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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 23-Apr-2024 at 5:09 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
21-Apr-2024xorbi12i 1515 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
(𝜑 ↔ 𝜓)    &   (𝜒 ↔ 𝜃)    ⇒   ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))
 
21-Apr-2024xorcom 1504 The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
 
21-Apr-2024an33rean 1479 Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
(((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) ↔ ((𝜑 ∧ 𝜏 ∧ 𝜌) ∧ ((𝜓 ∧ 𝜃) ∧ (𝜂 ∧ 𝜎) ∧ (𝜒 ∧ 𝜁))))
 
20-Apr-2024sn-elabg 39110 Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2144, ax-11 2160, ax-12 2176. It also avoids a disjoint variable condition on 𝑥 and 𝐴. Compare sbievw2 2106. (Contributed by SN, 20-Apr-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
 
20-Apr-2024vtoclg 3570 Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2176. (Revised by SN, 20-Apr-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   (𝐴 ∈ 𝑉 → 𝜓)
 
20-Apr-2024vtocl 3562 Implicit substitution of a class for a setvar variable. See also vtoclALT 3563. (Contributed by NM, 30-Aug-1993.) Remove dependency on ax-10 2144. (Revised by BJ, 29-Nov-2020.) (Proof shortened by SN, 20-Apr-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   𝜓
 
20-Apr-2024mpjao3dan 1427 Eliminate a three-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) (Proof shortened by Wolf Lammen, 20-Apr-2024.)
((𝜑 ∧ 𝜓) → 𝜒)    &   ((𝜑 ∧ 𝜃) → 𝜒)    &   ((𝜑 ∧ 𝜏) → 𝜒)    &   (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏))    ⇒   (𝜑 → 𝜒)
 
19-Apr-2024facp2 39104 The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.)
(𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2))))
 
19-Apr-2024fac2xp3 39103 Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.)
(𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3))))
 
17-Apr-2024rnmptc 6972 Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → ran 𝐹 = {𝐵})
 
17-Apr-2024ifpbi123d 1072 Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
(𝜑 → (𝜓 ↔ 𝜏))    &   (𝜑 → (𝜒 ↔ 𝜂))    &   (𝜑 → (𝜃 ↔ 𝜁))    ⇒   (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
 
16-Apr-2024cbvrabv2w 41400 A more general version of cbvrabv 3494. Version of cbvrabv2 41399 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by Gino Giotto, 16-Apr-2024.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
 
16-Apr-2024andiff 39102 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)
(𝜑 → (𝜒 → 𝜃))    &   (𝜓 → (𝜃 → 𝜒))    ⇒   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
 
14-Apr-2024bj-pw0ALT 34346 Alternate proof of pw0 4748. The proofs have a similar structure: pw0 4748 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 34346 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4748 and biconditional for bj-pw0ALT 34346) to translate the property ss0b 4354 into the wanted result. To translate a biconditional into a class equality, pw0 4748 uses abbii 2889 (which yields an equality of class abstractions), while bj-pw0ALT 34346 uses eqriv 2821 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2889, through its closed form abbi1 2887, is proved from eqrdv 2822, which is the deduction form of eqriv 2821. In the other direction, velpw 4547 and velsn 4586 are proved from the definitions of powerclass and singleton using elabg 3669, which is a version of abbii 2889 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 ∅ = {∅}
 
14-Apr-2024pwundif 4568 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) Remove use of ax-sep 5206, ax-nul 5213, ax-pr 5333 and shorten proof. (Revised by BJ, 14-Apr-2024.)
𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 
13-Apr-2024pwel 5285 Quantitative version of pwexg 5282: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5213 and ax-pr 5333 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
 
13-Apr-2024unieq 4852 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by BJ, 13-Apr-2024.)
(𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
 
13-Apr-2024pwunss 4562 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) Remove use of ax-sep 5206, ax-nul 5213, ax-pr 5333 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)
 
