ 𝑝 ∈ (dom 𝑂 ∖ {𝑍})(◡(𝑂‘𝑝) “ { 0 }))
 
5-Feb-2025irngnzply1lem 33043 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 = (0g‘(Poly1‘𝐸))    &    0 = (0g‘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-20250ringirng 33042 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 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   (𝜑 → ¬ 𝑅 ∈ NzRing)    ⇒   (𝜑 → (𝑅 IntgRing 𝑆) = ∅)
 
5-Feb-2025asclply1subcl 32934 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-20250ringmon1p 32910 There are no monic polynomials over a zero ring. (Contributed by Thierry Arnoux, 5-Feb-2025.)
𝑀 = (Monic1p‘𝑅)    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → (♯‘𝐵) = 1)    ⇒   (𝜑 → 𝑀 = ∅)
 
5-Feb-20250ringsubrg 32649 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-2025negsubsdi2d 27786 Distribution of negative over subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ( -us ‘(𝐴 -s 𝐵)) = (𝐵 -s 𝐴))
 
5-Feb-2025sltsub2d 27785 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-2025sltsub1d 27784 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-2025subaddsd 27778 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-2025subscld 27774 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-2025subsvald 27772 The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
 
5-Feb-2025negsdi 27763 Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us ‘𝐴) +s ( -us ‘𝐵)))
 
5-Feb-2025negsidd 27755 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 ( -us ‘𝐴)) = 0s )
 
5-Feb-2025adds42d 27732 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-2025adds4d 27731 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-2025addscan1d 27722 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-2025addscan2d 27721 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-2025sltadd1d 27720 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-2025sleadd2d 27718 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-2025ringcomlem 20167 Lemma for ringcom 20168. This (formerly) part of the proof for ringcom 20168 is also applicable for semirings (without using the commutativity of the addition given per definition of a semiring), see srgcom4lem 20107. (Contributed by Gérard Lang, 4-Dec-2014.) Variant of rglcom4d 20105 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-2025ringo2times 20163 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 20104 for rings. (Revised by AV, 5-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))
 
5-Feb-2025nadd42 8700 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-2025nadd4 8699 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-2025oenord1 42368 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 ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))
 
4-Feb-2025omnord1 42357 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 ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ·o 𝑐) ∈ (𝑏 ·o 𝑐))
 
4-Feb-2025mulslid 27837 Surreal one is a left identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
 
4-Feb-2025muls02 27836 Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → ( 0s ·s 𝐴) = 0s )
 
4-Feb-2025mulsrid 27808 Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)
 
4-Feb-2025muls01 27807 Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
 
4-Feb-2025mulsval 27804 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-2025mulsfn 27803 Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.)
·s Fn ( No × No )
 
4-Feb-2025df-muls 27802 Define surreal multiplication. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 4-Feb-2025.)
·s = norec2 ((𝑧 ∈ V, 𝑚 ∈ V ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘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-2025sltsub2 27783 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-2025sltsub1 27782 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-2025npcans 27781 Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 -s 𝐵) +s 𝐵) = 𝐴)
 
4-Feb-2025pncan3s 27780 Subtraction and addition of equals. (Contributed by Scott Fenton, 4-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s (𝐵 -s 𝐴)) = 𝐵)
 
4-Feb-2025pncans 27779 Cancellation law for surreal subtraction. (Contributed by Scott Fenton, 4-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((𝐴 +s 𝐵) -s 𝐵) = 𝐴)
 
4-Feb-2025right1s 27627 The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
( R ‘ 1s ) = ∅
 
4-Feb-2025left1s 27626 The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
( L ‘ 1s ) = { 0s }
 
4-Feb-2025old1 27607 The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
( O ‘1o) = { 0s }
 
3-Feb-2025cantnf2 42377 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-2025cantnfresb 42376 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 ∖ 2o) ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐹 ∈ dom (𝐴 CNF 𝐵))) → (((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶) ↔ ∀𝑥 ∈ (𝐵 ∖ 𝐶)(𝐹‘𝑥) = ∅))
 
3-Feb-2025subadds 27777 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-2025subsid 27776 Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → (𝐴 -s 𝐴) = 0s )
 
