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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 18-Feb-2025 at 5:14 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
15-Feb-2025rexanuz2nf 43718 A simple counterexample related to theorem rexanuz2 15234, demonstrating the necessity of its disjoint variable constraints. Here, 𝑗 appears free in 𝜑, showing that without these constraints, rexanuz2 15234 and similar theorems would not hold (see rexanre 15231 and rexanuz 15230). (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 43717 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 43716 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 43715 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 43495 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 43373 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 43372 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 43371 Commutative law for the negation of a biconditional. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜑 ↔ 𝜓)    ⇒    ¬ (𝜓 ↔ 𝜑)
 
15-Feb-2025nimnbi2 43370 If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜓 → 𝜑)    ⇒    ¬ (𝜑 ↔ 𝜓)
 
15-Feb-2025nimnbi 43369 If an implication is false, the biconditional is false. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
¬ (𝜑 → 𝜓)    ⇒    ¬ (𝜑 ↔ 𝜓)
 
15-Feb-2025eliund 43368 Membership in indexed union. (Contributed by Glauco Siliprandi, 15-Feb-2025.)
(𝜑 → ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)    ⇒   (𝜑 → 𝐴 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)
 
15-Feb-2025ghmqusker 32198 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 32196 In the case of theorem ghmqusker 32198, 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))
 
14-Feb-2025ghmquskerlem2 32197 Lemma for ghmqusker 32198. (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 32195 Lemma for ghmqusker 32198 (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 31625 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 31596 The image of a set is a set. Deduction version of imaexg 7852. (Contributed by Thierry Arnoux, 14-Feb-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 “ 𝐵) ∈ V)
 
14-Feb-2025rnexd 31595 The range of a set is a set. Deduction version of rnexd 31595. (Contributed by Thierry Arnoux, 14-Feb-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → ran 𝐴 ∈ V)
 
14-Feb-2025negsunif 27350 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)))
 
13-Feb-2025eqabr 2876 One direction of eqab 2877 is provable from fewer axioms. (Contributed by Wolf Lammen, 13-Feb-2025.)
(∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → 𝐴 = {𝑥 ∣ 𝜑})
 
12-Feb-2025eqab 2877 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 abbi 2808 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 5258 to inex1 5274 (look at the instance of zfauscl 5258 that occurs in the proof of inex1 5274). 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 4210 and cp 9827; the latter derives a formula containing wff variables from substitution instances of the class variables in its equivalent formulation cplem2 9826. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.

Usage of eqabw 2813 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))
 
9-Feb-2025cantnfub2 41642 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 40749 Value of the "variable selection" function. Use evlsevl 40740 for a simpler definition. (Contributed by SN, 9-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = (((𝐼 eval 𝑇)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
 
9-Feb-2025evladdval 40741 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 40740 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 40733 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 40723 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 32373 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 32372 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 32371, 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 32371 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 32370 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 32369 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-2025evls1fn 32267 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 40729 Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20173. TODO: Currently mhmvlin 21746 would have to be moved up. Investigate the usefulness of surrounding theorems like mndvcl 21740 and the difference between mhmvlin 21746, ofco 7640, and ofco2 21800. (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 40728 Show that the ring homomorphism in rhmmpl 40729 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 40727 Show that the ring homomorphism in rhmmpl 40729 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 40726 Lemma for rhmmpl 40729. A subproof of psrmulcllem 21355. (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 40725 Lemma for rhmmpl 40729. A subproof of psrmulcllem 21355. (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 32368 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 32273 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 40750 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)    &   𝐸 = (Base‘𝑇)    &    ✚ = (+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 40724 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 40722 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 40721 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-2025resrhm2b 40693 Restriction of the codomain of a (ring) homomorphism. resghm2b 19026 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-2025coexd 40652 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-2025mulsproplem2 34414 Lemma for surreal multiplication properties. Show that the core expression involved in surreal multiplication's recursive definition is a surreal under the inductive hypothesis. (Contributed by Scott Fenton, 7-Feb-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ∀𝑤 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) +no (( bday ‘𝑧) +no ( bday ‘𝑤))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) → ((𝑥 ·s 𝑦) ∈ No ∧ ((𝑥 <s 𝑦 ∧ 𝑧 <s 𝑤) → ((𝑦 ·s 𝑧) -s (𝑥 ·s 𝑧)) <s ((𝑦 ·s 𝑤) − (𝑥 ·s 𝑤))))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))    &   (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))    ⇒   (𝜑 → (((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) ∈ No )
 
7-Feb-2025mulsproplem1 34413 Lemma for surreal multiplication properties. In the next few lemmas, we aim to prove two properties of surreal multiplication at the same time. Here, we instantiate some quantifiers. (Contributed by Scott Fenton, 7-Feb-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No ∀𝑤 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) +no (( bday ‘𝑧) +no ( bday ‘𝑤))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))) → ((𝑥 ·s 𝑦) ∈ No ∧ ((𝑥 <s 𝑦 ∧ 𝑧 <s 𝑤) → ((𝑦 ·s 𝑧) -s (𝑥 ·s 𝑧)) <s ((𝑦 ·s 𝑤) − (𝑥 ·s 𝑤))))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → 𝑍 ∈ No )    &   (𝜑 → 𝑊 ∈ No )    &   (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝑌)) +no (( bday ‘𝑍) +no ( bday ‘𝑊))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) +no (( bday ‘𝐶) +no ( bday ‘𝐷))))    ⇒   (𝜑 → ((𝑋 ·s 𝑌) ∈ No ∧ ((𝑋 <s 𝑌 ∧ 𝑍 <s 𝑊) → ((𝑌 ·s 𝑍) -s (𝑋 ·s 𝑍)) <s ((𝑌 ·s 𝑊) − (𝑋 ·s 𝑊)))))
 
7-Feb-2025rspc4v 34299 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))
 
7-Feb-2025psrgrp 21366 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-2025eluzsubi 12796 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 12794 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 12793 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 12792 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 8644 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)))
 
6-Feb-2025oalim2cl 41610 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 27368 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 27367 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 14110 An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
(𝑁 ∈ ℤ → 𝑁 ≤ (𝑁↑2))
 
5-Feb-2025irngnzply1 32365 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 32364 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 32363 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 32281 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 32264 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 32064 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 27366 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 27365 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 27364 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 27358 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 27354 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 27352 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 27346 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 27340 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 27317 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 27316 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 27309 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 27308 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 27307 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 27305 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 20000 Lemma for ringcom 20001. This (formerly) part of the proof for ringcom 20001 is also applicable for semirings (without using the commutativity of the addition given per definition of a semiring), see srgcom4lem 19944. (Contributed by Gérard Lang, 4-Dec-2014.) Variant of rglcom4d 19942 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 19996 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 19941 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 8643 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 8642 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 41636 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 41625 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-2025mulsid2 34412 Surreal one is an identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
 
4-Feb-2025mulsid1 34411 Surreal one is an identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)
 
4-Feb-2025muls02 34410 Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → ( 0s ·s 𝐴) = 0s )
 
4-Feb-2025muls01 34409 Surreal multiplication by zero. (Contributed by Scott Fenton, 4-Feb-2025.)
(𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
 
4-Feb-2025mulsval 34408 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 34407 Surreal multiplication is a function over surreals. (Contributed by Scott Fenton, 4-Feb-2025.)
·s Fn ( No × No )
 
4-Feb-2025df-muls 34406 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 27363 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 27362 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 27361 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 27360 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 27359 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 27225 The right set of 1s is empty . (Contributed by Scott Fenton, 4-Feb-2025.)
( R ‘ 1s ) = ∅
 
4-Feb-2025left1s 27224 The left set of 1s is the singleton of 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
( L ‘ 1s ) = { 0s }
 
4-Feb-2025old1 27205 The only surreal older than 1o is 0s. (Contributed by Scott Fenton, 4-Feb-2025.)
( O ‘1o) = { 0s }
 
3-Feb-2025cantnf2 41645 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 41644 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 27357 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 27356 Subtraction of a surreal from itself. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → (𝐴 -s 𝐴) = 0s )
 
3-Feb-2025subsid1 27355 Identity law for subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → (𝐴 -s 0s ) = 𝐴)
 
3-Feb-2025subscl 27353 Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) ∈ No )
 
3-Feb-2025subsval 27351 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 27349 Surreal negation is a bijection. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No –1-1-onto→ No
 
3-Feb-2025negsfo 27348 Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No –onto→ No
 
3-Feb-2025negsf 27347 Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025.)
-us : No ⟶ No
 
3-Feb-2025negs11 27345 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 27344 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 27343 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 27342 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 27341 Every surreal has a negative. Note that this theorem, addscl 27291, addscom 27278, addsass 27313, addsid1 27276, and sltadd1im 27294 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 27339 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 27338 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 27337 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 27336 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 27335 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 27334 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 27333 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 27332 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 27331 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 27330 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-2025addsid2 27280 Surreal addition to zero is identity. (Contributed by Scott Fenton, 3-Feb-2025.)
(𝐴 ∈ No → ( 0s +s 𝐴) = 𝐴)
 
3-Feb-2025cuteq0 27171 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 27329 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 27328 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 27327 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 27326 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 8087 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 8086 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 8070 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 2817 Lemma for elisset 2819 and isset 3458. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3458. (Revised by WL, 2-Feb-2025.)
𝑥 ∈ 𝑉    ⇒   (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴)
 
1-Feb-2025smfinfdmmbl 45080 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 45079 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 45078 The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45049. 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 45077 The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 45049. 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 43367 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 41564 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 41563 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 41561 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 41560 For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025.)
(𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ 𝑥)
 
1-Feb-2025zaddcomlem 40906 Lemma for zaddcom 40907. (Contributed by SN, 1-Feb-2025.)
(((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
1-Feb-2025nn0addcom 40905 Addition is commutative for nonnegative integers. Proven without ax-mulcom 11115. (Contributed by SN, 1-Feb-2025.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
1-Feb-2025grpcominv2 40688 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 20001 Commutativity of the additive group of a ring. (See also lmodcom 20368.) 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 19997 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 19945 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 19944 Lemma for srgcom4 19945. This (formerly) part of the proof for ringcom 20001 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 19943 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 19941 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 19942 Restricted commutativity of the addition in a ring-like structure. This (formerly) part of the proof for ringcom 20001 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 19941 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 20001 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 18897 Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 18892. (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 7969 Version of elrel 5754 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 41641 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 40689 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 19350 A shorter proof of 0subg 18953 using df-od 19310. (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 19349 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 19335 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 18953 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 8084 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 5676 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 5443 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 41640 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 41639 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 41638 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 41637 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 41635 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 41632 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 41631 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 41630 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 41629 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 8534. (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 41628 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 8547. (Contributed by RP, 30-Jan-2025.)
(((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ (Lim 𝐵 ∧ 𝐵 ∈ 𝑉)) → Lim (𝐴 ↑o 𝐵))
 
30-Jan-20252omomeqom 41623 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 32280 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 32279 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 41627 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 8540. (Contributed by RP, 29-Jan-2025.)
(((𝐴 ∈ On ∧ 1o ∈ 𝐴) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵))
 
29-Jan-2025oege1 41626 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 8538. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ↑o 𝐵))
 
29-Jan-2025omnord1ex 41624 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 41622 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 41621 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 41620 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 41619 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 41618 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 8521. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → 𝐵 ⊆ (𝐴 ·o 𝐵))
 
29-Jan-2025omge1 41617 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 8520. (Contributed by RP, 29-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ≠ ∅) → 𝐴 ⊆ (𝐴 ·o 𝐵))
 
29-Jan-2025oaordnr 41616 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 41615 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 41614 Ordinal one plus omega is equal to omega. See oaabs 8594 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 8595 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 41613 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 41612 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 41611 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 41609 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 41608 Natural numbers are closed under ordinal addition, multiplication, and exponentiation. Theorem 2.20 of [Schloeder] p. 6. See nnacl 8558, nnmcl 8559, nnecl 8560. (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 41607 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 41606 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 41605 Ordinal one is both a left and right identity of ordinal multiplication. Lemma 2.15 of [Schloeder] p. 5. See om1 8489 and om1r 8490 for individual statements. (Contributed by RP, 29-Jan-2025.)
(𝐴 ∈ On → ((1o ·o 𝐴) = (𝐴 ·o 1o) ∧ (𝐴 ·o 1o) = 𝐴))
 
29-Jan-2025onintunirab 41547 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 41538 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 40687 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 40686 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-2025sn-grprinvd 40685 The right inverse of a group element. Deduction associated with grprinv 18801. (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 40684 The left inverse of a group element. Deduction associated with grplinv 18800. (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-2025grpassd 40683 A group operation is associative. (Contributed by SN, 29-Jan-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
29-Jan-2025sn-grpridd 40682 The identity element of a group is a right identity. Deduction associated with grprid 18781. (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-2025sn-grplidd 40681 The identity element of a group is a left identity. Deduction associated with grplid 18780. (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-2025riotaeqbidva 31424 Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3321 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-2025grpinvcld 18799 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)
 
28-Jan-2025onuniintrab 41546 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 7736. (Contributed by RP, 28-Jan-2025.)
((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
 
28-Jan-2025irngssv 32362 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 32361 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 32363). (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 32360 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 32359 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 32358 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 41604 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 41603 A limit ordinal is the union of its elements, indexed union version. Lemma 2.13 of [Schloeder] p. 5. See limuni 6378. (Contributed by RP, 27-Jan-2025.)
(Lim 𝐴 → 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥)
 
27-Jan-2025limexissup 41602 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 41601 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 41600 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 41570 Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9237. (Contributed by RP, 27-Jan-2025.)
((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴))
 
27-Jan-2025onsupnub 41569 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 41568 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 41567 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 41548 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 41544 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 41543 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 41540 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 41539 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 41537 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 3542 Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3515.) (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-2025vtoclg 3525 Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2171. (Revised by SN, 20-Apr-2024.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   (𝐴 ∈ 𝑉 → 𝜓)
 
26-Jan-2025vtoclf 3516 Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2393. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
Ⅎ𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   𝜓
 
25-Jan-2025onsupmaxb 41559 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 40911 Multiplication is commutative for integers. Proven without ax-mulcom 11115. From this result and grpcominv1 40687, we can show that rationals commute under multiplication without using ax-mulcom 11115. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
25-Jan-2025zmulcomlem 40910 Lemma for zmulcom 40911. (Contributed by SN, 25-Jan-2025.)
(((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
25-Jan-2025nn0mulcom 40909 Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11115. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
25-Jan-2025renegmulnnass 40908 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 40907 Addition is commutative for integers. Proven without ax-mulcom 11115. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
25-Jan-2025reelznn0nn 40904 elznn0nn 12513 restated using df-resub 40821. (Contributed by SN, 25-Jan-2025.)
(𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ)))
 
25-Jan-2025sn-nnne0 40903 nnne0 12187 without ax-mulcom 11115. (Contributed by SN, 25-Jan-2025.)
(𝐴 ∈ ℕ → 𝐴 ≠ 0)
 
25-Jan-2025sn-addgt0d 40902 The sum of positive numbers is positive. Proof of addgt0d 11730 without ax-mulcom 11115. (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 40901 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 40897 ltaddneg 11370 without ax-mulcom 11115. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
 
25-Jan-2025remulneg2d 40869 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 36034 A closed form of issetf 3459. The proof here is a modification of a subproof in vtoclgft 3509, 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 3507 Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3884. (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 45076 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 45075 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 45074 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 45073 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 44557 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 44553 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 43498 A natural number is an extended real. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
(𝑁 ∈ ℕ → 𝑁 ∈ ℝ*)
 
24-Jan-2025rnmptssdff 43494 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 43493 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 43492 The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 22803, 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 43491 Deduction version of fvmpt2 6959. (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 43365 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 43364 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 43363 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 43362 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 43361 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 43360 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 41566 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 32274 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 32270 Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. See evl1varpwval 21728. (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 32269 Univariate polynomial evaluation builder for an exponential. See also evl1expd 21711. (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 41565 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 41558 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 41557 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 41556 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 41555 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 41554 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 41553 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 41552 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 41551 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 41545 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 41542 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 41541 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 32272 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 32271 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 27248 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 27247 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 27244 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 27242 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 27240 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 3479 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 2714. (Revised by Wolf Lammen, 23-Jan-2025.)
Ⅎ𝑥𝜓    &   𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
 
22-Jan-2025onsupuni2 41550 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 27323 Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)
-s Fn ( No × No )
 
22-Jan-2025adds32d 27315 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 27314 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 27313 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 3494 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2171. (Revised by Gino Giotto, 12-Oct-2024.) (Proof shortened by Wolf Lammen, 22-Jan-2025.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
 
22-Jan-2025ceqsex 3492 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 32278 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 32268 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 31395 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 27312 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 27311 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 27310 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-2025sltadd2d 27306 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 27304 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 27303 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 27302 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 27301 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 27300 Addition to both sides of surreal less-than. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 ↔ (𝐶 +s 𝐴) <s (𝐶 +s 𝐵)))
 
21-Jan-2025sleadd2 27299 Addition to both sides of surreal less-than or equal. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵)))
 
21-Jan-2025sleadd1 27298 Addition to both sides of surreal less-than or equal. Theorem 5 of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶)))
 
21-Jan-2025sleadd2im 27297 Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐶 +s 𝐴) ≤s (𝐶 +s 𝐵) → 𝐴 ≤s 𝐵))
 
21-Jan-2025sleadd1im 27296 Surreal less-than or equal cancels under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 +s 𝐶) ≤s (𝐵 +s 𝐶) → 𝐴 ≤s 𝐵))
 
21-Jan-2025sltadd2im 27295 Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐶 +s 𝐴) <s (𝐶 +s 𝐵)))
 
21-Jan-2025sltadd1im 27294 Surreal less-than is preserved under addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → (𝐴 <s 𝐵 → (𝐴 +s 𝐶) <s (𝐵 +s 𝐶)))
 
21-Jan-2025addsfo 27293 Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)
+s :( No × No )–onto→ No
 
21-Jan-2025addsf 27292 Function statement for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
+s :( No × No )⟶ No
 
21-Jan-2025addscl 27291 Surreal numbers are closed under addition. Theorem 6(iii) of [Conway[ p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) ∈ No )
 
21-Jan-2025addscld 27290 Surreal numbers are closed under addition. Theorem 6(iii) of [Conway] p. 18. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    ⇒   (𝜑 → (𝑋 +s 𝑌) ∈ No )
 
21-Jan-2025addscut 27289 Demonstrate the cut properties of surreal addition. This gives us closure together with a pair of set-less-than relationships for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    ⇒   (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
 
21-Jan-2025addsprop 27288 Inductively show that surreal addition is closed and compatible with less-than. This proof follows from induction on the birthdays of the surreal numbers involved. This pattern occurs throughout surreal development. Theorem 3.1 of [Gonshor] p. 14. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))
 