13-Apr-2024pweq 4558 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 13-Apr-2024.)
(𝐴 = 𝐵 → 𝒫 𝐴 = 𝒫 𝐵)
 
13-Apr-2024sspwd 4557 The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
13-Apr-2024sspwi 4556 The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
𝐴 ⊆ 𝐵    ⇒   𝒫 𝐴 ⊆ 𝒫 𝐵
 
13-Apr-2024sspw 4555 The powerclass preserves inclusion. See sspwb 5345 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 5345 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
(𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 
13-Apr-2024olcnd 873 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2024.)
(𝜑 → (𝜓 ∨ 𝜒))    &   (𝜑 → ¬ 𝜒)    ⇒   (𝜑 → 𝜓)
 
12-Apr-2024bj-smgrpssmgmel 34555 Semigroups are magmas (elemental version). (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
(𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
 
12-Apr-2024bj-smgrpssmgm 34554 Semigroups are magmas. (Contributed by BJ, 12-Apr-2024.) (Proof modification is discouraged.)
Smgrp ⊆ Mgm
 
11-Apr-2024bj-mndsssmgrpel 34557 Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
(𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)
 
11-Apr-2024bj-mndsssmgrp 34556 Monoids are semigroups. (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)
Mnd ⊆ Smgrp
 
11-Apr-2024bj-prmoore 34411 A pair formed of two nested sets is a Moore collection. (Note that in the statement, if 𝐵 is a proper class, we are in the case of bj-snmoore 34409). A direct consequence is {∅, 𝐴} ∈ Moore.

More generally, any nonempty well-ordered chain of sets that is a set is a Moore collection.

We also have the biconditional ((𝐴 ∩ 𝐵) ∈ 𝑉 → ({𝐴, 𝐵} ∈ Moore ↔ (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))). (Contributed by BJ, 11-Apr-2024.)

((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → {𝐴, 𝐵} ∈ Moore)
 
11-Apr-20242falsed 379 Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.) (Proof shortened by Wolf Lammen, 11-Apr-2024.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)    ⇒   (𝜑 → (𝜓 ↔ 𝜒))
 
10-Apr-2024bj-snmooreb 34410 A singleton is a Moore collection, biconditional version. (Contributed by BJ, 9-Dec-2021.) (Proof shortened by BJ, 10-Apr-2024.)
(𝐴 ∈ V ↔ {𝐴} ∈ Moore)
 
10-Apr-2024bj-snmoore 34409 A singleton is a Moore collection. See bj-snmooreb 34410 for a biconditional version. (Contributed by BJ, 10-Apr-2024.)
(𝐴 ∈ 𝑉 → {𝐴} ∈ Moore)
 
10-Apr-2024mxidlnzrb 30985 A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)))
 
10-Apr-2024krull 30984 Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))
 
10-Apr-2024ssmxidl 30983 Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼 ⊆ 𝑚)
 
10-Apr-2024ssmxidllem 30982 The set 𝑃 used in the proof of ssmxidl 30983 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)}    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝐼 ≠ 𝐵)    &   (𝜑 → 𝑍 ⊆ 𝑃)    &   (𝜑 → 𝑍 ≠ ∅)    &   (𝜑 → [⊊] Or 𝑍)    ⇒   (𝜑 → ∪ 𝑍 ∈ 𝑃)
 
10-Apr-2024jcnd 165 Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
(𝜑 → 𝜓)    &   (𝜑 → ¬ 𝜒)    ⇒   (𝜑 → ¬ (𝜓 → 𝜒))
 
9-Apr-2024pridln1 30963 A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼)
 
9-Apr-2024rlmlsm 19982 Subgroup sum of the ring module. (Contributed by Thierry Arnoux, 9-Apr-2024.)
(𝑅 ∈ 𝑉 → (LSSum‘𝑅) = (LSSum‘(ringLMod‘𝑅)))
 