3-Feb-2025subsid1 27775 Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → (𝐴 -s 0s ) = 𝐴)
 
3-Feb-2025subscl 27773 Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) ∈ No )
 
3-Feb-2025subsval 27771 The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
 
3-Feb-2025negsf1o 27767 Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No –1-1-onto→ No
 
3-Feb-2025negsfo 27766 Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No –onto→ No
 
3-Feb-2025negsf 27765 Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No ⟶ No
 
3-Feb-2025negs11 27762 Surreal negation is one-to-one. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) = ( -us ‘𝐵) ↔ 𝐴 = 𝐵))
 
3-Feb-2025sleneg 27759 Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ( -us ‘𝐵) ≤s ( -us ‘𝐴)))
 
3-Feb-2025sltneg 27758 Negative of both sides of surreal less-than. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ( -us ‘𝐵) <s ( -us ‘𝐴)))
 
3-Feb-2025negnegs 27757 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 → ( -us ‘( -us ‘𝐴)) = 𝐴)
 
3-Feb-2025negsex 27756 Every surreal has a negative. Note that this theorem, addscl 27703, addscom 27688, addsass 27727, addsrid 27686, and sltadd1im 27707 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) = 0s )
 
3-Feb-2025negsid 27754 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 ( -us ‘𝐴)) = 0s )
 
3-Feb-2025negscut2 27753 The cut that defines surreal negation is legitimate. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
 
3-Feb-2025negscut 27752 The cut properties of surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
 
3-Feb-2025sltnegim 27751 The forward direction of the ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴)))
 
3-Feb-2025negscld 27750 The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → ( -us ‘𝐴) ∈ No )
 
3-Feb-2025negscl 27749 The surreals are closed under negation. Theorem 6(ii) of [Conway] p. 18. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → ( -us ‘𝐴) ∈ No )
 
3-Feb-2025negsprop 27748 Show closure and ordering properties of negation. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( -us ‘𝐴) ∈ No ∧ (𝐴 <s 𝐵 → ( -us ‘𝐵) <s ( -us ‘𝐴))))
 
3-Feb-2025negsproplem7 27747 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    ⇒   (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
 
3-Feb-2025negsproplem6 27746 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    &   (𝜑 → ( bday ‘𝐴) = ( bday ‘𝐵))    ⇒   (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
 
3-Feb-2025negsproplem5 27745 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    &   (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))    ⇒   (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
 
3-Feb-2025addslid 27690 Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴)
 
3-Feb-2025cuteq0 27570 Condition for a surreal cut to equal zero. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝜑 → 𝐴 <<s { 0s })    &   (𝜑 → { 0s } <<s 𝐵)    ⇒   (𝜑 → (𝐴 |s 𝐵) = 0s )
 
2-Feb-2025negsproplem4 27744 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    &   (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵))    ⇒   (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
 
2-Feb-2025negsproplem3 27743 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
 
2-Feb-2025negsproplem2 27742 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
 
2-Feb-2025negsproplem1 27741 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))    ⇒   (𝜑 → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))
 
2-Feb-2025xpord3indd 8143 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-2025xpord3inddlem 8142 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) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘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-2025ralxp3 8126 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-2025issetlem 2811 Lemma for elisset 2813 and isset 3485. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3485. (Revised by WL, 2-Feb-2025.)
𝑥 ∈ 𝑉    ⇒   (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴)
 
1-Feb-2025smfinfdmmbl 45863 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-2025smfinfdmmbllem 45862 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-2025finfdm2 45861 The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45832. 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-2025finfdm 45860 The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45832. 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-2025archd 44157 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-2025onexoegt 42295 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-2025onexlimgt 42294 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-2025onexomgt 42292 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-2025onexgt 42291 For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)
(𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
 
1-Feb-2025zaddcomlem 41626 Lemma for zaddcom 41627. (Contributed by SN, 1-Feb-2025.)
(((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
1-Feb-2025nn0addcom 41625 Addition is commutative for nonnegative integers. Proven without ax-mulcom 11176. (Contributed by SN, 1-Feb-2025.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
1-Feb-2025grpcominv2 41389 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 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))    ⇒   (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌))
 