21-Jan-2025addsproplem7 27287 Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → 𝑍 ∈ No )    &   (𝜑 → 𝑌 <s 𝑍)    ⇒   (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
 
21-Jan-2025addsproplem6 27286 Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → 𝑍 ∈ No )    &   (𝜑 → 𝑌 <s 𝑍)    &   (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))    ⇒   (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
 
21-Jan-2025addsproplem5 27285 Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → 𝑍 ∈ No )    &   (𝜑 → 𝑌 <s 𝑍)    &   (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌))    ⇒   (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
 
21-Jan-2025addsproplem4 27284 Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    &   (𝜑 → 𝑍 ∈ No )    &   (𝜑 → 𝑌 <s 𝑍)    &   (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍))    ⇒   (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
 
21-Jan-2025addsproplem3 27283 Lemma for surreal addition properties. Show the cut properties of surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    ⇒   (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))
 
21-Jan-2025addsproplem2 27282 Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝑋 ∈ No )    &   (𝜑 → 𝑌 ∈ No )    ⇒   (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}))
 
21-Jan-2025addsproplem1 27281 Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8345 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (Contributed by Scott Fenton, 21-Jan-2025.)
(𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (( bday ‘𝐴) +no ( bday ‘𝐶))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))    ⇒   (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
 
21-Jan-2025addsval2 27275 The value of surreal addition with different choices for each bound variable. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)})))
 
21-Jan-2025naddword1 8636 Weak-ordering principle for natural addition. (Contributed by Scott Fenton, 21-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐴 +no 𝐵))
 
20-Jan-2025ressply10g 32277 A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝑆 = (Poly1‘𝑅)    &   𝐻 = (𝑅 ↾s 𝑇)    &   𝑈 = (Poly1‘𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    &   𝑍 = (0g‘𝑆)    ⇒   (𝜑 → 𝑍 = (0g‘𝑈))
 
20-Jan-2025ply1ascl0 32276 The zero scalar as a polynomial. (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝑊 = (Poly1‘𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝑂 = (0g‘𝑅)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (𝐴‘𝑂) = 0 )
 
20-Jan-2025ressdeg1 32275 The degree of a univariate polynomial in a structure restriction. (Contributed by Thierry Arnoux, 20-Jan-2025.)
𝐻 = (𝑅 ↾s 𝑇)    &   𝐷 = ( deg1 ‘𝑅)    &   𝑈 = (Poly1‘𝐻)    &   𝐵 = (Base‘𝑈)    &   (𝜑 → 𝑃 ∈ 𝐵)    &   (𝜑 → 𝑇 ∈ (SubRing‘𝑅))    ⇒   (𝜑 → (𝐷‘𝑃) = (( deg1 ‘𝐻)‘𝑃))
 
20-Jan-2025nadd32 8641 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ((𝐴 +no 𝐶) +no 𝐵))
 
20-Jan-2025naddass 8640 Natural ordinal addition is associative. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))
 
20-Jan-2025naddasslem2 8639 Lemma for naddass 8640. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no (𝐵 +no 𝐶)) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥)})
 
20-Jan-2025naddasslem1 8638 Lemma for naddass 8640. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})
 
20-Jan-2025naddunif 8637 Uniformity theorem for natural addition. If 𝐴 is the upper bound of 𝑋 and 𝐵 is the upper bound of 𝑌, then (𝐴 +no 𝐵) can be expressed in terms of 𝑋 and 𝑌. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})    &   (𝜑 → 𝐵 = ∩ {𝑦 ∈ On ∣ 𝑌 ⊆ 𝑦})    ⇒   (𝜑 → (𝐴 +no 𝐵) = ∩ {𝑧 ∈ On ∣ (( +no “ (𝑋 × {𝐵})) ∪ ( +no “ ({𝐴} × 𝑌))) ⊆ 𝑧})
 
20-Jan-2025naddf 8627 Function statement for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
+no :(On × On)⟶On
 
20-Jan-2025naddov3 8626 Alternate expression for natural addition. (Contributed by Scott Fenton, 20-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) ⊆ 𝑥})
 
20-Jan-2025cofonr 8620 Inverse cofinality law for ordinals. Contrast with cofcutr 27243 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})    ⇒   (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)
 
20-Jan-2025cofon2 8619 Cofinality theorem for ordinals. If 𝐴 and 𝐵 are mutually cofinal, then their upper bounds agree. Compare cofcut2 27241 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝜑 → 𝐴 ∈ 𝒫 On)    &   (𝜑 → 𝐵 ∈ 𝒫 On)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)    &   (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)    ⇒   (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = ∩ {𝑏 ∈ On ∣ 𝐵 ⊆ 𝑏})
 
20-Jan-2025cofon1 8618 Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and the upper bound of 𝐴 dominates 𝐵, then their upper bounds are equal. Compare with cofcut1 27239 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝜑 → 𝐴 ∈ 𝒫 On)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)    &   (𝜑 → 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})    ⇒   (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} = ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
 
20-Jan-2025coflton 8617 Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27237 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝜑 → 𝐴 ⊆ On)    &   (𝜑 → 𝐵 ⊆ On)    &   (𝜑 → 𝐶 ⊆ On)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)    &   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)    ⇒   (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)
 
20-Jan-2025imaeqalov 7593 Substitute an operation value into a universal quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓))
 
20-Jan-2025imaeqexov 7592 Substitute an operation value into an existential quantifier over an image. (Contributed by Scott Fenton, 20-Jan-2025.)
(𝑥 = (𝑦𝐹𝑧) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ (𝐵 × 𝐶))𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓))
 
19-Jan-2025onsupuni 41549 The supremum of a set of ordinals is the union of that set. Lemma 2.10 of [Schloeder] p. 5. (Contributed by RP, 19-Jan-2025.)
((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → sup(𝐴, On, E ) = ∪ 𝐴)
 
19-Jan-2025r1sssucd 41530 Deductive form of r1sssuc 9719. (Contributed by Noam Pasman, 19-Jan-2025.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
 
19-Jan-2025bj-adjfrombun 35517 Adjunction from singleton and binary union. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
(𝑥 ∪ {𝑦}) ∈ V
 
19-Jan-2025bj-prfromadj 35516 Unordered pair from adjunction. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
{𝑥, 𝑦} ∈ V
 
19-Jan-2025bj-snfromadj 35515 Singleton from adjunction and empty set. (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
{𝑥} ∈ V
 
19-Jan-2025bj-adjg1 35514 Existence of the result of the adjunction (generalized only in the first term since this suffices for current applications). (Contributed by BJ, 19-Jan-2025.) (Proof modification is discouraged.)
(𝐴 ∈ 𝑉 → (𝐴 ∪ {𝑥}) ∈ V)
 
19-Jan-2025ax-bj-adj 35513 Axiom of adjunction. (Contributed by BJ, 19-Jan-2025.)
∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦))
 
19-Jan-2025df-minply 32367 Define the minimal polynomial builder function. (Contributed by Thierry Arnoux, 19-Jan-2025.)
minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑝 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑝)‘𝑥) = (0g‘𝑒)})))
 
18-Jan-2025oaomoecl 41599 The operations of addition, multiplication, and exponentiation are closed. Remark 2.8 of [Schloeder] p. 5. See oacl 8481, omcl 8482, oecl 8483. (Contributed by RP, 18-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On))
 
18-Jan-2025oe0suclim 41598 Closed form expression of the value of ordinal exponentiation for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.6 of [Schloeder] p. 4. See oe0 8468, oesuc 8473, oe0m1 8467, and oelim 8480. (Contributed by RP, 18-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ↑o 𝐵) = 1o) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐵) = ((𝐴 ↑o 𝐶) ·o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ↑o 𝐵) = if(∅ ∈ 𝐴, ∪ 𝑐 ∈ 𝐵 (𝐴 ↑o 𝑐), ∅))))
 
18-Jan-2025om0suclim 41597 Closed form expression of the value of ordinal multiplication for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.5 of [Schloeder] p. 4. See om0 8463, omsuc 8472, and omlim 8479. (Contributed by RP, 18-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ·o 𝐵) = ((𝐴 ·o 𝐶) +o 𝐴)) ∧ (Lim 𝐵 → (𝐴 ·o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 ·o 𝑐))))
 
18-Jan-2025oa0suclim 41596 Closed form expression of the value of ordinal addition for the cases when the second ordinal is zero, a successor ordinal, or a limit ordinal. Definition 2.3 of [Schloeder] p. 4. See oa0 8462, oasuc 8470, and oalim 8478. (Contributed by RP, 18-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 +o 𝐵) = 𝐴) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 +o 𝐵) = suc (𝐴 +o 𝐶)) ∧ (Lim 𝐵 → (𝐴 +o 𝐵) = ∪ 𝑐 ∈ 𝐵 (𝐴 +o 𝑐))))
 
18-Jan-2025onov0suclim 41595 Compactly express rules for binary operations on ordinals. (Contributed by RP, 18-Jan-2025.)
(𝐴 ∈ On → (𝐴 ⊗ ∅) = 𝐷)    &   ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊗ suc 𝐶) = 𝐸)    &   (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝐵) → (𝐴 ⊗ 𝐵) = 𝐹)    ⇒   ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐵 = ∅ → (𝐴 ⊗ 𝐵) = 𝐷) ∧ ((𝐵 = suc 𝐶 ∧ 𝐶 ∈ On) → (𝐴 ⊗ 𝐵) = 𝐸) ∧ (Lim 𝐵 → (𝐴 ⊗ 𝐵) = 𝐹)))
 
18-Jan-2025onsucf1o 41593 The successor operation is a bijective function between the ordinals and the class of succesor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
𝐹 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
 
18-Jan-2025onsucrn 41592 The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
𝐹 = (𝑥 ∈ On ↦ suc 𝑥)    ⇒   ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
 
18-Jan-2025onsucf1olem 41591 The successor operation is bijective between the ordinals and the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
((𝐴 ∈ On ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) → ∃!𝑏 ∈ On 𝐴 = suc 𝑏)
 
18-Jan-2025onsucf1lem 41590 For ordinals, the successor operation is injective, so there is at most one ordinal that a given ordinal could be the succesor of. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.)
(𝐴 ∈ On → ∃*𝑏 ∈ On 𝐴 = suc 𝑏)
 
18-Jan-2025bj-clex 35502 Two ways of stating that a class is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
(𝑥 ∈ 𝐴 ↔ 𝜑)    ⇒   (𝐴 ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
 
18-Jan-2025bj-abex 35501 Two ways of stating that the extension of a formula is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)
({𝑥 ∣ 𝜑} ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
 
17-Jan-2025dflim7 41594 A limit ordinal is a non-zero ordinal that contains all the successors of its elements. Lemma 1.18 of [Schloeder] p. 2. Closely related to dflim4 7784. (Contributed by RP, 17-Jan-2025.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∀𝑏 ∈ 𝐴 suc 𝑏 ∈ 𝐴 ∧ 𝐴 ≠ ∅))
 
17-Jan-2025orddif0suc 41589 For any distinct pair of ordinals, if the set difference between the greater and the successor of the lesser is empty, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 17-Jan-2025.)
((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → ((𝐵 ∖ suc 𝐴) = ∅ → 𝐵 = suc 𝐴))
 
17-Jan-2025prjspnn0 40946 A projective point is nonempty. (Contributed by SN, 17-Jan-2025.)
𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝐴 ∈ 𝑃)    ⇒   (𝜑 → 𝐴 ≠ ∅)
 
17-Jan-2025prjspnssbas 40945 A projective point spans a subset of the (nonzero) affine points. (Contributed by SN, 17-Jan-2025.)
𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ DivRing)    ⇒   (𝜑 → 𝑃 ⊆ 𝒫 𝐵)
 
17-Jan-2025flddrngd 40708 A field is a division ring. (Contributed by SN, 17-Jan-2025.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ DivRing)
 
17-Jan-2025rictr 40699 Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
((𝑅 ≃𝑟 𝑆 ∧ 𝑆 ≃𝑟 𝑇) → 𝑅 ≃𝑟 𝑇)
 
17-Jan-2025rimco 40696 The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025.)
((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 RingIso 𝑇))
 
17-Jan-2025bj-elpwgALT 35525 Alternate proof of elpwg 4563. See comment for bj-velpwALT 35524. (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
 
17-Jan-2025bj-velpwALT 35524 This theorem bj-velpwALT 35524 and the next theorem bj-elpwgALT 35525 are alternate proofs of velpw 4565 and elpwg 4563 respectively, where one proves first the setvar case and then generalizes using vtoclbg 3528 instead of proving first the general case using elab2g 3632 and then specifying. Here, this results in needing an extra DV condition, a longer combined proof and use of ax-12 2171. In other cases, that order is better (e.g., vsnex 5386 proved before snexg 5387). (Contributed by BJ, 17-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
 
17-Jan-2025intidg 5414 The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) Put in closed form and avoid ax-nul 5263. (Revised by BJ, 17-Jan-2025.)
(𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})
 
17-Jan-2025snelpwg 5399 A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 5263. (Revised by BJ, 17-Jan-2025.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
 
16-Jan-2025ordnexbtwnsuc 41588 For any distinct pair of ordinals, if there is no ordinal between the lesser and the greater, the greater is the successor of the lesser. Lemma 1.16 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
((𝐴 ∈ 𝐵 ∧ Ord 𝐵) → (∀𝑐 ∈ On ¬ (𝐴 ∈ 𝑐 ∧ 𝑐 ∈ 𝐵) → 𝐵 = suc 𝐴))
 
16-Jan-2025onsucss 41587 If one ordinal is less than another, then the successor of the first is less than or equal to the second. Lemma 1.13 of [Schloeder] p. 2. See ordsucss 7753. (Contributed by RP, 16-Jan-2025.)
(𝐴 ∈ On → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
 
16-Jan-2025limnsuc 41586 A limit ordinal is not an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
(Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
 
16-Jan-2025dflim6 41585 A limit ordinal is a non-zero ordinal which is not a succesor ordinal. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
 
16-Jan-2025onsucelab 41584 The successor of every ordinal is an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)
(𝐴 ∈ On → suc 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
 
16-Jan-2025ondif1i 41583 Ordinal zero is less than every non-zero ordinal, class difference version. Theorem 1.10 of [Schloeder] p. 2. See ondif1 8447. (Contributed by RP, 16-Jan-2025.)
(𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
 
16-Jan-2025ordne0gt0 41582 Ordinal zero is less than every non-zero ordinal. Theorem 1.10 of [Schloeder] p. 2. Closely related to ord0eln0 6372. (Contributed by RP, 16-Jan-2025.)
((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∅ ∈ 𝐴)
 
16-Jan-2025ordeldif1o 41581 Membership in the difference of ordinal and ordinal one. (Contributed by RP, 16-Jan-2025.)
(Ord 𝐴 → (𝐵 ∈ (𝐴 ∖ 1o) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ≠ ∅)))
 
16-Jan-2025ordeldifsucon 41580 Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025.)
((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐶 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)))
 
15-Jan-2025ordeldif 41579 Membership in the difference of ordinals. (Contributed by RP, 15-Jan-2025.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐶 ∈ (𝐴 ∖ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))
 
15-Jan-2025oneltri 41578 The elementhood relation on the ordinals is complete, so we have triality. Theorem 1.9(iii) of [Schloeder] p. 1. See ordtri3or 6349. (Contributed by RP, 15-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
 
15-Jan-2025oneptri 41577 The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵))
 
15-Jan-2025oneltr 41576 The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6363. (Contributed by RP, 15-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
 
15-Jan-2025oneptr 41575 The strict order on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 E 𝐵 ∧ 𝐵 E 𝐶) → 𝐴 E 𝐶))
 
15-Jan-2025epirron 41574 The strict order on the ordinals is irreflexive. Theorem 1.9(i) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
(𝐴 ∈ On → ¬ 𝐴 E 𝐴)
 
15-Jan-2025epsoon 41573 The ordinals are strictly and completely (linearly) ordered. Theorem 1.9 of [Schloeder] p. 1. Based on epweon 7709 and weso 5624. (Contributed by RP, 15-Jan-2025.)
E Or On
 
15-Jan-2025onepsuc 41572 Every ordinal is less than its successor, relationship version. Lemma 1.7 of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.)
(𝐴 ∈ On → 𝐴 E suc 𝐴)
 
15-Jan-2025onelord 41571 Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6342 and eloni 6327. (Contributed by RP, 15-Jan-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
 
15-Jan-20251fldgenq 32089 The field of rational numbers ℚ is generated by 1 in ℂfld, that is, ℚ is the prime field of ℂfld. (Contributed by Thierry Arnoux, 15-Jan-2025.)
(ℂfld fldGen {1}) = ℚ
 
15-Jan-2025fldgenidfld 32085 The subfield generated by a subfield is the subfield itself. (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ DivRing)    &   (𝜑 → 𝑆 ∈ (SubDRing‘𝐹))    ⇒   (𝜑 → (𝐹 fldGen 𝑆) = 𝑆)
 
15-Jan-2025fldgenss 32084 Generated subfields preserve subset ordering. ( see lspss 20445 and spanss 30290) (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ DivRing)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   (𝜑 → (𝐹 fldGen 𝑇) ⊆ (𝐹 fldGen 𝑆))
 
15-Jan-2025fldgenssid 32082 The field generated by a set of elements contains those elements. See lspssid 20446. (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ DivRing)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆))
 
15-Jan-2025sdrginvcl 32077 A sub-division-ring is closed under the ring inverse operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
𝐼 = (invr‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝐴 ∈ (SubDRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐴)
 
15-Jan-2025sdrgdvcl 32076 A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.)
/ = (/r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝐴 ∈ (SubDRing‘𝑅))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑌 ≠ 0 )    ⇒   (𝜑 → (𝑋 / 𝑌) ∈ 𝐴)
 
15-Jan-2025cshwsexa 14712 The class of (different!) words resulting by cyclically shifting something (not necessarily a word) is a set. (Contributed by AV, 8-Jun-2018.) (Revised by Mario Carneiro/AV, 25-Oct-2018.) (Proof shortened by SN, 15-Jan-2025.)
{𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} ∈ V
 
15-Jan-2025sels 5395 If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5397. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5256, ax-nul 5263, ax-pow 5320. (Revised by BTernaryTau, 15-Jan-2025.)
(𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
 
15-Jan-2025snexg 5387 A singleton built on a set is a set. Special case of snex 5388 which does not require ax-nul 5263 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5388 and shorten proof. (Revised by BJ, 15-Jan-2025.)
(𝐴 ∈ 𝑉 → {𝐴} ∈ V)
 
15-Jan-2025vsnex 5386 A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
{𝑥} ∈ V
 
15-Jan-2025iunid 5020 An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) (Proof shortened by SN, 15-Jan-2025.)
∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
 
15-Jan-2025rabeqc 3419 A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.)
(𝑥 ∈ 𝐴 → 𝜑)    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴
 
14-Jan-2025resisoeq45d 41682 Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)
(𝜑 → 𝐴 = 𝐶)    &   (𝜑 → 𝐵 = 𝐷)    ⇒   (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))
 