6-Apr-2024bj-unirel 34348 Quantitative version of uniexr 7488: if the union of a class is an element of a class, then that class is an element of the double powerclass of the union of this class. (Contributed by BJ, 6-Apr-2024.)
(∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝑉)
 
6-Apr-2024bj-sselpwuni 34347 Quantitative version of ssexg 5230: a subset of an element of a class is an element of the powerclass of the union of that class. (Contributed by BJ, 6-Apr-2024.)
((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝒫 ∪ 𝑉)
 
6-Apr-2024elpwunicl 30309 Closure of a set union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 21-Jun-2020.) (Proof shortened by BJ, 6-Apr-2024.)
(𝜑 → 𝐴 ∈ 𝒫 𝒫 𝐵)    ⇒   (𝜑 → ∪ 𝐴 ∈ 𝒫 𝐵)
 
6-Apr-2024vuniex 7468 The union of a setvar is a set. (Contributed by BJ, 3-May-2021.) (Revised by BJ, 6-Apr-2024.)
∪ 𝑥 ∈ V
 
5-Apr-2024bj-endmnd 34603 The monoid of endomorphisms on an object of a category is a monoid. (Contributed by BJ, 5-Apr-2024.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)
 
5-Apr-2024bj-endcomp 34602 Composition law of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
 
5-Apr-2024bj-endbase 34601 Base set of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
 
5-Apr-2024bj-endval 34600 Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.)
(𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})
 
4-Apr-2024df-bj-end 34599 The monoid of endomorphisms on an object of a category. (Contributed by BJ, 4-Apr-2024.)
End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
 
2-Apr-2024rexlimddvcbvw 40565 Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 40564. The equivalent of this theorem without the bound variable change is rexlimddv 3294. Version of rexlimddvcbv 40566 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜃)    &   ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))    ⇒   (𝜑 → 𝜓)
 
2-Apr-2024brabidgaw 35621 The law of concretion for a binary relation. Special case of brabga 5424. Version of brabidga 35622 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   (𝑥𝑅𝑦 ↔ 𝜑)
 
1-Apr-2024symgid 18532 The group identity element of the symmetric group on a set 𝐴. (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 1-Apr-2024.)
𝐺 = (SymGrp‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺))
 
31-Mar-2024symgvalstruct 18528 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.)
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⟩})
 
31-Mar-2024symgpssefmnd 18527 For a set 𝐴 with more than one element, the symmetric group on 𝐴 is a proper subset of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Mar-2024.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐺 = (SymGrp‘𝐴)    ⇒   ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘𝑀))
 
31-Mar-2024snsymgefmndeq 18526 The symmetric group on a singleton 𝐴 is identical with the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Mar-2024.)
(𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
 
31-Mar-2024efmndbas0 18059 The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 31-Mar-2024.)
(Base‘(EndoFMnd‘∅)) = {∅}
 
31-Mar-20240map0sn0 8452 The set of mappings of the empty set to the empty set is the singleton containing the empty set. (Contributed by AV, 31-Mar-2024.)
(∅ ↑m ∅) = {∅}
 
30-Mar-2024wl-axc11r 34774 Same as axc11r 2385, but using ax12 2444 instead of ax-12 2176 directly. This better reflects axiom usage in theorems dependent on it. (Contributed by NM, 25-Jul-2015.) Avoid direct use of ax-12 2176. (Revised by Wolf Lammen, 30-Mar-2024.)
(∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑))
 
30-Mar-2024symgtgp 22717 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.) (Proof shortened by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp)
 
30-Mar-2024pgrpsubgsymg 18540 Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.) (Revised by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐹 = (Base‘𝑃)    ⇒   (𝐴 ∈ 𝑉 → ((𝑃 ∈ Grp ∧ 𝐹 ⊆ 𝐵 ∧ (+g‘𝑃) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubGrp‘𝐺)))
 
30-Mar-2024symgsubmefmndALT 18534 The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. Alternate proof based on issubmndb 17973 and not on injsubmefmnd 18065 and sursubmefmnd 18064. (Contributed by AV, 18-Feb-2024.) (Revised by AV, 30-Mar-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀))
 