1-Feb-2025ringcom 20168 Commutativity of the additive group of a ring. (See also lmodcom 20662.) 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-2025ringadd2 20164 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-2025srgcom4 20108 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-2025srgcom4lem 20107 Lemma for srgcom4 20108. This (formerly) part of the proof for ringcom 20168 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-2025srgo2times 20106 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 20104 for semirings. (Revised by AV, 1-Feb-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ SRing ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))
 
1-Feb-2025rglcom4d 20105 Restricted commutativity of the addition in a ring-like structure. This (formerly) part of the proof for ringcom 20168 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-2025o2timesd 20104 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 20168 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-2025mulgnn0cld 19011 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 19006. (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-2025el2xpss 8025 Version of elrel 5797 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-2025cantnfub 42373 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-2025finsubmsubg 41390 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-20250subgALT 19477 A shorter proof of 0subg 19067 using df-od 19437. (Contributed by SN, 31-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
 
31-Jan-2025finodsubmsubg 19476 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-2025odm1inv 19462 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 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐴 ∈ 𝑋)    ⇒   (𝜑 → (((𝑂‘𝐴) − 1) · 𝐴) = (𝐼‘𝐴))
 
31-Jan-20250subg 19067 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
 
31-Jan-2025xpord3pred 8140 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) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘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-2025otelxp 5719 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-2025otthne 5485 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-2025cantnftermord 42372 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 ∖ 1o) ∧ 𝐷 ∈ (ω ∖ 1o))) → (𝐴 ∈ 𝐵 → ((ω ↑o 𝐴) ·o 𝐶) ∈ ((ω ↑o 𝐵) ·o 𝐷)))
 
30-Jan-2025oenass 42371 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-2025oenassex 42370 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.)
¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
 
30-Jan-2025oaomoencom 42369 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-2025oenord1ex 42367 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.)
¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
 
30-Jan-2025nnoeomeqom 42364 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 ∧ 1o ∈ 𝐴) → (𝐴 ↑o ω) = ω)
 
30-Jan-2025oeord2com 42363 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 ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 ↔ (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
 
30-Jan-2025oeord2i 42362 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 ∧ 1o ∈ 𝐴) ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
 
30-Jan-2025oeord2lim 42361 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 8589. (Contributed by RP, 30-Jan-2025.)
(((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶)))
 
30-Jan-2025rp-oelim2 42360 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 8602. (Contributed by RP, 30-Jan-2025.)
(((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑o 𝐵))
 
30-Jan-20252omomeqom 42355 Ordinal two times omega is omega. Lemma 3.17 of [Schloeder] p. 10. (Contributed by RP, 30-Jan-2025.)
(2o ·o ω) = ω
 
30-Jan-2025ressply1sub 32933 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-2025ressply1invg 32932 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 → ((invg‘𝑈)‘𝑋) = ((invg‘𝑃)‘𝑋))
 
29-Jan-2025oege2 42359 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 8595. (Contributed by RP, 29-Jan-2025.)
(((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
 
29-Jan-2025oege1 42358 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 8593. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑o 𝐵))
 
29-Jan-2025omnord1ex 42356 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.)
¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))
 
29-Jan-2025omord2com 42354 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-2025omord2i 42353 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-2025omord2lim 42352 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-2025omlim2 42351 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-2025omge2 42350 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 8576. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵))
 
29-Jan-2025omge1 42349 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 8575. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵))
 
29-Jan-2025oaordnr 42348 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-2025oaordnrex 42347 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.)
¬ (∅ ∈ 1o ↔ (∅ +o ω) ∈ (1o +o ω))
 
29-Jan-20251oaomeqom 42345 Ordinal one plus omega is equal to omega. See oaabs 8649 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 8650 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.)
(1o +o ω) = ω
 
29-Jan-2025oaord3 42344 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-2025oaordi3 42343 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-2025oaltublim 42342 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-2025onsucwordi 42340 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-2025nnamecl 42339 Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8613, nnmcl 8614, nnecl 8615. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 +o 𝐵) ∈ ω ∧ (𝐴 ·o 𝐵) ∈ ω ∧ (𝐴 ↑o 𝐵) ∈ ω))
 