14-Jan-2025isoeq145d 41681 Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)
(𝜑 → 𝐹 = 𝐺)    &   (𝜑 → 𝐴 = 𝐶)    &   (𝜑 → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷)))
 
14-Jan-2025sdomne0 41675 A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.)
(𝐵 ≺ 𝐴 → 𝐴 ≠ ∅)
 
14-Jan-2025omcl3g 41653 Closure law for ordinal multiplication. (Contributed by RP, 14-Jan-2025.)
(((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 ∈ 3o ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
 
13-Jan-2025mbfmbfm 32858 A measurable function to a Borel Set is measurable. (Contributed by Thierry Arnoux, 24-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)
(𝜑 → 𝐹 ∈ (dom 𝑀MblFnM(sigaGen‘𝐽)))    ⇒   (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
 
13-Jan-2025isanmbfm 32856 The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) Remove hypotheses. (Revised by SN, 13-Jan-2025.)
(𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇))    ⇒   (𝜑 → 𝐹 ∈ ∪ ran MblFnM)
 
13-Jan-2025fvssunirn 6875 The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by SN, 13-Jan-2025.)
(𝐹‘𝑋) ⊆ ∪ ran 𝐹
 
13-Jan-2025fvn0fvelrn 6873 If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by SN, 13-Jan-2025.)
((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
 
12-Jan-2025dfno2 41690 A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.)
No = {𝑓 ∈ 𝒫 (On × {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
 
12-Jan-2025omcl2 41652 Closure law for ordinal multiplication. (Contributed by RP, 12-Jan-2025.)
(((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o (ω ↑o 𝐷)) ∧ 𝐷 ∈ On))) → (𝐴 ·o 𝐵) ∈ 𝐶)
 
12-Jan-2025omabs2 41651 Ordinal multiplication by a larger ordinal is absorbed when the larger ordinal is either 2 or ω raised to some power of ω. (Contributed by RP, 12-Jan-2025.)
(((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ 𝐵 = 2o ∨ (𝐵 = (ω ↑o (ω ↑o 𝐶)) ∧ 𝐶 ∈ On))) → (𝐴 ·o 𝐵) = 𝐵)
 
12-Jan-2025bj-axadj 35512 Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj 35513). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝑥 ∪ {𝑦}) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦)))
 
12-Jan-2025bj-prex 35511 Existence of unordered pairs proved from ax-bj-sn 35504 and ax-bj-bun 35508. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
{𝐴, 𝐵} ∈ V
 
12-Jan-2025bj-prexg 35510 Existence of unordered pairs formed on sets, proved from ax-bj-sn 35504 and ax-bj-bun 35508. Contrary to bj-prex 35511, this proof is intuitionistically valid and does not require ax-nul 5263. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)
 
12-Jan-2025bj-unexg 35509 Existence of binary unions of sets, proved from ax-bj-bun 35508. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
 
12-Jan-2025ax-bj-bun 35508 Axiom of binary union. (Contributed by BJ, 12-Jan-2025.)
∀𝑥∀𝑦∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦))
 
12-Jan-2025bj-axbun 35507 Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 35508). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
((𝑥 ∪ 𝑦) ∈ V ↔ ∃𝑧∀𝑡(𝑡 ∈ 𝑧 ↔ (𝑡 ∈ 𝑥 ∨ 𝑡 ∈ 𝑦)))
 
12-Jan-2025bj-snex 35506 A singleton is a set. See also snex 5388, snexALT 5338. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 35504. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
{𝐴} ∈ V
 
12-Jan-2025bj-snexg 35505 A singleton built on a set is a set. Contrary to bj-snex 35506, this proof is intuitionistically valid and does not require ax-nul 5263. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5388 and prove it from ax-bj-sn 35504. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
(𝐴 ∈ 𝑉 → {𝐴} ∈ V)
 
12-Jan-2025ax-bj-sn 35504 Axiom of singleton. (Contributed by BJ, 12-Jan-2025.)
∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)
 
12-Jan-2025bj-axsn 35503 Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 35504). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥))
 
12-Jan-2025isrim 20165 An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    ⇒   (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
 
12-Jan-2025rabbiia 3411 Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
(𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
 
12-Jan-2025reuanid 3364 Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
(∃!𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥 ∈ 𝐴 𝜑)
 
12-Jan-2025rmoanid 3363 Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)
(∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥 ∈ 𝐴 𝜑)
 
11-Jan-2025primefldgen1 32088 The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    ⇒   (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 }))
 
11-Jan-2025fldgenfld 32087 A generated subfield is a field. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ Field)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝐹 ↾s (𝐹 fldGen 𝑆)) ∈ Field)
 
11-Jan-2025fldgenid 32086 The subfield of a field 𝐹 generated by the whole base set of 𝐹 is 𝐹 itself. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ DivRing)    ⇒   (𝜑 → (𝐹 fldGen 𝐵) = 𝐵)
 
11-Jan-2025fldgensdrg 32083 A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ DivRing)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹))
 
11-Jan-2025fldgenval 32081 Value of the field generating function: (𝐹 fldGen 𝑆) is the smallest sub-division-ring of 𝐹 containing 𝑆. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐹 ∈ DivRing)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎})
 
11-Jan-2025resrng 21025 The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.) (Proof shortened by Thierry Arnoux, 11-Jan-2025.)
ℝfld ∈ *-Ring
 
11-Jan-2025fldsdrgfld 20265 A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.)
((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRing‘𝐹)) → (𝐹 ↾s 𝐴) ∈ Field)
 
11-Jan-2025sucdom 9179 Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-pow 5320. (Revised by BTernaryTau, 4-Dec-2024.) (Proof shortened by BJ, 11-Jan-2025.)
(𝐴 ∈ ω → (𝐴 ≺ 𝐵 ↔ suc 𝐴 ≼ 𝐵))
 
11-Jan-2025onuniorsuc 7772 An ordinal number is either its own union (if zero or a limit ordinal) or the successor of its union. (Contributed by NM, 13-Jun-1994.) Put in closed form. (Revised by BJ, 11-Jan-2025.)
(𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
 
11-Jan-2025sucexeloni 7744 If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7746 does not require ax-un 7672. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.)
((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)
 
11-Jan-2025ordsuci 7743 The successor of an ordinal class is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) Extract and adapt from a subproof of onsuc 7746. (Revised by BTernaryTau, 6-Jan-2025.) (Proof shortened by BJ, 11-Jan-2025.)
(Ord 𝐴 → Ord suc 𝐴)
 
10-Jan-2025riccrng 40701 A ring is commutative if and only if an isomorphic ring is commutative. (Contributed by SN, 10-Jan-2025.)
(𝑅 ≃𝑟 𝑆 → (𝑅 ∈ CRing ↔ 𝑆 ∈ CRing))
 
10-Jan-2025riccrng1 40700 Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.)
((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ CRing) → 𝑆 ∈ CRing)
 
10-Jan-2025ricsym 40698 Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
(𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
 
10-Jan-2025brrici 40697 Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃𝑟 𝑆)
 
10-Jan-2025rimcnv 40695 The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → ◡𝐹 ∈ (𝑆 RingIso 𝑅))
 
10-Jan-2025rncrhmcl 40694 The range of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)
𝐶 = (𝑁 ↾s ran 𝐹)    &   (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   (𝜑 → 𝑀 ∈ CRing)    ⇒   (𝜑 → 𝐶 ∈ CRing)
 
10-Jan-2025ressbasss2 40669 The base set of a restriction to 𝐴 is a subset of 𝐴. (Contributed by SN, 10-Jan-2025.)
𝑅 = (𝑊 ↾s 𝐴)    ⇒   (Base‘𝑅) ⊆ 𝐴
 
10-Jan-2025ressbasssg 40668 The base set of a restriction to 𝐴 is a subset of 𝐴 and the base set 𝐵 of the original structure. (Contributed by SN, 10-Jan-2025.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐵 = (Base‘𝑊)    ⇒   (Base‘𝑅) ⊆ (𝐴 ∩ 𝐵)
 
10-Jan-2025rimrhm 20169 A ring isomorphism is a homomorphism. Compare gimghm 19054. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
 
10-Jan-2025isrim0 20156 A ring isomorphism is a homomorphism whose converse is also a homomorphism. Compare isgim2 19055. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
 
10-Jan-2025xpfi 9261 The Cartesian product of two finite sets is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 10-Jan-2025.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
 
10-Jan-2025elfvunirn 6874 A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6879). (Contributed by Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised by SN, 10-Jan-2025.)
(𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)
 
9-Jan-2025ply1fermltlchr 32283 Fermat's little theorem for polynomials in a commutative ring 𝐹 of characteristic 𝑃 prime: we have the polynomial equation (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴). (Contributed by Thierry Arnoux, 9-Jan-2025.)
𝑊 = (Poly1‘𝐹)    &   𝑋 = (var1‘𝐹)    &    + = (+g‘𝑊)    &   𝑁 = (mulGrp‘𝑊)    &    ↑ = (.g‘𝑁)    &   𝐶 = (algSc‘𝑊)    &   𝐴 = (𝐶‘((ℤRHom‘𝐹)‘𝐸))    &   𝑃 = (chr‘𝐹)    &   (𝜑 → 𝐹 ∈ CRing)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝐸 ∈ ℤ)    ⇒   (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
 
9-Jan-2025df-fldgen 32080 Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 32083). If the base structure is a field, this is a subfield (see fldgenfld 32087 and fldsdrgfld 20265). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.)
fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎})
 
8-Jan-2025dflim5 41649 A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)
(Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
 
8-Jan-2025succlg 41648 Closure law for ordinal successor. (Contributed by RP, 8-Jan-2025.)
((𝐴 ∈ 𝐵 ∧ (𝐵 = ∅ ∨ (𝐵 = (ω ·o 𝐶) ∧ 𝐶 ∈ (On ∖ 1o)))) → suc 𝐴 ∈ 𝐵)
 
8-Jan-2025omlimcl2 41562 The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.)
(((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))
 
8-Jan-2025infn0 9251 An infinite set is not empty. For a shorter proof using ax-un 7672, see infn0ALT 9252. (Contributed by NM, 23-Oct-2004.) Avoid ax-un 7672. (Revised by BTernaryTau, 8-Jan-2025.)
(ω ≼ 𝐴 → 𝐴 ≠ ∅)
 
8-Jan-2025cbvrexdva2 3324 Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 8-Jan-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
 
7-Jan-2025nnawordexg 41647 If an ordinal, 𝐵, is in a half-open interval between some 𝐴 and the next limit ordinal, 𝐵 is the sum of the 𝐴 and some natural number. This weakens the antecedent of nnawordex 8584. (Contributed by RP, 7-Jan-2025.)
((𝐴 ∈ On ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 +o ω)) → ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)
 
7-Jan-2025oawordex2 41646 If 𝐶 is between 𝐴 (inclusive) and (𝐴 +o 𝐵) (exclusive), there is an ordinal which equals 𝐶 when summed to 𝐴. This is a slightly different statement than oawordex 8504 or oawordeu 8502. (Contributed by RP, 7-Jan-2025.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 +o 𝐵))) → ∃𝑥 ∈ 𝐵 (𝐴 +o 𝑥) = 𝐶)
 
7-Jan-2025aks6d1c2p2 40549 Injective condition for countability argument assuming that 𝑁 is not a prime power. (Contributed by metakunt, 7-Jan-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   (𝜑 → 𝑄 ∈ ℙ)    &   (𝜑 → 𝑄 ∥ 𝑁)    &   (𝜑 → 𝑃 ≠ 𝑄)    ⇒   (𝜑 → 𝐸:(ℕ0 × ℕ0)–1-1→ℕ)
 
7-Jan-2025aks6d1c2p1 40548 In the AKS-theorem the subset defined by 𝐸 takes values in the positive integers. (Contributed by metakunt, 7-Jan-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    ⇒   (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
 
7-Jan-2025fldhmf1 40547 A field homomorphism is injective. This follows immediately from the definition of the ring homomorphism that sends the multiplicative identity to the multiplicative identity. (Contributed by metakunt, 7-Jan-2025.)
(𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝐿 ∈ Field)    &   (𝜑 → 𝐹 ∈ (𝐾 RingHom 𝐿))    &   𝐴 = (Base‘𝐾)    &   𝐵 = (Base‘𝐿)    ⇒   (𝜑 → 𝐹:𝐴–1-1→𝐵)
 
7-Jan-2025infsdomnn 9249 An infinite set strictly dominates a natural number. (Contributed by NM, 22-Nov-2004.) (Revised by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 7-Jan-2025.)
((ω ≼ 𝐴 ∧ 𝐵 ∈ ω) → 𝐵 ≺ 𝐴)
 
7-Jan-2025nnsdomg 9246 Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. In order to avoid the Axiom of Infinity, we include it as part of the antecedent. See nnsdom 9590 for the version without this sethood requirement. (Contributed by NM, 15-Jun-1998.) Avoid ax-pow 5320. (Revised by BTernaryTau, 7-Jan-2025.)
((ω ∈ V ∧ 𝐴 ∈ ω) → 𝐴 ≺ ω)
 
7-Jan-2025findcard3 9229 Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.) Avoid ax-pow 5320. (Revised by BTernaryTau, 7-Jan-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   (𝑦 ∈ Fin → (∀𝑥(𝑥 ⊊ 𝑦 → 𝜑) → 𝜒))    ⇒   (𝐴 ∈ Fin → 𝜏)
 
6-Jan-2025enp1i 9223 Proof induction for en2 9225 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.) Generalize to all ordinals and avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 6-Jan-2025.)
Ord 𝑀    &   𝑁 = suc 𝑀    &   ((𝐴 ∖ {𝑥}) ≈ 𝑀 → 𝜑)    &   (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   (𝐴 ≈ 𝑁 → ∃𝑥𝜓)
 
6-Jan-2025dif1ennn 9105 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. See also dif1ennnALT 9221. (Contributed by BTernaryTau, 6-Jan-2025.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
6-Jan-2025dif1en 9104 If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)
((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
6-Jan-2025ord3 8429 Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.)
Ord 3o
 
6-Jan-2025ordsuc 7748 A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7672. (Revised by BTernaryTau, 6-Jan-2025.)
(Ord 𝐴 ↔ Ord suc 𝐴)
 
5-Jan-2025et-sqrtnegnre 45104 The square root of a negative number is not a real number. (Contributed by Ender Ting, 5-Jan-2025.)
((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ¬ (√‘𝐴) ∈ ℝ)
 
5-Jan-2025smfdivdmmbl2 45072 If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator. It is required only for the function at the denominator. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐺    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹:𝐴⟶𝑉)    &   (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → dom 𝐺 ∈ 𝑆)    &   𝐷 = {𝑥 ∈ dom 𝐺 ∣ (𝐺‘𝑥) ≠ 0}    &   𝐻 = (𝑥 ∈ (dom 𝐹 ∩ 𝐷) ↦ ((𝐹‘𝑥) / (𝐺‘𝑥)))    ⇒   (𝜑 → dom 𝐻 ∈ 𝑆)
 
5-Jan-2025smfpimne2 45071 Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value is in the subspace sigma-algebra induced by its domain. Notice that 𝐴 is not assumed to be an extended real. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐹    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≠ 𝐴} ∈ (𝑆 ↾t 𝐷))
 
5-Jan-2025smfpimne 45070 Given a function measurable w.r.t. to a sigma-algebra, the preimage of reals that are different from a value in the extended reals is in the subspace of sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐹    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑 → 𝐴 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≠ 𝐴} ∈ (𝑆 ↾t 𝐷))
 
5-Jan-2025smfdivdmmbl 45069 If a functions and a sigma-measurable function have domains in the sigma-algebra, the domain of the division of the two functions is in the sigma-algebra. This is the third statement of Proposition 121H of [Fremlin1] p. 39 . Note: While the theorem in the book assumes both functions are sigma-measurable, this assumption is unnecessary for the part concerning their division, for the function at the numerator (it is needed only for the function at the denominator). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐵    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ 𝑊)    &   (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ (SMblFn‘𝑆))    &   𝐸 = {𝑥 ∈ 𝐵 ∣ 𝐷 ≠ 0}    ⇒   (𝜑 → (𝐴 ∩ 𝐸) ∈ 𝑆)
 
5-Jan-2025smfdmmblpimne 45068 If a measurable function w.r.t. to a sigma-algebra has domain in the sigma-algebra, the set of elements that are not mapped to a given real, is in the sigma-algebra (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → 𝐶 ∈ ℝ)    &   𝐷 = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶}    ⇒   (𝜑 → 𝐷 ∈ 𝑆)
 
5-Jan-2025smffmptf 45035 A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝑆 ∈ SAlg)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
 
5-Jan-2025fmptdff 43490 A version of fmptd 7062 using bound-variable hypothesis instead of a distinct variable condition for 𝜑. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐶    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    &   𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   (𝜑 → 𝐹:𝐴⟶𝐶)
 
5-Jan-2025fvmptelcdmf 43489 The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐶    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)    ⇒   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
 
5-Jan-2025fmptff 43488 Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    &   𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)    ⇒   (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
 
5-Jan-2025dmmpt1 43487 The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐵    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) = 𝐵)
 
5-Jan-2025ssrabdf 43315 Subclass of a restricted class abstraction (deduction form). (Contributed by Glauco Siliprandi, 5-Jan-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    &   Ⅎ𝑥𝜑    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝜓)    ⇒   (𝜑 → 𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜓})
 
5-Jan-2025ofoacom 41661 Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (ω ↑m 𝐴) ∧ 𝐺 ∈ (ω ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
 
5-Jan-2025ofoaass 41660 Component-wise addition of ordinal-yielding functions is associative. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ 𝐺 ∈ (𝐵 ↑m 𝐴) ∧ 𝐻 ∈ (𝐵 ↑m 𝐴))) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
 
5-Jan-2025ofoaid2 41659 Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → ((𝐴 × {∅}) ∘f +o 𝐹) = 𝐹)
 
5-Jan-2025ofoaid1 41658 Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝐹 ∈ (𝐵 ↑m 𝐴)) → (𝐹 ∘f +o (𝐴 × {∅})) = 𝐹)
 
5-Jan-2025ofoacl 41657 Closure law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) ∧ (𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐴))) → (𝐹 ∘f +o 𝐺) ∈ (𝐶 ↑m 𝐴))
 
5-Jan-2025ofoafo 41656 Addition operator for functions from a set into a power of omega is an onto binary operator. (Contributed by RP, 5-Jan-2025.)
((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ On ∧ 𝐶 = (ω ↑o 𝐵))) → ( ∘f +o ↾ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))):((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐴))–onto→(𝐶 ↑m 𝐴))
 
5-Jan-2025ofoaf 41655 Addition operator for functions from sets into power of omega results in a function from the intersection of sets to that power of omega. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 = (ω ↑o 𝐷))) → ( ∘f +o ↾ ((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐸 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐸 ↑m 𝐶))
 