30-Mar-20240symgefmndeq 18525 The symmetric group on the empty set is identical with the monoid of endofunctions on the empty set. (Contributed by AV, 30-Mar-2024.)
(EndoFMnd‘∅) = (SymGrp‘∅)
 
30-Mar-2024symgov 18515 The value of the group operation of the symmetric group on 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Revised by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
 
30-Mar-2024symgressbas 18513 The symmetric group on 𝐴 characterized as structure restriction of the monoid of endofunctions on 𝐴 to its base set. (Contributed by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝑀 = (EndoFMnd‘𝐴)    ⇒   𝐺 = (𝑀 ↾s 𝐵)
 
30-Mar-2024symgbasmap 18508 A permutation (element of the symmetric group) is a mapping (or set exponentiation) from a set into itself. (Contributed by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐹 ∈ 𝐵 → 𝐹 ∈ (𝐴 ↑m 𝐴))
 
30-Mar-2024symgbasex 18503 The base set of the symmetric group over a set 𝐴 exists. (Contributed by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → 𝐵 ∈ V)
 
30-Mar-2024permsetex 18501 The set of permutations of a set 𝐴 exists. (Contributed by AV, 30-Mar-2024.)
(𝐴 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V)
 
30-Mar-2024nf1oconst 7063 A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.)
((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1-onto→𝐶)
 
30-Mar-2024nf1const 7062 A constant function from at least two elements is not one-to-one. (Contributed by AV, 30-Mar-2024.)
((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶)
 
30-Mar-2024dral1v 2386 Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2460 with a disjoint variable condition, which does not require ax-13 2389. Remark: the corresponding versions for dral2 2459 and drex2 2463 are instances of albidv 1920 and exbidv 1921 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2177. (Revised by Wolf Lammen, 30-Mar-2024.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
29-Mar-2024symgplusg 18514 The group operation of a symmetric group is the function composition. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Proof shortened by AV, 19-Feb-2024.) (Revised by AV, 29-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (𝐴 ↑m 𝐴)    &    + = (+g‘𝐺)    ⇒    + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
 
29-Mar-2024symgbas 18502 The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) (Proof shortened by AV, 29-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
 
29-Mar-2024elefmndbas2 18042 Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.) (Proof shortened by AV, 29-Mar-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐴))
 
29-Mar-2024efmndbasabf 18040 The base set of the monoid of endofunctions on class 𝐴 is the set of functions from 𝐴 into itself. (Contributed by AV, 29-Mar-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   𝐵 = {𝑓 ∣ 𝑓:𝐴⟶𝐴}
 
29-Mar-2024fvmptd 6778 Deduction version of fvmpt 6771. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐹‘𝐴) = 𝐶)
 
29-Mar-2024fvmptdf 6777 Deduction version of fvmptd 6778 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.)
(𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐶    ⇒   (𝜑 → (𝐹‘𝐴) = 𝐶)
 
29-Mar-2024csbie2df 4395 Conversion of implicit substitution to explicit class substitution. This version of csbiedf 3916 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. Deduction form of csbie2 3925. (Contributed by AV, 29-Mar-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝐶)    &   (𝜑 → Ⅎ𝑥𝐷)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    &   ((𝜑 ∧ 𝑦 = 𝐴) → 𝐶 = 𝐷)    ⇒   (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐷)
 
28-Mar-2024symgval 18500 The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 28-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}    ⇒   𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
 
28-Mar-2024df-symg 18499 Define the symmetric group on set 𝑥. We represent the group as the set of one-to-one onto functions from 𝑥 to itself under function composition, and topologize it as a function space assuming the set is discrete. This definition is based on the fact that a symmetric group is a restriction of the monoid of endofunctions. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 28-Mar-2024.)
SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
 