29-Jan-2025oasubex 42338 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-2025oe0rif 42337 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, ∅, 1o))
 
29-Jan-2025om1om1r 42336 Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8544 and om1r 8545 for individual statements. (Contributed by RP, 29-Jan-2025.)
(𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))
 
29-Jan-2025onintunirab 42278 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-2025unielss 42269 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-2025grpcominv1 41388 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 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))    ⇒   (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))
 
29-Jan-2025grpasscan2d 41387 An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
 
29-Jan-2025riotaeqbidva 32003 Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3325 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-2025grprinvd 18916 The right inverse of a group element. Deduction associated with grprinv 18911. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )
 
29-Jan-2025grplinvd 18915 The left inverse of a group element. Deduction associated with grplinv 18910. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )
 
29-Jan-2025grpinvcld 18909 A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑁‘𝑋) ∈ 𝐵)
 
29-Jan-2025grpridd 18891 The identity element of a group is a right identity. Deduction associated with grprid 18889. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 0 ) = 𝑋)
 
29-Jan-2025grplidd 18890 The identity element of a group is a left identity. Deduction associated with grplid 18888. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 0 + 𝑋) = 𝑋)
 
29-Jan-2025grpassd 18867 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)))
 
28-Jan-2025onuniintrab 42277 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 7791. (Contributed by RP, 28-Jan-2025.)
((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
 
28-Jan-2025irngssv 33041 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 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   (𝜑 → (𝑅 IntgRing 𝑆) ⊆ 𝐵)
 
28-Jan-2025irngss 33040 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 33042). (Contributed by Thierry Arnoux, 28-Jan-2025.)
𝑂 = (𝑅 evalSub1 𝑆)    &   𝑈 = (𝑅 ↾s 𝑆)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝑅 ∈ NzRing)    ⇒   (𝜑 → 𝑆 ⊆ (𝑅 IntgRing 𝑆))
 
28-Jan-2025elirng 33039 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 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )))
 
28-Jan-2025irngval 33038 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 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))
 
28-Jan-2025df-irng 33037 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 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡((𝑟 evalSub1 𝑠)‘𝑓) “ {(0g‘𝑟)}))
 
27-Jan-2025limexissupab 42335 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-2025limiun 42334 A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6424. (Contributed by RP, 27-Jan-2025.)
(Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)
 
27-Jan-2025limexissup 42333 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-2025onsssupeqcond 42332 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-2025onsupsucismax 42331 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-2025onfisupcl 42301 Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9295. (Contributed by RP, 27-Jan-2025.)
((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴))
 
27-Jan-2025onsupnub 42300 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-2025onsuplub 42299 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-2025onsupeqnmax 42298 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-2025onsupnmax 42279 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-2025onsupneqmaxlim0 42275 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-2025onmaxnelsup 42274 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-2025ssunib 42271 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-2025unielid 42270 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-2025uniel 42268 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-2025vtoclegft 3573 Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3549.) (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-2025vtoclf 3550 Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2392. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
Ⅎ𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   𝜓
 
26-Jan-2025vtoclg 3541 Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2169. (Revised by SN, 20-Apr-2024.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   (𝐴 ∈ 𝑉 → 𝜓)
 
25-Jan-2025onsupmaxb 42290 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-2025zmulcom 41631 Multiplication is commutative for integers. Proven without ax-mulcom 11176. From this result and grpcominv1 41388, we can show that rationals commute under multiplication without using ax-mulcom 11176. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
25-Jan-2025zmulcomlem 41630 Lemma for zmulcom 41631. (Contributed by SN, 25-Jan-2025.)
(((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
25-Jan-2025nn0mulcom 41629 Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11176. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
25-Jan-2025renegmulnnass 41628 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-2025zaddcom 41627 Addition is commutative for integers. Proven without ax-mulcom 11176. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
25-Jan-2025reelznn0nn 41624 elznn0nn 12576 restated using df-resub 41541. (Contributed by SN, 25-Jan-2025.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ)))
 
25-Jan-2025sn-nnne0 41623 nnne0 12250 without ax-mulcom 11176. (Contributed by SN, 25-Jan-2025.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)
 