5-Jan-2025ofoafg 41654 Addition operator for functions from sets into ordinals results in a function from the intersection of sets into an ordinal. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 = (𝐴 ∩ 𝐵)) ∧ (𝐷 ∈ On ∧ 𝐸 ∈ On ∧ 𝐹 = ∪ 𝑑 ∈ 𝐷 (𝑑 +o 𝐸))) → ( ∘f +o ↾ ((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))):((𝐷 ↑m 𝐴) × (𝐸 ↑m 𝐵))⟶(𝐹 ↑m 𝐶))
 
5-Jan-2025oacl2g 41650 Closure law for ordinal addition. Here we show that ordinal addition is closed within the empty set or any ordinal power of omega. (Contributed by RP, 5-Jan-2025.)
(((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ (𝐶 = ∅ ∨ (𝐶 = (ω ↑o 𝐷) ∧ 𝐷 ∈ On))) → (𝐴 +o 𝐵) ∈ 𝐶)
 
5-Jan-2025rexdif1en 9102 If a set is equinumerous to a nonzero ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals and avoid ax-un 7672. (Revised by BTernaryTau, 5-Jan-2025.)
((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
 
5-Jan-2025dif1enlem 9100 Lemma for rexdif1en 9102 and dif1en 9104. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7672. (Revised by BTernaryTau, 5-Jan-2025.)
(((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
 
5-Jan-2025imbibi 392 The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.)
(((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))
 
4-Jan-2025en1eqsn 9218 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 4-Jan-2025.)
((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})
 
4-Jan-2025f1finf1o 9215 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.) Avoid ax-pow 5320. (Revised by BTernaryTau, 4-Jan-2025.)
((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐹:𝐴–1-1→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵))
 
3-Jan-2025naddcnfass 41669 Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) → ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
 
3-Jan-2025naddcnfid2 41668 Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → ((𝑋 × {∅}) ∘f +o 𝐹) = 𝐹)
 
3-Jan-2025naddcnfid1 41667 Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) → (𝐹 ∘f +o (𝑋 × {∅})) = 𝐹)
 
3-Jan-2025glbconN 37839 De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) New df-riota 7313. (Revised by SN, 3-Jan-2025.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝐺 = (glb‘𝐾)    &    ⊥ = (oc‘𝐾)    ⇒   ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))
 
3-Jan-2025nfra2w 3282 Similar to Lemma 24 of [Monk2] p. 114, except that quantification is restricted. Once derived from hbra2VD 43132. Version of nfra2 3349 with a disjoint variable condition not requiring ax-13 2370. (Contributed by Alan Sare, 31-Dec-2011.) Reduce axiom usage. (Revised by Gino Giotto, 24-Sep-2024.) (Proof shortened by Wolf Lammen, 3-Jan-2025.)
Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
 
2-Jan-2025naddcnfcom 41666 Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
 
2-Jan-2025naddcnfcl 41665 Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) → (𝐹 ∘f +o 𝐺) ∈ 𝑆)
 
2-Jan-2025naddcnffo 41664 Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)–onto→𝑆)
 
2-Jan-2025naddcnffn 41663 Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
 
2-Jan-2025naddcnff 41662 Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (ω CNF 𝑋)) → ( ∘f +o ↾ (𝑆 × 𝑆)):(𝑆 × 𝑆)⟶𝑆)
 
2-Jan-2025isinf 9204 Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.) Avoid ax-pow 5320. (Revised by BTernaryTau, 2-Jan-2025.)
(¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
 
2-Jan-2025ominf 9202 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.) Avoid ax-pow 5320. (Revised by BTernaryTau, 2-Jan-2025.)
¬ ω ∈ Fin
 
2-Jan-2025dtru 5393 Given any set (the "𝑦 " in the statement), not all sets are equal to it. The same statement without disjoint variable condition is false since it contradicts stdpc6 2031. The same comments and revision history concerning axiom usage as in exneq 5392 apply. (Contributed by NM, 7-Nov-2006.) Extract exneq 5392 as an intermediate result. (Revised by BJ, 2-Jan-2025.)
¬ ∀𝑥 𝑥 = 𝑦
 
2-Jan-2025exneq 5392 Given any set (the "𝑦 " in the statement), there exists a set not equal to it.

The same statement without disjoint variable condition is false, since we do not have ∃𝑥¬ 𝑥 = 𝑥. This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext 2707, ax-sep 5256, or ax-pow 5320 nor auxiliary logical axiom schemes ax-10 2137 to ax-13 2370. See dtruALT 5343 for a shorter proof using more axioms, and dtruALT2 5325 for a proof using ax-pow 5320 instead of ax-pr 5384. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-8 2108. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5384 instead of ax-pow 5320. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5393. (Revised by BJ, 2-Jan-2025.)

∃𝑥 ¬ 𝑥 = 𝑦
 
2-Jan-2025exexneq 5391 There exist two different sets. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-8 2108. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5384 instead of ax-pow 5320. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5393. (Revised by BJ, 2-Jan-2025.)
∃𝑥∃𝑦 ¬ 𝑥 = 𝑦
 
2-Jan-2025ralcom13 3277 Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.) (Proof shortened by Wolf Lammen, 2-Jan-2025.)
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑)
 
1-Jan-2025snssg 4744 The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))
 
1-Jan-2025snssb 4743 Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
 
31-Dec-2024mpets 37304 Member Partition-Equivalence Theorem in its shortest possible form: it shows that member partitions and comember equivalence relations are literally the same. Cf. pet 37313, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
MembParts = CoMembErs
 
31-Dec-2024cpet 37300 The conventional form of Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have been calling disjoint or partition what we call element disjoint or member partition, see also cpet2 37299. Cf. mpet 37301, mpet2 37302 and mpet3 37298 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 37313 and pet2 37312 for Partition-Equivalence Theorem with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)
( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
 
31-Dec-2024eqvrelcossid 37256 The cosets by the identity class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
EqvRel ≀ I
 
31-Dec-2024eqvrelcoss0 37250 The cosets by the null class are in equivalence relation. (Contributed by Peter Mazsa, 31-Dec-2024.)
EqvRel ≀ ∅
 
31-Dec-2024eldisjn0elb 37207 Two forms of disjoint elements when the empty set is not an element of the class. (Contributed by Peter Mazsa, 31-Dec-2024.)
(( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
 
31-Dec-2024suceqsneq 36694 One-to-one relationship between the successor operation and the singleton. (Contributed by Peter Mazsa, 31-Dec-2024.)
(𝐴 ∈ 𝑉 → (suc 𝐴 = suc 𝐵 ↔ {𝐴} = {𝐵}))
 
30-Dec-2024muldmmbl2 45067 If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐺    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → dom 𝐹 ∈ 𝑆)    &   (𝜑 → dom 𝐺 ∈ 𝑆)    &   𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))    ⇒   (𝜑 → dom 𝐻 ∈ 𝑆)
 
30-Dec-2024muldmmbl 45066 If two functions have domains in the sigma-algebra, the domain of their multiplication also belongs to the sigma-algebra. This is the second statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their multiplication. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    ⇒   (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 · 𝐷)) ∈ 𝑆)
 
30-Dec-2024adddmmbl2 45065 If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐺    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → dom 𝐹 ∈ 𝑆)    &   (𝜑 → dom 𝐺 ∈ 𝑆)    &   𝐻 = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥) + (𝐺‘𝑥)))    ⇒   (𝜑 → dom 𝐻 ∈ 𝑆)
 
30-Dec-2024adddmmbl 45064 If two functions have domains in the sigma-algebra, the domain of their addition also belongs to the sigma-algebra. This is the first statement of Proposition 121H of [Fremlin1], p. 39. Note: While the theorem in the book assumes the functions are sigma-measurable, this assumption is unnecessary for the part concerning their addition. (Contributed by Glauco Siliprandi, 30-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    ⇒   (𝜑 → dom (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ (𝐶 + 𝐷)) ∈ 𝑆)
 
30-Dec-2024cpet2 37299 The conventional form of the Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have called disjoint or partition what we call element disjoint or member partition, see also cpet 37300. Together with cpet 37300, mpet 37301 mpet2 37302, this is what we used to think of as the partition equivalence theorem (but cf. pet2 37312 with general 𝑅). (Contributed by Peter Mazsa, 30-Dec-2024.)
(( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
 
30-Dec-2024fences3 37292 Implication of eqvrelqseqdisj2 37291 and n0eldmqseq 37111, see comment of fences 37306. (Contributed by Peter Mazsa, 30-Dec-2024.)
(( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
 
30-Dec-2024eldisjim2 37247 Alternate form of eldisjim 37246. (Contributed by Peter Mazsa, 30-Dec-2024.)
( ElDisj 𝐴 → EqvRel ∼ 𝐴)
 
30-Dec-2024eqvreldmqs2 37138 Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024.)
(( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
 
30-Dec-2024n0elim 37112 Implication of that the empty set is not an element of a class. (Contributed by Peter Mazsa, 30-Dec-2024.)
(¬ ∅ ∈ 𝐴 → (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)
 
30-Dec-2024pr2ne 9940 If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 30-Dec-2024.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))
 
30-Dec-2024enpr2 9938 An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 8993. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 30-Dec-2024.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
 
30-Dec-20241sdom 9192 A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8974.) (Contributed by Mario Carneiro, 12-Jan-2013.) Avoid ax-un 7672. (Revised by BTernaryTau, 30-Dec-2024.)
(𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
 
30-Dec-2024rex2dom 9190 A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)
((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦) → 2o ≼ 𝐴)
 
29-Dec-2024dffun2 6506 Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 
29-Dec-2024cnvsym 6066 Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
(◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
 
29-Dec-2024cotrg 6061 Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr 6064 for the main application. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6064. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) Avoid ax-11 2154. (Revised by BTernaryTau, 29-Dec-2024.)
((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
 
28-Dec-2024onunisuc 6427 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.) Generalize from onunisuci 6437. (Revised by BJ, 28-Dec-2024.)
(𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
 
28-Dec-2024unisucg 6395 A transitive class is equal to the union of its successor, closed form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) Generalize from unisuc 6396. (Revised by BJ, 28-Dec-2024.)
(𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
 
28-Dec-2024unisucs 6394 The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6396. (Revised by BJ, 28-Dec-2024.)
(𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
 
28-Dec-2024dftr5 5226 An alternate way of defining a transitive class. Definition 1.1 of [Schloeder] p. 1. (Contributed by NM, 20-Mar-2004.) Avoid ax-11 2154. (Revised by BTernaryTau, 28-Dec-2024.)
(Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
 
28-Dec-2024dftr2c 5225 Variant of dftr2 5224 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 2154. (Contributed by BTernaryTau, 28-Dec-2024.)
(Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
 
28-Dec-2024unissb 4900 Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.) Avoid ax-11 2154. (Revised by BTernaryTau, 28-Dec-2024.)
(∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
 
28-Dec-2024alcomw 2047 Weak version of alcom 2156 and biconditional form of alcomiw 2046. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 28-Dec-2024.)
(𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))    ⇒   (∀𝑥∀𝑦𝜑 ↔ ∀𝑦∀𝑥𝜑)
 
25-Dec-2024partimeq 37271 Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 37141. (Contributed by Peter Mazsa, 25-Dec-2024.)
(𝑅 ∈ 𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))
 
23-Dec-2024rmxyelqirr 41219 The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.)
((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ0 ∃𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})
 
23-Dec-20241sdom2dom 9191 Strict dominance over 1 is the same as dominance over 2. (Contributed by BTernaryTau, 23-Dec-2024.)
(1o ≺ 𝐴 ↔ 2o ≼ 𝐴)
 
23-Dec-2024enpr2d 8993 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-un 7672. (Revised by BTernaryTau, 23-Dec-2024.)
(𝜑 → 𝐴 ∈ 𝐶)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   (𝜑 → {𝐴, 𝐵} ≈ 2o)
 
23-Dec-2024en2prd 8992 Two unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})
 
23-Dec-2024tz6.12-1 6865 Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
 
23-Dec-2024tz6.12c 6864 Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
 
23-Dec-2024relssdmrn 6220 A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.) (Proof shortened by SN, 23-Dec-2024.)
(Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
 
23-Dec-2024cnvsymOLD 6067 Obsolete proof of cnvsym 6066 as of 29-Dec-2024. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by SN, 23-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))
 
23-Dec-2024exel 5390 There exist two sets, one a member of the other.

This theorem looks similar to el 5394, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1811, ax-6 1971, and ax-pr 5384. This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq 5391. (Contributed by SN, 23-Dec-2024.)

∃𝑦∃𝑥 𝑥 ∈ 𝑦
 
23-Dec-2024rexlimivw 3148 Weaker version of rexlimiv 3145. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2024.)
(𝜑 → 𝜓)    ⇒   (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)
 
23-Dec-2024rexlimiva 3144 Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.) Shorten dependent theorems. (Revised by Wolf lammen, 23-Dec-2024.)
((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)    ⇒   (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)
 
22-Dec-2024ssabdv 40641 Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.)
(𝜑 → (𝑥 ∈ 𝐴 → 𝜓))    ⇒   (𝜑 → 𝐴 ⊆ {𝑥 ∣ 𝜓})
 
22-Dec-2024ss2ab1 40640 Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4016 using fewer axioms. (Contributed by SN, 22-Dec-2024.)
(∀𝑥(𝜑 → 𝜓) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝜓})
 
22-Dec-2024mainpart 37305 Partition with general 𝑅 also imply member partition. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
(𝑅 Part 𝐴 → MembPart 𝐴)
 
22-Dec-2024partimcomember 37297 Partition with general 𝑅 (in addition to the member partition cf. mpet 37301 and mpet2 37302) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.)
(𝑅 Part 𝐴 → CoMembEr 𝐴)
 
22-Dec-2024abssdv 4025 Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.) (Proof shortened by SN, 22-Dec-2024.)
(𝜑 → (𝜓 → 𝑥 ∈ 𝐴))    ⇒   (𝜑 → {𝑥 ∣ 𝜓} ⊆ 𝐴)
 
22-Dec-2024r19.29 3117 Restricted quantifier version of 19.29 1876. See also r19.29r 3119. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
 
22-Dec-2024r19.35 3111 Restricted quantifier version of 19.35 1880. (Contributed by NM, 20-Sep-2003.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
(∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
 
22-Dec-2024ralcom3 3100 A commutation law for restricted universal quantifiers that swaps the domains of the restriction. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Wolf Lammen, 22-Dec-2024.)
(∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 → 𝜑))
 
21-Dec-2024et-equeucl 45103 Alternative proof that equality is left-Euclidean, using ax7 2019 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.)
(𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
 
21-Dec-2024salrestss 44592 A sigma-algebra restricted to one of its elements is a subset of the original sigma-algebra. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐸 ∈ 𝑆)    ⇒   (𝜑 → (𝑆 ↾t 𝐸) ⊆ 𝑆)
 
21-Dec-2024pimxrneun 43714 The preimage of a set of extended reals that does not contain a value 𝐶 is the union of the preimage of the elements smaller than 𝐶 and the preimage of the subset of elements larger than 𝐶. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝐶} = ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}))
 
21-Dec-2024mpteq2dfa 43486 Slightly more general equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
 
21-Dec-2024dmmptif 43485 Domain of the mapping operation. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Ⅎ𝑥𝐴    &   𝐵 ∈ V    &   𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   dom 𝐹 = 𝐴
 
21-Dec-2024fnmptif 43484 Functionality and domain of an ordered-pair class abstraction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Ⅎ𝑥𝐴    &   𝐵 ∈ V    &   𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   𝐹 Fn 𝐴
 
21-Dec-2024dmmptdff 43434 The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐵    &   𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → dom 𝐴 = 𝐵)
 
21-Dec-2024toprestsubel 43359 A subset is open in the topology it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ⊆ ∪ 𝐽)    ⇒   (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))
 
21-Dec-2024restsubel 43358 A subset belongs in the space it generates via restriction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑 → 𝐽 ∈ 𝑉)    &   (𝜑 → ∪ 𝐽 ∈ 𝐽)    &   (𝜑 → 𝐴 ⊆ ∪ 𝐽)    ⇒   (𝜑 → 𝐴 ∈ (𝐽 ↾t 𝐴))
 
21-Dec-2024restopnssd 43357 A topology restricted to an open set is a subset of the original topology. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ∈ 𝐽)    ⇒   (𝜑 → (𝐽 ↾t 𝐴) ⊆ 𝐽)
 
21-Dec-2024restopn3 43356 If 𝐴 is open, then 𝐴 is open in the restriction to itself. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ (𝐽 ↾t 𝐴))
 
21-Dec-2024ss2rabdf 43355 Deduction of restricted abstraction subclass from implication. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})
 
21-Dec-2024inopnd 43354 The intersection of two open sets of a topology is an open set. (Contributed by Glauco Siliprandi, 21-Dec-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ∈ 𝐽)    &   (𝜑 → 𝐵 ∈ 𝐽)    ⇒   (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝐽)
 
20-Dec-2024smfpimgtxrmpt 45016 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝑆 ∈ SAlg)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → 𝐿 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴))
 
20-Dec-2024smfpimgtxrmptf 45015 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝑆 ∈ SAlg)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → 𝐿 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴))
 
20-Dec-2024smfpimltxrmpt 44990 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝑆 ∈ SAlg)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → 𝑅 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴))
 
20-Dec-2024smfpimltxrmptf 44989 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝑆 ∈ SAlg)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆))    &   (𝜑 → 𝑅 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑅} ∈ (𝑆 ↾t 𝐴))
 
20-Dec-2024pimgtmnf 44954 Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)
 
20-Dec-2024pimgtmnff 44953 Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < 𝐵} = 𝐴)
 
20-Dec-2024pimltpnf 44935 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)
 
20-Dec-2024pimltpnff 44934 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < +∞} = 𝐴)
 
19-Dec-2024isarep1 6590 Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof shortened by SN, 19-Dec-2024.)
(𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
 
19-Dec-2024funimaexg 6587 Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.) Shorten proof and avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.)
((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
 
19-Dec-2024funmo 6516 A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) (Proof shortened by SN, 19-Dec-2024.)
(Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
 
19-Dec-2024dffun3 6510 Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof shortened by SN, 19-Dec-2024.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
 
19-Dec-2024dffun6 6509 Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.)
(Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
 
19-Dec-2024dffun2OLD 6507 Obsolete version of dffun2 6506 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧)))
 
19-Dec-2024cotrgOLD 6062 Obsolete version of cotrg 6061 as of 29-Dec-2024. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalized from its special instance cotr 6064. (Revised by Richard Penner, 24-Dec-2019.) (Proof shortened by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧))
 
19-Dec-2024difopab 5786 Difference of two ordered-pair class abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof shortened by SN, 19-Dec-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
 
15-Dec-2024smfpimgtxr 45011 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑 → 𝐴 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷))
 
15-Dec-2024smfpimltxr 44978 Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹 ∈ (SMblFn‘𝑆))    &   𝐷 = dom 𝐹    &   (𝜑 → 𝐴 ∈ ℝ*)    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))
 