27-Mar-2024bnj1018g 32239 Version of bnj1018 32240 with less disjoint variable conditions, but requiring ax-13 2389. (Contributed by Gino Giotto, 27-Mar-2024.) (New usage is discouraged.)
(𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))    &   (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))    &   (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))    &   (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))    &   (𝜑′ ↔ [𝑝 / 𝑛]𝜑)    &   (𝜓′ ↔ [𝑝 / 𝑛]𝜓)    &   (𝜒′ ↔ [𝑝 / 𝑛]𝜒)    &   (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)    &   (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)    &   (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)    &   𝐷 = (ω ∖ {∅})    &   𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    &   𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    &   (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))    &   ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)    &   ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))    ⇒   ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
 
27-Mar-2024bnj985v 32229 Version of bnj985 32230 with an additional disjoint variable condition, not requiring ax-13 2389. (Contributed by Gino Giotto, 27-Mar-2024.) (New usage is discouraged.)
(𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))    &   (𝜒′ ↔ [𝑝 / 𝑛]𝜒)    &   (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)    &   𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}    &   𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})    ⇒   (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″)
 
23-Mar-2024fundcmpsurbijinj 43577 Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖∃𝑝∃𝑞((𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1-onto→𝑞 ∧ 𝑖:𝑞–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
 
23-Mar-2024cosq34lt1 25115 Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 23-Mar-2024.)
(𝐴 ∈ (π[,)(2 · π)) → (cos‘𝐴) < 1)
 
23-Mar-2024cos02pilt1 25114 Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 23-Mar-2024.)
(𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)
 
23-Mar-2024o2p2e4 8169 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6200. For the usual proof using complex numbers, see 2p2e4 11775. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5193, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)
(2o +o 2o) = 4o
 
23-Mar-2024funfvima2d 6997 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))
 
23-Mar-2024elabd 3672 Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝜒)    &   (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})
 
22-Mar-2024fundcmpsurinjpreimafv 43575 Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ(𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
 
22-Mar-2024fundcmpsurbijinjpreimafv 43574 Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto 𝑃, a bijective function from 𝑃 and an injective function into the codomain of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑖((𝑔:𝐴–onto→𝑃 ∧ ℎ:𝑃–1-1-onto→(𝐹 “ 𝐴) ∧ 𝑖:(𝐹 “ 𝐴)–1-1→𝐵) ∧ 𝐹 = ((𝑖 ∘ ℎ) ∘ 𝑔)))
 
22-Mar-2024imasetpreimafvbij 43573 The mapping 𝐻 is a bijective function betwen the set 𝑃 of all preimages of values of function 𝐹 and the range of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–1-1-onto→(𝐹 “ 𝐴))
 
22-Mar-2024imasetpreimafvbijlemfo 43572 Lemma for imasetpreimafvbij 43573: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐻:𝑃–onto→(𝐹 “ 𝐴))
 
22-Mar-2024imasetpreimafvbijlemf1 43571 Lemma for imasetpreimafvbij 43573: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   (𝐹 Fn 𝐴 → 𝐻:𝑃–1-1→(𝐹 “ 𝐴))
 
22-Mar-2024imasetpreimafvbijlemf 43568 Lemma for imasetpreimafvbij 43573: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   (𝐹 Fn 𝐴 → 𝐻:𝑃⟶(𝐹 “ 𝐴))
 
22-Mar-2024uniimaelsetpreimafv 43563 The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∪ (𝐹 “ 𝑆) ∈ ran 𝐹)
 
17-Mar-2024fundcmpsurinjimaid 43578 Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective function onto the image (𝐹 “ 𝐴) of the domain of 𝐹 and an injective function from the image (𝐹 “ 𝐴). (Contributed by AV, 17-Mar-2024.)
𝐼 = (𝐹 “ 𝐴)    &   𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))    &   𝐻 = ( I ↾ 𝐼)    ⇒   (𝐹:𝐴⟶𝐵 → (𝐺:𝐴–onto→𝐼 ∧ 𝐻:𝐼–1-1→𝐵 ∧ 𝐹 = (𝐻 ∘ 𝐺)))
 