25-Jan-2025sn-addgt0d 41622 The sum of positive numbers is positive. Proof of addgt0d 11793 without ax-mulcom 11176. (Contributed by SN, 25-Jan-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 0 < 𝐵)    ⇒   (𝜑 → 0 < (𝐴 + 𝐵))
 
25-Jan-2025sn-addlt0d 41621 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-2025sn-ltaddneg 41617 ltaddneg 11433 without ax-mulcom 11176. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
 
25-Jan-2025remulneg2d 41589 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-2025wl-issetft 36747 A closed form of issetf 3487. The proof here is a modification of a subproof in vtoclgft 3539, where it could be used to shorten the proof. (Contributed by Wolf Lammen, 25-Jan-2025.)
(Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))
 
25-Jan-2025sbhypf 3537 Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3921. (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-2025smfsupdmmbl 45859 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-2025smfsupdmmbllem 45858 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-2025fsupdm2 45857 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-2025fsupdm 45856 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-2025saliinclf 45340 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-2025saliunclf 45336 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-2025nnxr 44282 A natural number is an extended real. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
(𝑁 ∈ ℕ → 𝑁 ∈ ℝ*)
 
24-Jan-2025rnmptssdff 44278 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-2025rnmptssff 44277 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-2025rn1st 44276 The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 23176, 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-2025fvmpt2df 44275 Deduction version of fvmpt2 7008. (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-2025iindif2f 44155 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-2025r19.28zf 44154 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-2025r19.3rzf 44153 Restricted quantification of wff not containing quantified variable. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    ⇒   (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
 
24-Jan-2025iinss2d 44152 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-2025iunssdf 44151 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-2025rabidd 44150 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-2025onsupeqmax 42297 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-2025evls1muld 32923 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-2025evls1varpwval 32919 Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 22101. (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-2025evls1expd 32918 Univariate polynomial evaluation builder for an exponential. See also evl1expd 22084. (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-2025oninfex2 42296 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-2025oninfcl2 42289 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-2025oninfunirab 42288 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-2025oninfint 42287 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-2025onuniintrab2 42286 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-2025onsupex3 42285 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-2025onsupcl3 42284 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-2025onsupintrab2 42283 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-2025onsupintrab 42282 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-2025onsupcl2 42276 The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
(𝐴 ∈ 𝒫 On → ∪ 𝐴 ∈ On)
 
23-Jan-2025rp-unirabeq 42273 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-2025rp-intrabeq 42272 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-2025ressply1evl 32921 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-2025evls1fpws 32920 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 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑥))))))
 
23-Jan-2025cofcutrtime2d 27654 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-2025cofcutrtime1d 27653 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-2025cofcutr1d 27650 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-2025cofcut2d 27648 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-2025cofcut1d 27646 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-2025ceqsal 3508 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 2708. (Revised by Wolf Lammen, 23-Jan-2025.)
Ⅎ𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
 
22-Jan-2025onsupuni2 42281 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-2025subsfn 27738 Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)
-s Fn ( No × No )
 
22-Jan-2025adds32d 27729 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-2025addsassd 27728 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-2025addsass 27727 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-2025ceqsexv 3524 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2169. (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-2025ceqsex 3522 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-2025ressply1mon1p 32931 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 = (Monic1p‘𝑅)    &   𝑁 = (Monic1p‘𝐻)    ⇒   (𝜑 → 𝑁 = (𝐵 ∩ 𝑀))
 
21-Jan-2025evls1scafv 32917 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-202513an22anass 31973 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-2025addsasslem2 27726 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-2025addsasslem1 27725 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-2025addsunif 27724 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-2025addsuniflem 27723 Lemma for addsunif 27724. State the whole theorem with extra distinct variable conditions. (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-2025sltadd2d 27719 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-2025sleadd1d 27717 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-2025addscan1 27716 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-2025addscan2 27715 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-2025sltadd1 27714 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-2025sltadd2 27713 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)))

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).

FrontBackDetail
ZFC T-shirt front ZFC T-shirt back ZFC T-shirt 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.

OldNewDescription
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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