15-Dec-2024pimltpnf2 44944 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)
 
15-Dec-2024pimltpnf2f 44943 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)
 
15-Dec-2024pimgtpnf2 44937 Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)
 
15-Dec-2024pimltmnf2 44929 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)
 
15-Dec-2024pimltmnf2f 44928 Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -∞, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.)
Ⅎ𝑥𝐹    &   Ⅎ𝑥𝐴    &   (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < -∞} = ∅)
 
14-Dec-2024fzuntgd 41720 Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝐿 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ≤ 𝑀)    &   (𝜑 → 𝑀 ≤ (𝐿 + 1))    &   (𝜑 → 𝐿 ≤ 𝑁)    ⇒   (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
14-Dec-2024fzunt1d 41719 Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝐿 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ≤ 𝑀)    &   (𝜑 → 𝑀 ≤ 𝐿)    &   (𝜑 → 𝐿 ≤ 𝑁)    ⇒   (𝜑 → ((𝐾...𝐿) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
14-Dec-2024fzuntd 41718 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ≤ 𝑀)    &   (𝜑 → 𝑀 ≤ 𝑁)    ⇒   (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
14-Dec-2024fzunt 41717 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.)
(((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁)) → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))
 
13-Dec-2024nlim4 41707 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
¬ Lim 4o
 
13-Dec-2024nlim3 41706 3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
¬ Lim 3o
 
13-Dec-2024nlim2NEW 41705 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
¬ Lim 2o
 
13-Dec-2024nlim1NEW 41704 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
¬ Lim 1o
 
13-Dec-2024nlimsuc 41703 A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
(𝐴 ∈ On → ¬ Lim suc 𝐴)
 
13-Dec-2024wksonproplem 28652 Lemma for theorems for properties of walks between two vertices, e.g., trlsonprop 28656. (Contributed by AV, 16-Jan-2021.) Remove is-walk hypothesis. (Revised by SN, 13-Dec-2024.)
𝑉 = (Vtx‘𝐺)    &   (((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))    &   𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑂‘𝑔)𝑏)𝑝 ∧ 𝑓(𝑄‘𝑔)𝑝)}))    ⇒   (𝐹(𝐴(𝑊‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂‘𝐺)𝐵)𝑃 ∧ 𝐹(𝑄‘𝐺)𝑃)))
 
13-Dec-2024mptmpoopabovd 8014 The operation value of a function value of a collection of ordered pairs of related elements. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
(𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)})
 
13-Dec-2024mptmpoopabbrd 8013 The operation value of a function value of a collection of ordered pairs of elements related in two ways. (Contributed by Alexander van Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable condition on 𝐷, 𝑓, ℎ to remove hypotheses. (Revised by SN, 13-Dec-2024.)
(𝜑 → 𝐺 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ (𝐴‘𝐺))    &   (𝜑 → 𝑌 ∈ (𝐵‘𝐺))    &   ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝜏 ↔ 𝜃))    &   (𝑔 = 𝐺 → (𝜒 ↔ 𝜏))    &   𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {⟨𝑓, ℎ⟩ ∣ (𝜒 ∧ 𝑓(𝐷‘𝑔)ℎ)}))    ⇒   (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {⟨𝑓, ℎ⟩ ∣ (𝜃 ∧ 𝑓(𝐷‘𝐺)ℎ)})
 
13-Dec-2024fvmptopab 7411 The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) Add disjoint variable condition on 𝐹, 𝑥, 𝑦 to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.)
(𝑧 = 𝑍 → (𝜑 ↔ 𝜓))    &   𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)})    ⇒   (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}
 
13-Dec-2024opabresex2 7409 Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.) Add disjoint variable conditions betweem 𝑊, 𝐺 and 𝑥, 𝑦 to remove hypotheses. (Revised by SN, 13-Dec-2024.)
{⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V
 
13-Dec-2024nfralw 3294 Bound-variable hypothesis builder for restricted quantification. Version of nfral 3347 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 1-Sep-1999.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 13-Dec-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
 
12-Dec-2024sdom1 9186 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 12-Dec-2024.)
(𝐴 ≺ 1o ↔ 𝐴 = ∅)
 
11-Dec-2024wksv 28567 The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.)
{⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
 
11-Dec-2024abrexexg 7893 Existence of a class abstraction of existentially restricted sets. The class 𝐵 can be thought of as an expression in 𝑥 (which is typically a free variable in the class expression substituted for 𝐵) and the class abstraction appearing in the statement as the class of values 𝐵 as 𝑥 varies through 𝐴. If the "domain" 𝐴 is a set, then the abstraction is also a set. Therefore, this statement is a kind of Replacement. This can be seen by tracing back through the path axrep6g 5250, axrep6 5249, ax-rep 5242. See also abrexex2g 7897. There are partial converses under additional conditions, see for instance abnexg 7690. (Contributed by NM, 3-Nov-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2137, ax-11 2154, ax-12 2171, ax-pr 5384, ax-un 7672 and shorten proof. (Revised by SN, 11-Dec-2024.)
(𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
 
11-Dec-2024ssrel 5738 A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5256, ax-nul 5263, ax-pr 5384. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2171. (Revised by SN, 11-Dec-2024.)
(Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
 
11-Dec-2024elopaelxp 5721 Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.) Avoid ax-sep 5256, ax-nul 5263, ax-pr 5384. (Revised by SN, 11-Dec-2024.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
 
11-Dec-2024elopabr 5518 Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
(𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)
 
11-Dec-2024elopabw 5483 Membership in a class abstraction of ordered pairs. Weaker version of elopab 5484 with a sethood antecedent, avoiding ax-sep 5256, ax-nul 5263, and ax-pr 5384. Originally a subproof of elopab 5484. (Contributed by SN, 11-Dec-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
 
11-Dec-2024axrep6g 5250 axrep6 5249 in class notation. It is equivalent to both ax-rep 5242 and abrexexg 7893, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.)
((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)
 
11-Dec-2024dfiun2g 4990 Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 11-Dec-2024.)
(∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
 
11-Dec-2024r19.21v 3176 Restricted quantifier version of 19.21v 1942. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 11-Dec-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
 
10-Dec-2024sltn0 27234 If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅))
 
10-Dec-2024cbvreuw 3383 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3399 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2137. (Revised by Wolf Lammen, 10-Dec-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
8-Dec-20241sdom2 9184 Ordinal 1 is strictly dominated by ordinal 2. For a shorter proof requiring ax-un 7672, see 1sdom2ALT 9185. (Contributed by NM, 4-Apr-2007.) Avoid ax-un 7672. (Revised by BTernaryTau, 8-Dec-2024.)
1o ≺ 2o
 
8-Dec-2024rexcom 3273 Commutation of restricted existential quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2024.)
(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
 
7-Dec-20240sdom1dom 9182 Strict dominance over 0 is the same as dominance over 1. For a shorter proof requiring ax-un 7672, see 0sdom1domALT . (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7672. (Revised by BTernaryTau, 7-Dec-2024.)
(∅ ≺ 𝐴 ↔ 1o ≼ 𝐴)
 
7-Dec-2024ssct 8995 Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 7-Dec-2024.)
((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ ω) → 𝐴 ≼ ω)
 
7-Dec-2024domssr 8939 If 𝐶 is a superset of 𝐵 and 𝐵 dominates 𝐴, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶)
 
7-Dec-2024domssl 8938 If 𝐴 is a subset of 𝐵 and 𝐶 dominates 𝐵, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)
((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
7-Dec-2024f1dom4g 8905 The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 8912 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
(((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
7-Dec-2024f1oen4g 8904 The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 8911 does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
(((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
5-Dec-2024sb8f 2349 Substitution of variable in universal quantifier. Version of sb8 2519 with a disjoint variable condition, not requiring ax-10 2137 or ax-13 2370. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2137. (Revised by SN, 5-Dec-2024.)
Ⅎ𝑦𝜑    ⇒   (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
5-Dec-2024sb8v 2348 Substitution of variable in universal quantifier. Version of sb8f 2349 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2171. (Contributed by SN, 5-Dec-2024.)
(∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
4-Dec-2024sucdom2 9150 Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 4-Dec-2024.)
(𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)
 
4-Dec-2024undom 9003 Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 4-Dec-2024.)
(((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))
 
3-Dec-2024eqvreldisj1 37286 The elements of the quotient set of an equivalence relation are disjoint (cf. eqvreldisj2 37287, eqvreldisj3 37288). (Contributed by Mario Carneiro, 10-Dec-2016.) (Revised by Peter Mazsa, 3-Dec-2024.)
( EqvRel 𝑅 → ∀𝑥 ∈ (𝐴 / 𝑅)∀𝑦 ∈ (𝐴 / 𝑅)(𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
 
3-Dec-2024fvprc 6834 A function's value at a proper class is the empty set. See fvprcALT 6835 for a proof that uses ax-pow 5320 instead of ax-pr 5384. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5320. (Revised by BTernaryTau, 3-Aug-2024.) (Proof shortened by BTernaryTau, 3-Dec-2024.)
(¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
 
3-Dec-2024f1un 6804 The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.)
(((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))
 
3-Dec-2024dtruOLD 5398 Obsolete proof of dtru 5393 as of 01-Jan-2025. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2370. (Revised by BJ, 31-May-2019.) Avoid ax-12 2171. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5384 instead of ax-pow 5320. (Revised by BTernaryTau, 3-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ∀𝑥 𝑥 = 𝑦
 
2-Dec-2024onomeneq 9172 An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.) Avoid ax-pow 5320. (Revised by BTernaryTau, 2-Dec-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
 
2-Dec-2024el 5394 Any set is an element of some other set. See elALT 5397 for a shorter proof using more axioms, and see elALT2 5324 for a proof that uses ax-9 2116 and ax-pow 5320 instead of ax-pr 5384. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5384 instead of ax-9 2116 and ax-pow 5320. (Revised by BTernaryTau, 2-Dec-2024.)
∃𝑦 𝑥 ∈ 𝑦
 
1-Dec-2024frrlem16 9694 Lemma for general well-founded recursion. Establish a subset relation. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.)
(((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
 
1-Dec-2024snnen2o 9181 A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 1-Dec-2024.)
¬ {𝐴} ≈ 2o
 
1-Dec-20242onn 8588 The ordinal 2 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7672, see 2onnALT 8589. (Contributed by NM, 28-Sep-2004.) Avoid ax-un 7672. (Revised by BTernaryTau, 1-Dec-2024.)
2o ∈ ω
 
1-Dec-20241onn 8586 The ordinal 1 is a natural number. For a shorter proof using Peano's postulates that depends on ax-un 7672, see 1onnALT 8587. Lemma 2.2 of [Schloeder] p. 4. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7672. (Revised by BTernaryTau, 1-Dec-2024.)
1o ∈ ω
 
1-Dec-20242ellim 8445 A limit ordinal contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
(Lim 𝐴 → 2o ∈ 𝐴)
 
1-Dec-20241ellim 8444 A limit ordinal contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
(Lim 𝐴 → 1o ∈ 𝐴)
 
1-Dec-2024ord2eln012 8443 An ordinal that is not 0, 1, or 2 contains 2. (Contributed by BTernaryTau, 1-Dec-2024.)
(Ord 𝐴 → (2o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o)))
 
1-Dec-2024ord1eln01 8442 An ordinal that is not 0 or 1 contains 1. (Contributed by BTernaryTau, 1-Dec-2024.)
(Ord 𝐴 → (1o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o)))
 
1-Dec-2024nlim2 8436 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
¬ Lim 2o
 
1-Dec-2024nlim1 8435 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
¬ Lim 1o
 
1-Dec-2024f1cdmsn 7228 If a one-to-one function with a nonempty domain has a singleton as its codomain, its domain must also be a singleton. (Contributed by BTernaryTau, 1-Dec-2024.)
((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
 
30-Nov-20242on 8426 Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) Avoid ax-un 7672. (Revised by BTernaryTau, 30-Nov-2024.)
2o ∈ On
 
30-Nov-20241on 8424 Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) Avoid ax-un 7672. (Revised by BTernaryTau, 30-Nov-2024.)
1o ∈ On
 
30-Nov-2024onsuc 7746 The successor of an ordinal number is an ordinal number. Closed form of onsuci 7774. Forward implication of onsucb 7752. Proposition 7.24 of [TakeutiZaring] p. 41. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) (Proof shortened by BTernaryTau, 30-Nov-2024.)
(𝐴 ∈ On → suc 𝐴 ∈ On)
 
30-Nov-2024sucexeloniOLD 7745 Obsolete version of sucexeloni 7744 as of 6-Jan-2025. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)
 
30-Nov-2024epweon 7709 The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7672, see epweonALT 7710. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7672. (Revised by BTernaryTau, 30-Nov-2024.)
E We On
 
30-Nov-2024elex2 2816 If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2116, ax-ext 2707, df-clab 2714. (Revised by Wolf Lammen, 30-Nov-2024.)
(𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵)
 
29-Nov-2024nndomog 9160 Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 9177 when both are natural numbers. (Contributed by NM, 17-Jun-1998.) Generalize from nndomo 9177. (Revised by RP, 5-Nov-2023.) Avoid ax-pow 5320. (Revised by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
 
29-Nov-2024sdom0 9052 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 29-Nov-2024.)
¬ 𝐴 ≺ ∅
 
29-Nov-20240sdomg 9048 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
 
29-Nov-2024dom0 9046 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
 
29-Nov-20240domg 9044 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5320, ax-un 7672. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
 
29-Nov-2024en0r 8960 The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.)
(∅ ≈ 𝐴 ↔ 𝐴 = ∅)
 
29-Nov-2024brdomi 8898 Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-un 7672. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
 
29-Nov-2024brdomg 8896 Dominance relation. (Contributed by NM, 15-Jun-1998.) Extract brdom2g 8895 as an intermediate result. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
29-Nov-2024brdom2g 8895 Dominance relation. This variation of brdomg 8896 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 8896. (Revised by BTernaryTau, 29-Nov-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
 
29-Nov-2024peano1 7825 Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. Note: Unlike most textbooks, our proofs of peano1 7825 through peano5 7830 do not use the Axiom of Infinity. Unlike Takeuti and Zaring, they also do not use the Axiom of Regularity. (Contributed by NM, 15-May-1994.) Avoid ax-un 7672. (Revised by BTernaryTau, 29-Nov-2024.)
∅ ∈ ω
 
28-Nov-2024phpeqd 9159 Corollary of the Pigeonhole Principle using equality. Strengthening of php 9154 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) Avoid ax-pow. (Revised by BTernaryTau, 28-Nov-2024.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐴 ≈ 𝐵)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
27-Nov-2024frmin 9685 Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6301 and tz7.5 6338. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
(((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 
26-Nov-2024php3 9156 Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5320. (Revised by BTernaryTau, 26-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
 
25-Nov-2024domsdomtrfi 9149 Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 9056). (Contributed by BTernaryTau, 25-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)
 
25-Nov-2024sdomdomtrfi 9148 Transitivity of strict dominance and dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike sdomdomtr 9054). (Contributed by BTernaryTau, 25-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶)
 
25-Nov-2024predres 6293 Predecessor class is unaffected by restriction to the base class. (Contributed by Scott Fenton, 25-Nov-2024.)
Pred(𝑅, 𝐴, 𝑋) = Pred((𝑅 ↾ 𝐴), 𝐴, 𝑋)
 
25-Nov-2024predprc 6292 The predecessor of a proper class is empty. (Contributed by Scott Fenton, 25-Nov-2024.)
(¬ 𝑋 ∈ V → Pred(𝑅, 𝐴, 𝑋) = ∅)
 
25-Nov-2024predrelss 6291 Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024.)
(𝑅 ⊆ 𝑆 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑆, 𝐴, 𝑋))
 
24-Nov-2024ssdomfi2 9144 A set dominates its finite subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8940). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ≼ 𝐵)
 
24-Nov-2024domtrfir 9141 Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 8947). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
24-Nov-2024domtrfi 9140 Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 8947). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
24-Nov-2024domtrfil 9139 Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 8947). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
 
24-Nov-2024f1domfi2 9129 If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g 8909). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
 
24-Nov-2024rabid2 3436 An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 24-Nov-2024.)
(𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
24-Nov-2024clelsb2 2865 Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2123). (Contributed by Jim Kingdon, 22-Nov-2018.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 24-Nov-2024.)
([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)
 
23-Nov-2024natglobalincr 45106 Local monotonicity on half-open integer range implies global monotonicity. Inference form. (Contributed by Ender Ting, 23-Nov-2024.)
∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))    &   𝑇 ∈ ℤ    ⇒   ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ ((𝑘 + 1)...𝑇)(𝐵‘𝑘) < (𝐵‘𝑡)
 
23-Nov-2024prjcrv0 40957 The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.)
𝑌 = ((0...𝑁) mPoly 𝐾)    &    0 = (0g‘𝑌)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ Field)    ⇒   (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)
 
23-Nov-2024prjcrvval 40956 Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
𝐻 = ((0...𝑁) mHomP 𝐾)    &   𝐸 = ((0...𝑁) eval 𝐾)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &    0 = (0g‘𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝐹 ∈ ∪ ran 𝐻)    ⇒   (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})
 
23-Nov-2024prjcrvfval 40955 Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
𝐻 = ((0...𝑁) mHomP 𝐾)    &   𝐸 = ((0...𝑁) eval 𝐾)    &   𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &    0 = (0g‘𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ Field)    ⇒   (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
 
23-Nov-2024df-prjcrv 40954 Define the projective curve function. This takes a homogeneous polynomial and outputs the homogeneous coordinates where the polynomial evaluates to zero (the "zero set"). (In other words, scalar multiples are collapsed into the same projective point. See mhphf4 40760 and prjspvs 40934). (Contributed by SN, 23-Nov-2024.)
ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
 
23-Nov-2024mhphf4 40760 A homogeneous polynomial defines a homogeneous function; this is mhphf3 40759 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.)
𝑄 = (𝐼 eval 𝑆)    &   𝐻 = (𝐼 mHomP 𝑆)    &   𝐾 = (Base‘𝑆)    &   𝐹 = (𝑆 freeLMod 𝐼)    &   𝑀 = (Base‘𝐹)    &    ∙ = ( ·𝑠 ‘𝐹)    &    · = (.r‘𝑆)    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝐿 ∈ 𝐾)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ 𝑀)    ⇒   (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
23-Nov-2024mhphf3 40759 A homogeneous polynomial defines a homogeneous function; this is mhphf2 40758 with the finite support restriction (frlmpws 21156, frlmbas 21161) on the assignments 𝐴 from variables to values. See comment of mhphf2 40758. (Contributed by SN, 23-Nov-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐹 = (𝑆 freeLMod 𝐼)    &   𝑀 = (Base‘𝐹)    &    ∙ = ( ·𝑠 ‘𝐹)    &    · = (.r‘𝑆)    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐿 ∈ 𝑅)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ 𝑀)    ⇒   (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
23-Nov-2024evl0 40731 The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.)
𝑄 = (𝐼 eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑅)    &   𝑂 = (0g‘𝑅)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → (𝑄‘ 0 ) = ((𝐵 ↑m 𝐼) × {𝑂}))
 