13-Mar-2024fundcmpsurinjALT 43579 Alternate proof of fundcmpsurinj 43576, based on fundcmpsurinjimaid 43578: Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
 
13-Mar-2024fundcmpsurinj 43576 Every function 𝐹:𝐴⟶𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → ∃𝑔∃ℎ∃𝑝(𝑔:𝐴–onto→𝑝 ∧ ℎ:𝑝–1-1→𝐵 ∧ 𝐹 = (ℎ ∘ 𝑔)))
 
12-Mar-2024uniimaprimaeqfv 43549 The union of the image of the preimage of a function value is the function value. (Contributed by AV, 12-Mar-2024.)
((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋))
 
10-Mar-2024preimafvelsetpreimafv 43555 The preimage of a function value is an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ∈ 𝑃)
 
10-Mar-2024elsetpreimafvb 43551 The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
 
10-Mar-2024setpreimafvex 43550 The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   (𝐴 ∈ 𝑉 → 𝑃 ∈ V)
 
9-Mar-2024elsetpreimafvrab 43561 An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → 𝑆 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐹‘𝑋)})
 
9-Mar-2024eqfvelsetpreimafv 43560 If an element of the domain of the function has the same function value as an element of the preimage of a function value, then it is an element of the same preimage. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋)) → 𝑌 ∈ 𝑆))
 
9-Mar-2024elsetpreimafvbi 43558 An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑆 ↔ (𝑌 ∈ 𝐴 ∧ (𝐹‘𝑌) = (𝐹‘𝑋))))
 
8-Mar-2024elsetpreimafveq 43564 If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑅)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑆 = 𝑅))
 
8-Mar-2024fvelsetpreimafv 43554 There is an element in a preimage 𝑆 of function values so that 𝑆 is the preimage of the function value at this element. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → ∃𝑥 ∈ 𝑆 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
 
8-Mar-2024elsetpreimafvssdm 43553 An element of the class 𝑃 of all preimages of function values is a subset of the domain of the function. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) → 𝑆 ⊆ 𝐴)
 
8-Mar-2024elsetpreimafv 43552 An element of the class 𝑃 of all preimages of function values. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   (𝑆 ∈ 𝑃 → ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)}))
 
7-Mar-2024preimafvn0 43547 The preimage of a function value is not empty. (Contributed by AV, 7-Mar-2024.)
((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ≠ ∅)
 
7-Mar-2024preimafvsnel 43546 The preimage of a function value at 𝑋 contains 𝑋. (Contributed by AV, 7-Mar-2024.)
((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
 
6-Mar-20240nelsetpreimafv 43557 The empty set is not an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 6-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
 
5-Mar-2024imasetpreimafvbijlemfv1 43570 Lemma for imasetpreimafvbij 43573: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝑃) → ∃𝑦 ∈ 𝑋 (𝐻‘𝑋) = (𝐹‘𝑦))
 
5-Mar-2024imasetpreimafvbijlemfv 43569 Lemma for imasetpreimafvbij 43573: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))
 
5-Mar-2024imaelsetpreimafv 43562 The image of an element of the preimage of a function value is the singleton consisting of the function value at one of its elements. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆) → (𝐹 “ 𝑆) = {(𝐹‘𝑋)})
 
5-Mar-2024elsetpreimafveqfv 43559 The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   ((𝐹 Fn 𝐴 ∧ (𝑆 ∈ 𝑃 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐹‘𝑋) = (𝐹‘𝑌))
 
5-Mar-2024preimafvsspwdm 43556 The class 𝑃 of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    ⇒   (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)
 
5-Mar-2024uniimafveqt 43548 The union of the image of a subset 𝑆 of the domain of a function with elements having the same function value is the function value at one of the elements of 𝑆. (Contributed by AV, 5-Mar-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ 𝑆) = (𝐹‘𝑋)))
 