23-Nov-2024mplascl0 40730 The zero scalar as a polynomial. (Contributed by SN, 23-Nov-2024.)
𝑊 = (𝐼 mPoly 𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝑂 = (0g‘𝑅)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → (𝐴‘𝑂) = 0 )
 
23-Nov-2024abbi1sn 40644 Originally part of uniabio 6463. Convert a theorem about df-iota 6448 to one about dfiota2 6449, without ax-10 2137, ax-11 2154, ax-12 2171. Although, eu6 2572 uses ax-10 2137 and ax-12 2171. (Contributed by SN, 23-Nov-2024.)
(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
 
23-Nov-2024recvs 24509 The field of the real numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) (Proof shortened by SN, 23-Nov-2024.)
𝑅 = (ringLMod‘ℝfld)    ⇒   𝑅 ∈ ℂVec
 
23-Nov-2024fldcrngd 20197 A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
(𝜑 → 𝑅 ∈ Field)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
23-Nov-2024iotaval 6467 Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 23-Nov-2024.)
(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
 
23-Nov-2024nfrabw 3440 A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3443 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 13-Oct-2003.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Nov-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    ⇒   Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
 
23-Nov-2024moel 3375 "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2154. (Revised by Wolf Lammen, 23-Nov-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦)
 
23-Nov-2024rmobidva 3368 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) Avoid ax-6 1971, ax-7 2011, ax-12 2171. (Revised by Wolf Lammen, 23-Nov-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒))
 
22-Nov-2024tworepnotupword 45115 Concatenation of identical singletons is never an increasing sequence. (Contributed by Ender Ting, 22-Nov-2024.)
𝐴 ∈ V    ⇒    ¬ (⟨“𝐴”⟩ ++ ⟨“𝐴”⟩) ∈ UpWord 𝑆
 
22-Nov-2024singoutnupword 45112 Singleton with character out of range 𝑆 is not an increasing sequence for that range. (Contributed by Ender Ting, 22-Nov-2024.)
𝐴 ∈ V    ⇒   (¬ 𝐴 ∈ 𝑆 → ¬ ⟨“𝐴”⟩ ∈ UpWord 𝑆)
 
22-Nov-2024natlocalincr 45105 Global monotonicity on half-open range implies local monotonicity. Inference form. (Contributed by Ender Ting, 22-Nov-2024.)
∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘) < (𝐵‘𝑡))    ⇒   ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘) < (𝐵‘(𝑘 + 1))
 
22-Nov-2024et-ltneverrefl 45102 Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11264. (New usage is discouraged.)
¬ 𝐴 < 𝐴
 
22-Nov-2024domnsymfi 9147 If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 9043). (Contributed by BTernaryTau, 22-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵) → ¬ 𝐵 ≺ 𝐴)
 
21-Nov-2024upwordsseti 45114 Strictly increasing sequences with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.)
𝑆 ∈ V    ⇒   UpWord 𝑆 ∈ V
 
21-Nov-2024upwordsing 45113 Singleton is an increasing sequence for any compatible range. (Contributed by Ender Ting, 21-Nov-2024.)
𝐴 ∈ 𝑆    ⇒   ⟨“𝐴”⟩ ∈ UpWord 𝑆
 
21-Nov-2024singoutnword 45111 Singleton with character out of range 𝑉 is not a word for that range. (Contributed by Ender Ting, 21-Nov-2024.)
𝐴 ∈ V    ⇒   (¬ 𝐴 ∈ 𝑉 → ¬ ⟨“𝐴”⟩ ∈ Word 𝑉)
 
21-Nov-2024nfreuw 3387 Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3406 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 30-Oct-2010.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2707. (Revised by Wolf Lammen, 21-Nov-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
 
21-Nov-2024nfrmow 3386 Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3405 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by NM, 16-Jun-2017.) Avoid ax-13 2370. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-9 2116, ax-ext 2707. (Revised by Wolf Lammen, 21-Nov-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
 
21-Nov-2024eeor 2329 Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    ⇒   (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 
21-Nov-2024aaan 2327 Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    ⇒   (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
 
20-Nov-2024php2 9155 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.) Avoid ax-pow 5320. (Revised by BTernaryTau, 20-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → 𝐵 ≺ 𝐴)
 
20-Nov-20242ralor 3219 Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Wolf Lammen, 20-Nov-2024.)
(∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∨ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∨ ∀𝑦 ∈ 𝐵 𝜓))
 
20-Nov-2024sbrim 2300 Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Avoid ax-10 2137. (Revised by Gino Giotto, 20-Nov-2024.)
Ⅎ𝑥𝜑    ⇒   ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
19-Nov-2024upwordisword 45110 Any increasing sequence is a sequence. (Contributed by Ender Ting, 19-Nov-2024.)
(𝐴 ∈ UpWord 𝑆 → 𝐴 ∈ Word 𝑆)
 
19-Nov-2024upwordnul 45109 Empty set is an increasing sequence for every range. (Contributed by Ender Ting, 19-Nov-2024.)
∅ ∈ UpWord 𝑆
 
19-Nov-2024df-upword 45108 Strictly increasing sequence is a sequence, adjacent elements of which increase. (Contributed by Ender Ting, 19-Nov-2024.)
UpWord 𝑆 = {𝑤 ∣ (𝑤 ∈ Word 𝑆 ∧ ∀𝑘 ∈ (0..^((♯‘𝑤) − 1))(𝑤‘𝑘) < (𝑤‘(𝑘 + 1)))}
 
19-Nov-2024moeu2 36823 Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.)
(∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑))
 
19-Nov-2024fri 5593 A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (inference form). (Contributed by BJ, 16-Nov-2024.) (Proof shortened by BJ, 19-Nov-2024.)
(((𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
 
18-Nov-2024mopickr 36824 "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2636) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.)
((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑))
 
18-Nov-2024php 9154 Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of phplem1 9151, phplem2 9152, nneneq 9153, and this final piece of the proof. (Contributed by NM, 29-May-1998.) Avoid ax-pow 5320. (Revised by BTernaryTau, 18-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵)
 
18-Nov-2024wfr3 8283 The principle of Well-Ordered Recursion, part 3 of 3. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in wfr1 8281 and wfr2 8282 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
 
18-Nov-2024wfr1 8281 The Principle of Well-Ordered Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a set-like well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-ordered recursion". The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
 
18-Nov-2024wfr2a 8280 A weak version of wfr2 8282 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.) (Proof shortened by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
18-Nov-2024wfrresex 8279 Show without using the axiom of replacement that the restriction of the well-ordered recursion generator to a predecessor class is a set. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
 
18-Nov-2024csbwrecsg 8252 Move class substitution in and out of the well-founded recursive function generator. (Contributed by ML, 25-Oct-2020.) (Revised by Scott Fenton, 18-Nov-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs(𝑅, 𝐷, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))
 
18-Nov-2024fprresex 8241 The restriction of a function defined by well-founded recursion to the predecessor of an element of its domain is a set. Avoids the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)) ∈ V)
 
18-Nov-2024fprfung 8240 A "function" defined by well-founded recursion is indeed a function when the relation is a partial order. Avoids the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
 
18-Nov-2024frrdmss 8238 Show without using the axiom of replacement that the domain of the well-founded recursion generator is a subclass of 𝐴. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   dom 𝐹 ⊆ 𝐴
 
18-Nov-2024frrrel 8237 Show without using the axiom of replacement that the well-founded recursion generator gives a relation. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   Rel 𝐹
 
18-Nov-2024fpr2 8235 Law of well-founded recursion over a partial order, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.) (Proof shortened by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
18-Nov-2024fpr2a 8233 Weak version of fpr2 8235 which is useful for proofs that avoid the axiom of replacement. (Contributed by Scott Fenton, 18-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
18-Nov-2024csbfrecsg 8215 Move class substitution in and out of the well-founded recursive function generator. (Contributed by Scott Fenton, 18-Nov-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌frecs(𝑅, 𝐷, 𝐹) = frecs(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝐹))
 
18-Nov-2024drnf1v 2368 Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2441 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))
 
18-Nov-2024dral1v 2365 Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2437 with a disjoint variable condition, which does not require ax-13 2370. Remark: the corresponding versions for dral2 2436 and drex2 2440 are instances of albidv 1923 and exbidv 1924 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2172. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
18-Nov-2024equsexv 2259 An equivalence related to implicit substitution. Version of equsex 2416 with a disjoint variable condition, which does not require ax-13 2370. See equsexvw 2008 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2258. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.)
Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
 
17-Nov-2024bj-rdg0gALT 35542 Alternate proof of rdg0g 8373. More direct since it bypasses tz7.44-1 8352 and rdg0 8367 (and vtoclg 3525, vtoclga 3534). (Contributed by NM, 25-Apr-1995.) More direct proof. (Revised by BJ, 17-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
 
17-Nov-2024wfrfun 8278 The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → Fun 𝐹)
 
17-Nov-2024wfrdmcl 8277 The predecessor class of an element of the well-ordered recursion generator's domain is a subset of its domain. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
 
17-Nov-2024wfrdmss 8276 The domain of the well-ordered recursion generator is a subclass of 𝐴. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   dom 𝐹 ⊆ 𝐴
 
17-Nov-2024wfrrel 8275 The well-ordered recursion generator generates a relation. Avoids the axiom of replacement. (Contributed by Scott Fenton, 8-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = wrecs(𝑅, 𝐴, 𝐺)    ⇒   Rel 𝐹
 
17-Nov-2024nfwrecs 8247 Bound-variable hypothesis builder for the well-ordered recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Ⅎ𝑥𝑅    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐹    ⇒   Ⅎ𝑥wrecs(𝑅, 𝐴, 𝐹)
 
17-Nov-2024wrecseq123 8245 General equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))
 
17-Nov-2024frrdmcl 8239 Show without using the axiom of replacement that for a "function" defined by well-founded recursion, the predecessor class of an element of its domain is a subclass of its domain. (Contributed by Scott Fenton, 21-Apr-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹)
 
17-Nov-2024wfis2fg 6310 Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Ⅎ𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
 
17-Nov-2024wfisg 6307 Well-Ordered Induction Schema. If a property passes from all elements less than 𝑦 of a well-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
(𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
 
17-Nov-2024wfi 6304 The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
(((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)
 
17-Nov-2024tz6.26 6301 All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
(((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 
17-Nov-2024cbvmptv 5218 Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid auxiliary axioms . See cbvmptvg 5220 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Nov-2024.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
 
17-Nov-2024cbvopab1v 5184 Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.) Reduce axiom usage. (Revised by Gino Giotto, 17-Nov-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
16-Nov-2024frd 5592 A nonempty subset of an 𝑅-well-founded class has an 𝑅-minimal element (deduction form). (Contributed by BJ, 16-Nov-2024.)
(𝜑 → 𝑅 Fr 𝐴)    &   (𝜑 → 𝐵 ⊆ 𝐴)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐵 ≠ ∅)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥)
 
16-Nov-2024dffr6 5591 Alternate definition of df-fr 5588. See dffr5 34327 for a definition without dummy variables (but note that their equivalence uses ax-sep 5256). (Contributed by BJ, 16-Nov-2024.)
(𝑅 Fr 𝐴 ↔ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
 
15-Nov-20241strbas 17100 The base set of a constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Proof shortened by AV, 15-Nov-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
 
15-Nov-20241strstr1 17099 A constructed one-slot structure. (Contributed by AV, 15-Nov-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩}    ⇒   𝐺 Struct ⟨(Base‘ndx), (Base‘ndx)⟩
 
14-Nov-2024aks4d1 40546 Lemma 4.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf, existence of a polynomially bounded number by the digit size of 𝑁 that asserts the polynomial subspace that we need to search to guarantee that 𝑁 is prime. Eventually we want to show that the polynomial searching space is bounded by degree 𝐵. (Contributed by metakunt, 14-Nov-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → ∃𝑟 ∈ (1...𝐵)((𝑁 gcd 𝑟) = 1 ∧ ((2 logb 𝑁)↑2) < ((odℤ‘𝑟)‘𝑁)))
 
14-Nov-2024aks4d1p9 40545 Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))
 
14-Nov-2024aks4d1lem1 40519 Technical lemma to reduce proof size. (Contributed by metakunt, 14-Nov-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
 
13-Nov-2024aks4d1p8d3 40543 The remainder of a division with its maximal prime power is coprime with that prime power. (Contributed by metakunt, 13-Nov-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    ⇒   (𝜑 → ((𝑁 / (𝑃↑(𝑃 pCnt 𝑁))) gcd (𝑃↑(𝑃 pCnt 𝑁))) = 1)
 
13-Nov-2024aks4d1p8d2 40542 Any prime power dividing a positive integer is less than that integer if that integer has another prime factor. (Contributed by metakunt, 13-Nov-2024.)
(𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑄 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑅)    &   (𝜑 → 𝑄 ∥ 𝑅)    &   (𝜑 → ¬ 𝑃 ∥ 𝑁)    &   (𝜑 → 𝑄 ∥ 𝑁)    ⇒   (𝜑 → (𝑃↑(𝑃 pCnt 𝑅)) < 𝑅)
 
12-Nov-2024prstcocval 47081 Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ⊥ = (oc‘𝐶))
 
12-Nov-2024prstcleval 47078 Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → ≤ = (le‘𝐶))
 
12-Nov-2024zlmtset 32545 Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 12-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐽 = (TopSet‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
 
12-Nov-2024setsmsbas 23828 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝑋 = (Base‘𝑀))    &   (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   (𝜑 → 𝑋 = (Base‘𝐾))
 
12-Nov-2024matvsca 21764 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ( ·𝑠 ‘𝐺) = ( ·𝑠 ‘𝐴))
 
12-Nov-2024matsca 21762 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    ⇒   ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
 
12-Nov-2024sravsca 20648 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
 
12-Nov-2024srasca 20646 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
 
12-Nov-2024odubas 18180 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐷 = (ODual‘𝑂)    &   𝐵 = (Base‘𝑂)    ⇒   𝐵 = (Base‘𝐷)
 
12-Nov-2024slotsdifocndx 17299 The index of the slot for the orthocomplementation is not the index of other slots. Formerly part of proof for prstcocval 47081. (Contributed by AV, 12-Nov-2024.)
((oc‘ndx) ≠ (comp‘ndx) ∧ (oc‘ndx) ≠ (Hom ‘ndx))
 
12-Nov-2024slotsdifplendx2 17298 The index of the slot for the "less than or equal to" ordering is not the index of other slots. Formerly part of proof for prstcleval 47078. (Contributed by AV, 12-Nov-2024.)
((le‘ndx) ≠ (comp‘ndx) ∧ (le‘ndx) ≠ (Hom ‘ndx))
 
12-Nov-2024slotsdifipndx 17216 The slot for the scalar is not the index of other slots. Formerly part of proof for srasca 20646 and sravsca 20648. (Contributed by AV, 12-Nov-2024.)
(( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
 
12-Nov-2024ssdomfi 9143 A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8940). (Contributed by BTernaryTau, 12-Nov-2024.)
(𝐵 ∈ Fin → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵))
 
11-Nov-2024mpteq1df 43451 An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
 
11-Nov-2024mhphf2 40758 A homogeneous polynomial defines a homogeneous function; this is mhphf 40757 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 21172 but without the finite support restriction (frlmpws 21156, frlmbas 21161) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21313) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.)

𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &    ∙ = ( ·𝑠 ‘((ringLMod‘𝑆) ↑s 𝐼))    &    · = (.r‘𝑆)    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐿 ∈ 𝑅)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
11-Nov-2024zlmds 32543 Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 11-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐷 = (dist‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
 
11-Nov-2024setsmsds 23830 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 11-Nov-2024.)
(𝜑 → 𝑋 = (Base‘𝑀))    &   (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   (𝜑 → (dist‘𝑀) = (dist‘𝐾))
 
11-Nov-2024thlle 21102 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    ≤ = (le‘𝐼)    ⇒    ≤ = (le‘𝐾)
 
11-Nov-2024thlbas 21100 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    ⇒   𝐶 = (Base‘𝐾)
 
11-Nov-2024cnfldfunALT 20809 The field of complex numbers is a function. Alternate proof of cnfldfun 20808 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Fun ℂfld
 
11-Nov-2024fldidom 20775 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)
 
11-Nov-2024slotsdifdsndx 17275 The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 20809. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (dist‘ndx) ∧ (le‘ndx) ≠ (dist‘ndx))
 
11-Nov-2024plendxnocndx 17265 The slot for the orthocomplementation is not the slot for the order in an extensible structure. Formerly part of proof for thlle 21102. (Contributed by AV, 11-Nov-2024.)
(le‘ndx) ≠ (oc‘ndx)
 
11-Nov-2024basendxnocndx 17264 The slot for the orthocomplementation is not the slot for the base set in an extensible structure. Formerly part of proof for thlbas 21100. (Contributed by AV, 11-Nov-2024.)
(Base‘ndx) ≠ (oc‘ndx)
 
11-Nov-2024slotsdifplendx 17256 The index of the slot for the distance is not the index of other slots. Formerly part of proof for cnfldfunALT 20809. (Contributed by AV, 11-Nov-2024.)
((*𝑟‘ndx) ≠ (le‘ndx) ∧ (TopSet‘ndx) ≠ (le‘ndx))
 
11-Nov-2024tsetndxnstarvndx 17240 The slot for the topology is not the slot for the involution in an extensible structure. Formerly part of proof for cnfldfunALT 20809. (Contributed by AV, 11-Nov-2024.)
(TopSet‘ndx) ≠ (*𝑟‘ndx)
 
11-Nov-2024nneneq 9153 Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5320. (Revised by BTernaryTau, 11-Nov-2024.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))
 
11-Nov-2024ofeqd 7619 Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
(𝜑 → 𝑅 = 𝑆)    ⇒   (𝜑 → ∘f 𝑅 = ∘f 𝑆)
 
11-Nov-2024iunopab 5516 Move indexed union inside an ordered-pair class abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.) Avoid ax-sep 5256, ax-nul 5263, ax-pr 5384. (Revised by SN, 11-Nov-2024.)
∪ 𝑧 ∈ 𝐴 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 𝜑}
 
11-Nov-2024mpteq2ia 5208 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
(𝑥 ∈ 𝐴 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)
 
11-Nov-2024mpteq2dva 5205 Slightly more general equality inference for the maps-to notation. (Contributed by Scott Fenton, 25-Apr-2012.) Remove dependency on ax-10 2137. (Revised by SN, 11-Nov-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
 
11-Nov-2024mpteq2da 5203 Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
 
11-Nov-2024mpteq1i 5201 An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) Remove all disjoint variable conditions. (Revised by SN, 11-Nov-2024.)
𝐴 = 𝐵    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
 
11-Nov-2024mpteq1 5198 An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Proof shortened by SN, 11-Nov-2024.)
(𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
 
11-Nov-2024mpteq12dva 5194 An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove dependency on ax-10 2137, ax-12 2171. (Revised by SN, 11-Nov-2024.)
(𝜑 → 𝐴 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
11-Nov-2024mpteq12df 5191 An equality inference for the maps-to notation. Compare mpteq12dv 5196. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof shortened by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐶)    &   (𝜑 → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
11-Nov-2024mpteq12da 5190 An equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Remove dependency on ax-10 2137. (Revised by SN, 11-Nov-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
10-Nov-2024aks4d1p8 40544 Show that 𝑁 and 𝑅 are coprime for AKS existence theorem, with eliminated hypothesis. (Contributed by metakunt, 10-Nov-2024.) (Proof sketch by Thierry Arnoux.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → (𝑁 gcd 𝑅) = 1)
 
10-Nov-2024aks4d1p8d1 40541 If a prime divides one number 𝑀, but not another number 𝑁, then it divides the quotient of 𝑀 and the gcd of 𝑀 and 𝑁. (Contributed by Thierry Arnoux, 10-Nov-2024.)
(𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑀)    &   (𝜑 → ¬ 𝑃 ∥ 𝑁)    ⇒   (𝜑 → 𝑃 ∥ (𝑀 / (𝑀 gcd 𝑁)))
 
10-Nov-2024slotsdifunifndx 17282 The index of the slot for the uniform set is not the index of other slots. Formerly part of proof for cnfldfunALT 20809. (Contributed by AV, 10-Nov-2024.)
(((+g‘ndx) ≠ (UnifSet‘ndx) ∧ (.r‘ndx) ≠ (UnifSet‘ndx) ∧ (*𝑟‘ndx) ≠ (UnifSet‘ndx)) ∧ ((le‘ndx) ≠ (UnifSet‘ndx) ∧ (dist‘ndx) ≠ (UnifSet‘ndx)))
 
9-Nov-2024bj-flddrng 35760 Fields are division rings (elemental version). (Contributed by BJ, 9-Nov-2024.)
(𝐹 ∈ Field → 𝐹 ∈ DivRing)
 
9-Nov-2024bj-dfid2ALT 35536 Alternate version of dfid2 5533. (Contributed by BJ, 9-Nov-2024.) (Proof modification is discouraged.) Use df-id 5531 instead to make the semantics of the construction df-opab 5168 clearer. (New usage is discouraged.)
I = {⟨𝑥, 𝑥⟩ ∣ ⊤}
 
9-Nov-2024ttgval 27817 Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    &    − = (-g‘𝐻)    &    · = ( ·𝑠 ‘𝐻)    &   𝐼 = (Itv‘𝐺)    ⇒   (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})))
 
9-Nov-2024lngndxnitvndx 27385 The slot for the line is not the slot for the Interval (segment) in an extensible structure. Formerly part of proof for ttgval 27817. (Contributed by AV, 9-Nov-2024.)
(LineG‘ndx) ≠ (Itv‘ndx)
 
9-Nov-2024rescabs 17718 Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 9-Nov-2024.)
(𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝐽 Fn (𝑇 × 𝑇))    &   (𝜑 → 𝑆 ∈ 𝑊)    &   (𝜑 → 𝑇 ⊆ 𝑆)    ⇒   (𝜑 → ((𝐶 ↾cat 𝐻) ↾cat 𝐽) = (𝐶 ↾cat 𝐽))
 
7-Nov-2024ressbas 17118 Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐵 = (Base‘𝑊)    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) = (Base‘𝑅))
 
7-Nov-2024setsnid 17081 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 7-Nov-2024.)
𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ 𝐷    ⇒   (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨𝐷, 𝐶⟩))
 
6-Nov-2024sn-iotalemcor 40643 Corollary of sn-iotalem 40642. Compare sb8iota 6460. (Contributed by SN, 6-Nov-2024.)
(℩𝑥𝜑) = (℩𝑦{𝑥 ∣ 𝜑} = {𝑦})
 
6-Nov-2024sn-iotalem 40642 An unused lemma showing that many equivalences involving df-iota 6448 are potentially provable without ax-10 2137, ax-11 2154, ax-12 2171. (Contributed by SN, 6-Nov-2024.)
{𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧}}
 
6-Nov-2024eqimssd 40635 Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → 𝐴 ⊆ 𝐵)
 
6-Nov-2024hlhilsmul 40407 Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &    · = (.r‘𝐸)    ⇒   (𝜑 → · = (.r‘𝑅))
 
6-Nov-2024hlhilsplus 40405 Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &    + = (+g‘𝐸)    ⇒   (𝜑 → + = (+g‘𝑅))
 
6-Nov-2024hlhilsbase 40403 The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   𝐶 = (Base‘𝐸)    ⇒   (𝜑 → 𝐶 = (Base‘𝑅))
 
6-Nov-2024hlhilslem 40401 Lemma for hlhilsbase 40403 etc. (Contributed by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((HLHil‘𝐾)‘𝑊)    &   𝑅 = (Scalar‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   𝐹 = Slot (𝐹‘ndx)    &   (𝐹‘ndx) ≠ (*𝑟‘ndx)    &   𝐶 = (𝐹‘𝐸)    ⇒   (𝜑 → 𝐶 = (𝐹‘𝑅))
 
6-Nov-2024oppradd 20058 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &    + = (+g‘𝑅)    ⇒    + = (+g‘𝑂)
 
6-Nov-2024opprbas 20056 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   𝐵 = (Base‘𝑂)
 
6-Nov-2024opprlem 20054 Lemma for opprbas 20056 and oppradd 20058. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (oppr‘𝑅)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (.r‘ndx)    ⇒   (𝐸‘𝑅) = (𝐸‘𝑂)
 
6-Nov-2024symgvalstruct 19178 The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (Proof shortened by AV, 6-Nov-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}    &   𝑀 = (𝐴 ↑m 𝐴)    &    + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))    ⇒   (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 
6-Nov-2024frmdplusg 18664 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) (Proof shortened by AV, 6-Nov-2024.)
𝑀 = (freeMnd‘𝐼)    &   𝐵 = (Base‘𝑀)    &    + = (+g‘𝑀)    ⇒    + = ( ++ ↾ (𝐵 × 𝐵))
 
6-Nov-2024iotaex 6469 Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 6-Nov-2024.)
(℩𝑥𝜑) ∈ V
 
6-Nov-2024iotassuni 6468 The ℩ class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.) Remove dependency on ax-10 2137, ax-11 2154, ax-12 2171. (Revised by SN, 6-Nov-2024.)
(℩𝑥𝜑) ⊆ ∪ {𝑥 ∣ 𝜑}
 
6-Nov-2024iotanul2 6466 Version of iotanul 6474 using df-iota 6448 instead of dfiota2 6449. (Contributed by SN, 6-Nov-2024.)
(¬ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∅)
 
6-Nov-2024iotauni2 6465 Version of iotauni 6471 using df-iota 6448 instead of dfiota2 6449. (Contributed by SN, 6-Nov-2024.)
(∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
 
6-Nov-2024iotaval2 6464 Version of iotaval 6467 using df-iota 6448 instead of dfiota2 6449. (Contributed by SN, 6-Nov-2024.)
({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
 
5-Nov-2024dfid2 5533 Alternate definition of the identity relation. Instance of dfid3 5534 not requiring auxiliary axioms. (Contributed by NM, 15-Mar-2007.) Reduce axiom usage. (Revised by Gino Giotto, 4-Nov-2024.) (Proof shortened by BJ, 5-Nov-2024.)

Use df-id 5531 instead to make the semantics of the constructor df-opab 5168 clearer. (New usage is discouraged.)

I = {⟨𝑥, 𝑥⟩ ∣ 𝑥 = 𝑥}
 
5-Nov-2024r19.30 3123 Restricted quantifier version of 19.30 1884. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 5-Nov-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
 
4-Nov-2024phplem2 9152 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 4-Nov-2024.)
𝐴 ∈ V    ⇒   ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵 → 𝐴 ≈ 𝐵))
 
4-Nov-2024sbthfi 9146 Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 9037). (Contributed by BTernaryTau, 4-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
 
4-Nov-2024sbthfilem 9145 Lemma for sbthfi 9146. (Contributed by BTernaryTau, 4-Nov-2024.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}    &   𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))    &   𝐵 ∈ V    ⇒   ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ≈ 𝐵)
 
4-Nov-2024r19.12 3297 Restricted quantifier version of 19.12 2320. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) Avoid ax-13 2370, ax-ext 2707. (Revised by Wolf Lammen, 17-Jun-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
 
4-Nov-2024r19.29vva 3207 A commonly used pattern based on r19.29 3117, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)    &   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   (𝜑 → 𝜒)
 
4-Nov-2024reximdvai 3162 Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
 
4-Nov-2024r19.29d2r 3137 Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)    &   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
 
4-Nov-2024ralrexbid 3109 Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.) (Proof shortened by Wolf Lammen, 4-Nov-2024.)
(𝜑 → (𝜓 ↔ 𝜃))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))
 
4-Nov-2024exexw 2054 Existential quantification over a given variable is idempotent. Weak version of bj-exexbiex 35165, requiring fewer axioms. (Contributed by Gino Giotto, 4-Nov-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥𝜑 ↔ ∃𝑥∃𝑥𝜑)
 
3-Nov-2024znmul 20947 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌))
 
3-Nov-2024znadd 20945 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌))
 
3-Nov-2024znbas2 20943 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    ⇒   (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌))
 
3-Nov-2024znbaslem 20941 Lemma for znbas 20950. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpan‘ℤring)    &   𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   𝑌 = (ℤ/nℤ‘𝑁)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (le‘ndx)    ⇒   (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌))
 
3-Nov-2024zlmmulr 20923 Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r‘𝐺)    ⇒    · = (.r‘𝑊)
 
3-Nov-2024zlmplusg 20921 Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g‘𝐺)    ⇒    + = (+g‘𝑊)
 
3-Nov-2024zlmbas 20919 Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)    ⇒   𝐵 = (Base‘𝑊)
 
3-Nov-2024zlmlem 20917 Lemma for zlmbas 20919 and zlmplusg 20921. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Scalar‘ndx)    &   (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)    ⇒   (𝐸‘𝐺) = (𝐸‘𝑊)
 
3-Nov-2024nelb 3222 A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
(¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
 
3-Nov-2024rexbi 3107 Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-2024.) (Proof shortened by Wolf Lammen, 3-Nov-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))
 
2-Nov-2024psrvscafval 21358 The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &    ∙ = ( ·𝑠 ‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r‘𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    ⇒    ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
 
2-Nov-2024zlmsca 20925 Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
𝑊 = (ℤMod‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊))
 
2-Nov-2024rexab 3652 Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
(𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒))
 
2-Nov-2024ralab 3649 Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) Reduce axiom usage. (Revised by Gino Giotto, 2-Nov-2024.)
(𝑦 = 𝑥 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∀𝑥(𝜓 → 𝜒))
 
1-Nov-2024mnringvscad 42494 The scalar product of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐵 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐵)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → ( ·𝑠 ‘𝑉) = ( ·𝑠 ‘𝐹))
 
1-Nov-2024mnringscad 42492 The scalar ring of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → 𝑅 = (Scalar‘𝐹))
 
1-Nov-2024mnringaddgd 42487 The additive operation of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → (+g‘𝑉) = (+g‘𝐹))
 
1-Nov-2024mnringbased 42481 The base set of a monoid ring. (Contributed by Rohan Ridenour, 14-May-2024.) (Proof shortened by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   𝐵 = (Base‘𝑉)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → 𝐵 = (Base‘𝐹))
 
1-Nov-2024mnringnmulrd 42479 Components of a monoid ring other than its ring product match its underlying free module. (Contributed by Rohan Ridenour, 14-May-2024.) (Revised by AV, 1-Nov-2024.)
𝐹 = (𝑅 MndRing 𝑀)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (.r‘ndx)    &   𝐴 = (Base‘𝑀)    &   𝑉 = (𝑅 freeLMod 𝐴)    &   (𝜑 → 𝑅 ∈ 𝑈)    &   (𝜑 → 𝑀 ∈ 𝑊)    ⇒   (𝜑 → (𝐸‘𝑉) = (𝐸‘𝐹))
 
1-Nov-2024opsrsca 21460 The scalar ring of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → 𝑅 = (Scalar‘𝑂))
 
1-Nov-2024opsrvsca 21458 The scalar product operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑂))
 
1-Nov-2024opsrmulr 21456 The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → (.r‘𝑆) = (.r‘𝑂))
 
1-Nov-2024opsrplusg 21454 The addition operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → (+g‘𝑆) = (+g‘𝑂))
 
1-Nov-2024opsrbas 21452 The base set of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 30-Aug-2015.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    ⇒   (𝜑 → (Base‘𝑆) = (Base‘𝑂))
 
1-Nov-2024opsrbaslem 21450 Get a component of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 1-Nov-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)    &   (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (le‘ndx)    ⇒   (𝜑 → (𝐸‘𝑆) = (𝐸‘𝑂))
 
1-Nov-2024plendxnvscandx 17255 The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. Formerly part of proof for opsrvsca 21458. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ ( ·𝑠 ‘ndx)
 
1-Nov-2024plendxnscandx 17254 The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. Formerly part of proof for opsrsca 21460. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (Scalar‘ndx)
 
1-Nov-2024plendxnmulrndx 17253 The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr 21456. (Contributed by AV, 1-Nov-2024.)
(le‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024fnimafnex 41702 The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.)
𝐹 Fn 𝐵    ⇒   (𝐹 “ (𝐺‘𝐴)) ∈ V
 
31-Oct-2024mendvscafval 41503 Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠 ‘𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)    ⇒   ( ·𝑠 ‘𝐴) = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
 
31-Oct-2024mendsca 41502 The module endomorphism algebra has the same scalars as the underlying module. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &   𝑆 = (Scalar‘𝑀)    ⇒   𝑆 = (Scalar‘𝐴)
 
31-Oct-2024mendmulrfval 41500 Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    ⇒   (.r‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
 
31-Oct-2024mendplusgfval 41498 Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 31-Oct-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g‘𝑀)    ⇒   (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))
 
31-Oct-2024aks4d1p7 40540 Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → ∃𝑝 ∈ ℙ (𝑝 ∥ 𝑅 ∧ ¬ 𝑝 ∥ 𝑁))
 
31-Oct-2024aks4d1p7d1 40539 Technical step in AKS lemma 4.1 (Contributed by metakunt, 31-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   (𝜑 → ∀𝑝 ∈ ℙ (𝑝 ∥ 𝑅 → 𝑝 ∥ 𝑁))    ⇒   (𝜑 → 𝑅 ∥ (𝑁↑(⌊‘(2 logb 𝐵))))
 
31-Oct-2024resvmulr 32130 .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &    · = (.r‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → · = (.r‘𝐻))
 
31-Oct-2024resvvsca 32128 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &    · = ( ·𝑠 ‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻))
 
31-Oct-2024resvplusg 32126 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &    + = (+g‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → + = (+g‘𝐻))
 
31-Oct-2024resvbas 32124 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 ↾v 𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → 𝐵 = (Base‘𝐻))
 
31-Oct-2024resvlem 32122 Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝑅 = (𝑊 ↾v 𝐴)    &   𝐶 = (𝐸‘𝑊)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Scalar‘ndx)    ⇒   (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
 
31-Oct-2024nrgtrg 24054 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) (Proof shortened by AV, 31-Oct-2024.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
 
31-Oct-2024tngip 24009 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    , = (·𝑖‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇))
 
31-Oct-2024tngvsca 24007 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    · = ( ·𝑠 ‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → · = ( ·𝑠 ‘𝑇))
 
31-Oct-2024tngsca 24005 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐹 = (Scalar‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 𝐹 = (Scalar‘𝑇))
 
31-Oct-2024tngmulr 24003 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    · = (.r‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → · = (.r‘𝑇))
 
31-Oct-2024tng0 24002 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    0 = (0g‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 0 = (0g‘𝑇))
 
31-Oct-2024tngplusg 24000 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    + = (+g‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → + = (+g‘𝑇))
 
31-Oct-2024tngbas 23998 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐵 = (Base‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 𝐵 = (Base‘𝑇))
 
31-Oct-2024tnglem 23996 Lemma for tngbas 23998 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (TopSet‘ndx)    &   (𝐸‘ndx) ≠ (dist‘ndx)    ⇒   (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇))
 
31-Oct-2024indistpsALT 22363 The indiscrete topology on a set 𝐴 expressed as a topological space. Here we show how to derive the structural version indistps 22361 from the direct component assignment version indistps2 22362. (Contributed by NM, 24-Oct-2012.) (Revised by AV, 31-Oct-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), {∅, 𝐴}⟩}    ⇒   𝐾 ∈ TopSp
 
31-Oct-2024eltpsg 22292 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by AV, 31-Oct-2024.)
𝐾 = {⟨(Base‘ndx), 𝐴⟩, ⟨(TopSet‘ndx), 𝐽⟩}    ⇒   (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ TopSp)
 
31-Oct-2024dsndxnmulrndx 17272 The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(dist‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnmulrndx 17239 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (.r‘ndx)
 
31-Oct-2024tsetndxnbasendx 17237 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
(TopSet‘ndx) ≠ (Base‘ndx)
 
31-Oct-2024basendxlttsetndx 17236 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
(Base‘ndx) < (TopSet‘ndx)
 
31-Oct-2024tsetndxnn 17235 The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
(TopSet‘ndx) ∈ ℕ
 
31-Oct-2024oveqprc 17064 Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like resvlem 32122. (Contributed by AV, 31-Oct-2024.)
(𝐸‘∅) = ∅    &   𝑍 = (𝑋𝑂𝑌)    &   Rel dom 𝑂    ⇒   (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑍))
 
31-Oct-2024fveqprc 17063 Lemma for showing the equality of values for functions like slot extractors 𝐸 at a proper class. Extracted from several former proofs of lemmas like zlmlem 20917. (Contributed by AV, 31-Oct-2024.)
(𝐸‘∅) = ∅    &   𝑌 = (𝐹‘𝑋)    ⇒   (¬ 𝑋 ∈ V → (𝐸‘𝑋) = (𝐸‘𝑌))
 
31-Oct-2024ttrclse 9663 If 𝑅 is set-like over 𝐴, then the transitive closure of the restriction of 𝑅 to 𝐴 is set-like over 𝐴.

This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.)

(𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)
 
31-Oct-2024ttrclselem2 9662 Lemma for ttrclse 9663. Show that a suc 𝑁 element long chain gives membership in the 𝑁-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.)
𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ((𝑁 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑁 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑁) = 𝑋) ∧ ∀𝑎 ∈ suc 𝑁(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (𝐹‘𝑁)))
 
31-Oct-2024ttrclselem1 9661 Lemma for ttrclse 9663. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)
𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)
 
31-Oct-2024rdg0n 8380 If 𝐴 is a proper class, then the recursive function generator at ∅ is the empty set. (Contributed by Scott Fenton, 31-Oct-2024.)
(¬ 𝐴 ∈ V → (rec(𝐹, 𝐴)‘∅) = ∅)
 
31-Oct-2024ralcom4 3269 Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom dependencies. (Revised by BJ, 13-Jun-2019.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
(∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
 
31-Oct-2024ralbida 3253 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
Ⅎ𝑥𝜑    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
 
31-Oct-2024reximia 3084 Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Wolf Lammen, 31-Oct-2024.)
(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
 
30-Oct-2024aks4d1p6 40538 The maximal prime power exponent is smaller than the binary logarithm floor of 𝐵. (Contributed by metakunt, 30-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑅)    &   𝐾 = (𝑃 pCnt 𝑅)    ⇒   (𝜑 → 𝐾 ≤ (⌊‘(2 logb 𝐵)))
 
30-Oct-2024aks4d1p5 40537 Show that 𝑁 and 𝑅 are coprime for AKS existence theorem. Precondition will be eliminated in further theorem. (Contributed by metakunt, 30-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    &   (((𝜑 ∧ 1 < (𝑁 gcd 𝑅)) ∧ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴) → ¬ (𝑅 / (𝑁 gcd 𝑅)) ∥ 𝐴)    ⇒   (𝜑 → (𝑁 gcd 𝑅) = 1)
 
30-Oct-2024basendxltedgfndx 27944 The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)
(Base‘ndx) < (.ef‘ndx)
 
30-Oct-2024isposix 18214 Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.) (Proof shortened by AV, 30-Oct-2024.)
𝐵 ∈ V    &    ≤ ∈ V    &   𝐾 = {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), ≤ ⟩}    &   (𝑥 ∈ 𝐵 → 𝑥 ≤ 𝑥)    &   ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))    &   ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))    ⇒   𝐾 ∈ Poset
 
30-Oct-2024plendxnbasendx 17251 The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
(le‘ndx) ≠ (Base‘ndx)
 
30-Oct-2024basendxltplendx 17250 The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
(Base‘ndx) < (le‘ndx)
 
30-Oct-2024plendxnn 17249 The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
(le‘ndx) ∈ ℕ
 
30-Oct-2024pm13.181 3026 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
 
29-Oct-2024cchhllem 27835 Lemma for chlbas and chlvsca . (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.)
𝐶 = (((subringAlg ‘ℂfld)‘ℝ) sSet ⟨(·𝑖‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · (∗‘𝑦)))⟩)    &   𝐸 = Slot (𝐸‘ndx)    &   (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   (𝐸‘ℂfld) = (𝐸‘𝐶)
 
29-Oct-2024ttgds 27828 The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &   𝐷 = (dist‘𝐻)    ⇒   𝐷 = (dist‘𝐺)
 
29-Oct-2024ttgvsca 27826 The scalar product of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &    · = ( ·𝑠 ‘𝐻)    ⇒    · = ( ·𝑠 ‘𝐺)
 
29-Oct-2024ttgplusg 27823 The addition operation of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &    + = (+g‘𝐻)    ⇒    + = (+g‘𝐺)
 
29-Oct-2024ttgbas 27821 The base set of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &   𝐵 = (Base‘𝐻)    ⇒   𝐵 = (Base‘𝐺)
 
29-Oct-2024ttglem 27819 Lemma for ttgbas 27821, ttgvsca 27826 etc. (Contributed by Thierry Arnoux, 15-Apr-2019.) (Revised by AV, 29-Oct-2024.)
𝐺 = (toTG‘𝐻)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (LineG‘ndx)    &   (𝐸‘ndx) ≠ (Itv‘ndx)    ⇒   (𝐸‘𝐻) = (𝐸‘𝐺)
 
29-Oct-2024slotslnbpsd 27384 The slots Base, +g, ·𝑠 and dist are different from the slot LineG. Formerly part of ttglem 27819 and proofs using it. (Contributed by AV, 29-Oct-2024.)
(((LineG‘ndx) ≠ (Base‘ndx) ∧ (LineG‘ndx) ≠ (+g‘ndx)) ∧ ((LineG‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (LineG‘ndx) ≠ (dist‘ndx)))
 
29-Oct-2024slotsinbpsd 27383 The slots Base, +g, ·𝑠 and dist are different from the slot Itv. Formerly part of ttglem 27819 and proofs using it. (Contributed by AV, 29-Oct-2024.)
(((Itv‘ndx) ≠ (Base‘ndx) ∧ (Itv‘ndx) ≠ (+g‘ndx)) ∧ ((Itv‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (Itv‘ndx) ≠ (dist‘ndx)))
 
29-Oct-2024tngds 24011 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof shortened by AV, 29-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    − = (-g‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇))
 
29-Oct-2024srads 20654 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (dist‘𝑊) = (dist‘𝐴))
 
29-Oct-2024sratset 20651 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))
 
29-Oct-2024sramulr 20644 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (.r‘𝑊) = (.r‘𝐴))
 
29-Oct-2024sraaddg 20642 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (+g‘𝑊) = (+g‘𝐴))
 
29-Oct-2024srabase 20640 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   (𝜑 → (Base‘𝑊) = (Base‘𝐴))
 
29-Oct-2024sralem 20638 Lemma for srabase 20640 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
(𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑 → 𝑆 ⊆ (Base‘𝑊))    &   𝐸 = Slot (𝐸‘ndx)    &   (Scalar‘ndx) ≠ (𝐸‘ndx)    &   ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)    &   (·𝑖‘ndx) ≠ (𝐸‘ndx)    ⇒   (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))
 
29-Oct-2024dsndxntsetndx 17274 The slot for the distance function is not the slot for the topology in an extensible structure. Formerly part of proof for tngds 24011. (Contributed by AV, 29-Oct-2024.)
(dist‘ndx) ≠ (TopSet‘ndx)
 
29-Oct-2024slotsdnscsi 17273 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot dist. Formerly part of sralem 20638 and proofs using it. (Contributed by AV, 29-Oct-2024.)
((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024slotstnscsi 17241 The slots Scalar, ·𝑠 and ·𝑖 are different from the slot TopSet. Formerly part of sralem 20638 and proofs using it. (Contributed by AV, 29-Oct-2024.)
((TopSet‘ndx) ≠ (Scalar‘ndx) ∧ (TopSet‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (TopSet‘ndx) ≠ (·𝑖‘ndx))
 
29-Oct-2024ipndxnmulrndx 17215 The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 19904. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024ipndxnplusgndx 17214 The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
(·𝑖‘ndx) ≠ (+g‘ndx)
 
29-Oct-2024vscandxnmulrndx 17204 The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for rmodislmod 20390. (Contributed by AV, 29-Oct-2024.)
( ·𝑠 ‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024scandxnmulrndx 17199 The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. Formerly part of proof for mgpsca 19904. (Contributed by AV, 29-Oct-2024.)
(Scalar‘ndx) ≠ (.r‘ndx)
 
29-Oct-2024pm13.18 3025 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 29-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)
 
28-Oct-2024aks4d1p4 40536 There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 28-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )    ⇒   (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
 
28-Oct-2024edgfndxid 27942 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)
(𝐺 ∈ 𝑉 → (.ef‘𝐺) = (𝐺‘(.ef‘ndx)))
 
28-Oct-2024tuslem 23618 Lemma for tusbas 23620, tusunif 23621, and tustopn 23623. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof shortened by AV, 28-Oct-2024.)
𝐾 = (toUnifSp‘𝑈)    ⇒   (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
 
28-Oct-2024estrreslem1 18024 Lemma 1 for estrres 18027. (Contributed by AV, 14-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
(𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → 𝐵 = (Base‘𝐶))
 
28-Oct-2024slotsbhcdif 17296 The slots Base, Hom and comp are different. (Contributed by AV, 5-Mar-2020.) (Proof shortened by AV, 28-Oct-2024.)
((Base‘ndx) ≠ (Hom ‘ndx) ∧ (Base‘ndx) ≠ (comp‘ndx) ∧ (Hom ‘ndx) ≠ (comp‘ndx))
 
28-Oct-2024unifndxntsetndx 17281 The slot for the uniform set is not the slot for the topology in an extensible structure. Formerly part of proof for tuslem 23618. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ≠ (TopSet‘ndx)
 
28-Oct-2024basendxltunifndx 17279 The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. Formerly part of proof for tuslem 23618. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (UnifSet‘ndx)
 
28-Oct-2024unifndxnn 17278 The index of the slot for the uniform set in an extensible structure is a positive integer. Formerly part of proof for tuslem 23618. (Contributed by AV, 28-Oct-2024.)
(UnifSet‘ndx) ∈ ℕ
 
28-Oct-2024dsndxnbasendx 17270 The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
(dist‘ndx) ≠ (Base‘ndx)
 
28-Oct-2024basendxltdsndx 17269 The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. Formerly part of proof for tmslem 23837. (Contributed by AV, 28-Oct-2024.)
(Base‘ndx) < (dist‘ndx)
 
28-Oct-2024dsndxnn 17268 The index of the slot for the distance in an extensible structure is a positive integer. Formerly part of proof for tmslem 23837. (Contributed by AV, 28-Oct-2024.)
(dist‘ndx) ∈ ℕ
 
28-Oct-2024basendxnmulrndx 17176 The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) (Proof shortened by AV, 28-Oct-2024.)
(Base‘ndx) ≠ (.r‘ndx)
 
28-Oct-2024wunress 17131 Closure of structure restriction in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 28-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑊 ∈ 𝑈)    ⇒   (𝜑 → (𝑊 ↾s 𝐴) ∈ 𝑈)
 
28-Oct-2024predpo 6277 Property of the predecessor class for partial orders. (Contributed by Scott Fenton, 28-Apr-2012.) (Proof shortened by Scott Fenton, 28-Oct-2024.)
((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)))
 
28-Oct-2024predtrss 6276 If 𝑅 is transitive over 𝐴 and 𝑌𝑅𝑋, then Pred(𝑅, 𝐴, 𝑌) is a subclass of Pred(𝑅, 𝐴, 𝑋). (Contributed by Scott Fenton, 28-Oct-2024.)
((((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∧ 𝑋 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋))
 
28-Oct-2024necon3ai 2968 Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝐴 ≠ 𝐵 → ¬ 𝜑)
 
28-Oct-2024sbabel 2941 Theorem to move a substitution in and out of a class abstraction. (Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 28-Oct-2024.)
Ⅎ𝑥𝐴    ⇒   ([𝑦 / 𝑥]{𝑧 ∣ 𝜑} ∈ 𝐴 ↔ {𝑧 ∣ [𝑦 / 𝑥]𝜑} ∈ 𝐴)
 
27-Oct-2024aks4d1p3 40535 There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 27-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)
 
27-Oct-2024aks4d1p2 40534 Technical lemma for existence of non-divisor. (Contributed by metakunt, 27-Oct-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵)))
 
27-Oct-2024grpplusg 17169 The operation of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ( + ∈ 𝑉 → + = (+g‘𝐺))
 
27-Oct-2024grpbase 17167 The base set of a constructed group. (Contributed by Mario Carneiro, 2-Aug-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 27-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   (𝐵 ∈ 𝑉 → 𝐵 = (Base‘𝐺))
 
27-Oct-2024grpstrndx 17166 A constructed group is a structure. Version not depending on the implementation of the indices. (Contributed by AV, 27-Oct-2024.)
𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩}    ⇒   𝐺 Struct ⟨(Base‘ndx), (+g‘ndx)⟩
 
27-Oct-2024df-wrecs 8243 Define the well-ordered recursive function generator. This function takes the usual expressions from recursion theorems and forms a unified definition. Specifically, given a function 𝐹, a relation 𝑅, and a base set 𝐴, this definition generates a function 𝐺 = wrecs(𝑅, 𝐴, 𝐹) that has property that, at any point 𝑥 ∈ 𝐴, (𝐺‘𝑥) = (𝐹‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑥))). See wfr1 8281, wfr2 8282, and wfr3 8283. (Contributed by Scott Fenton, 7-Jun-2018.) (Revised by BJ, 27-Oct-2024.)
wrecs(𝑅, 𝐴, 𝐹) = frecs(𝑅, 𝐴, (𝐹 ∘ 2nd ))
 
27-Oct-2024opco2 8056 Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵))
 
27-Oct-2024opco1 8055 Value of an operation precomposed with the projection on the first component. (Contributed by Mario Carneiro, 28-May-2014.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = (𝐹‘𝐴))
 
27-Oct-2024predexg 6271 The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
(𝐴 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑋) ∈ V)
 
26-Oct-2024sticksstones22 40576 Non-exhaustive sticks and stones. (Contributed by metakunt, 26-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∣ (𝑓:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑓‘𝑖) ≤ 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + (♯‘𝑆))C(♯‘𝑆)))
 
26-Oct-2024dfttrcl2 9660 When 𝑅 is a set and a relation, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → t++𝑅 = ∩ {𝑧 ∣ (𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
 
26-Oct-2024ttrclexg 9659 If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.)
(𝑅 ∈ 𝑉 → t++𝑅 ∈ V)
 
26-Oct-2024rnttrcl 9658 The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
ran t++𝑅 = ran 𝑅
 
26-Oct-2024dmttrcl 9657 The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
dom t++𝑅 = dom 𝑅
 
26-Oct-2024nfttrcld 9646 Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
(𝜑 → Ⅎ𝑥𝑅)    ⇒   (𝜑 → Ⅎ𝑥t++𝑅)
 
26-Oct-2024nfopab 5174 Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.) (Revised by Scott Fenton, 26-Oct-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
26-Oct-2024nfopabd 5173 Bound-variable hypothesis builder for class abstraction. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑧𝜓)    ⇒   (𝜑 → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜓})
 
26-Oct-2024sbceqal 3805 Class version of one implication of equvelv 2034. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.)
(𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
 
26-Oct-2024sbcim1 3795 Distribution of class substitution over implication. One direction of sbcimg 3790 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 26-Oct-2024.)
([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
 
26-Oct-2024sbievg 2359 Substitution applied to expressions linked by implicit substitution. The proof was part of a former cbvabw 2810 version. (Contributed by GG and WL, 26-Oct-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
 
25-Oct-2024hbab1 2722 Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
(𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
 
25-Oct-2024nfsbv 2323 If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is disjoint from both 𝑥 and 𝑦. Version of nfsb 2525 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2370. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Oct-2024.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧[𝑦 / 𝑥]𝜑
 
24-Oct-2024sticksstones21 40575 Lift sticks and stones to arbitrary finite non-empty sets. (Contributed by metakunt, 24-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∣ (𝑓:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (𝑓‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + ((♯‘𝑆) − 1))C((♯‘𝑆) − 1)))
 
24-Oct-2024sticksstones20 40574 Lift sticks and stones to arbitrary finite non-empty sets. (Contributed by metakung, 24-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → (♯‘𝑆) = 𝐾)    ⇒   (𝜑 → (♯‘𝐵) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
24-Oct-2024eldifsucnn 8610 Condition for membership in the difference of ω and a nonzero finite ordinal. (Contributed by Scott Fenton, 24-Oct-2024.)
(𝐴 ∈ ω → (𝐵 ∈ (ω ∖ suc 𝐴) ↔ ∃𝑥 ∈ (ω ∖ 𝐴)𝐵 = suc 𝑥))
 
24-Oct-2024eqtr3 2762 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
 
24-Oct-2024eqtr2 2760 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
 
23-Oct-2024sticksstones19 40573 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
23-Oct-2024sticksstones18 40572 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
23-Oct-2024sticksstones17 40571 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
23-Oct-2024eqeq12 2753 Equality relationship among four classes. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
23-Oct-2024eqeq12d 2752 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
23-Oct-2024eqeqan12d 2750 A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2758. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜓 → 𝐶 = 𝐷)    ⇒   ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
21-Oct-2024unifndxnbasendx 17280 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(UnifSet‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024ipndxnbasendx 17213 The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(·𝑖‘ndx) ≠ (Base‘ndx)
 
21-Oct-2024scandxnbasendx 17197 The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
(Scalar‘ndx) ≠ (Base‘ndx)
 
20-Oct-2024sticksstones16 40570 Sticks and stones with collapsed definitions for positive integers. (Contributed by metakunt, 20-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
20-Oct-2024ttrclss 9656 If 𝑅 is a subclass of 𝑆 and 𝑆 is transitive, then the transitive closure of 𝑅 is a subclass of 𝑆. (Contributed by Scott Fenton, 20-Oct-2024.)
((𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → t++𝑅 ⊆ 𝑆)
 
20-Oct-2024cottrcl 9655 Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
(𝑅 ∘ t++𝑅) ⊆ t++𝑅
 
20-Oct-2024ttrclco 9654 Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
(t++𝑅 ∘ 𝑅) ⊆ t++𝑅
 
20-Oct-2024ttrclresv 9653 The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)
t++(𝑅 ↾ V) = t++𝑅
 
19-Oct-2024resseqnbas 17122 The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
𝑅 = (𝑊 ↾s 𝐴)    &   𝐶 = (𝐸‘𝑊)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Base‘ndx)    ⇒   (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅))
 
18-Oct-2024rmodislmod 20390 The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 20324 of a left module, see also islmod 20326. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
𝑉 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = ( ·𝑠 ‘𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    ⨣ = (+g‘𝐹)    &    × = (.r‘𝐹)    &    1 = (1r‘𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &    ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩)    ⇒   (𝐹 ∈ CRing → 𝐿 ∈ LMod)
 
18-Oct-2024mgpress 19911 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
𝑆 = (𝑅 ↾s 𝐴)    &   𝑀 = (mulGrp‘𝑅)    ⇒   ((𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑀 ↾s 𝐴) = (mulGrp‘𝑆))
 
18-Oct-2024setsplusg 19128 The other components of an extensible structure remain unchanged if the +g component is set/substituted. (Contributed by Stefan O'Rear, 26-Aug-2015.) Generalisation of the former oppglem and mgplem. (Revised by AV, 18-Oct-2024.)
𝑂 = (𝑅 sSet ⟨(+g‘ndx), 𝑆⟩)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (+g‘ndx)    ⇒   (𝐸‘𝑅) = (𝐸‘𝑂)
 
18-Oct-2024rescbas 17712 Base set of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.)
𝐷 = (𝐶 ↾cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → 𝑆 = (Base‘𝐷))
 
18-Oct-2024oppcbas 17599 Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 18-Oct-2024.)
𝑂 = (oppCat‘𝐶)    &   𝐵 = (Base‘𝐶)    ⇒   𝐵 = (Base‘𝑂)
 
18-Oct-2024dsndxnplusgndx 17271 The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds 19909. (Contributed by AV, 18-Oct-2024.)
(dist‘ndx) ≠ (+g‘ndx)

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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