5-Mar-2024wl-cbvalsbi 34789 Change bounded variables in a special case. The reverse direction seems to involve ax-11 2160. My hope is that I will in some future be able to prove mo3 2647 with reversed quantifiers not using ax-11 2160. See also the remark in mo4 2649, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.)
(∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
 
5-Mar-2024iuneqconst 4933 Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)
(𝑥 = 𝑋 → 𝐵 = 𝐶)    ⇒   ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
 
4-Mar-2024fundcmpsurinjlem2 43566 Lemma 2 for fundcmpsurinj 43576. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐺:𝐴–onto→𝑃)
 
4-Mar-2024fundcmpsurinjlem1 43565 Lemma 1 for fundcmpsurinj 43576. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐺 = (𝑥 ∈ 𝐴 ↦ (◡𝐹 “ {(𝐹‘𝑥)}))    ⇒   ran 𝐺 = 𝑃
 
3-Mar-2024fundcmpsurinjlem3 43567 Lemma 3 for fundcmpsurinj 43576. (Contributed by AV, 3-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}    &   𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))    ⇒   ((Fun 𝐹 ∧ 𝑋 ∈ 𝑃) → (𝐻‘𝑋) = ∪ (𝐹 “ 𝑋))
 
3-Mar-2024mendvscafval 39796 Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-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))
 
3-Mar-2024mendmulrfval 39793 Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    ⇒   (.r‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘ 𝑦))
 
3-Mar-2024mendplusgfval 39791 Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g‘𝑀)    ⇒   (+g‘𝐴) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ∘f + 𝑦))
 
2-Mar-2024clwwlknonmpo 27871 (ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)
(ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
 
2-Mar-2024pcofval 23617 The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) (Proof shortened by AV, 2-Mar-2024.)
(*𝑝‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))
 
2-Mar-2024marepvfval 21177 First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)    ⇒   𝑄 = (𝑚 ∈ 𝐵, 𝑣 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))
 
2-Mar-2024marrepfval 21172 First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g‘𝑅)    ⇒   𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
 
2-Mar-2024ipffval 20795 The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖‘𝑊)    &    · = (·if‘𝑊)    ⇒    · = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥 , 𝑦))
 
2-Mar-2024psrmulr 20167 The 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-Mar-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r‘𝑅)    &    ∙ = (.r‘𝑆)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    ⇒    ∙ = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘f − 𝑥)))))))
 
2-Mar-2024scaffval 19655 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    ∙ = ( ·sf ‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    ⇒    ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
 
2-Mar-2024dvrfval 19437 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr‘𝑅)    &    / = (/r‘𝑅)    ⇒    / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))
 
2-Mar-2024oppglsm 18770 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.)
𝑂 = (oppg‘𝐺)    &    ⊕ = (LSSum‘𝐺)    ⇒   (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇)
 
2-Mar-2024plusffval 17861 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    ⨣ = (+𝑓‘𝐺)    ⇒    ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
 
2-Mar-2024xpccofval 17435 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐾 = (Hom ‘𝑇)    &    · = (comp‘𝐶)    &    ∙ = (comp‘𝐷)    &   𝑂 = (comp‘𝑇)    ⇒   𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐾𝑦), 𝑓 ∈ (𝐾‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩ · (1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩ ∙ (2nd ‘𝑦))(2nd ‘𝑓))⟩))
 
1-Mar-2024xpchomfval 17432 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐾 = (Hom ‘𝑇)    ⇒   𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))
 
1-Mar-2024natfval 17219 Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    ⇒   𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ 𝐵 ((𝑟‘𝑥)𝐽(𝑠‘𝑥)) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝑎‘𝑦)(⟨(𝑟‘𝑥), (𝑟‘𝑦)⟩ · (𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(⟨(𝑟‘𝑥), (𝑠‘𝑥)⟩ · (𝑠‘𝑦))(𝑎‘𝑥))})
 
1-Mar-2024comfffval 16971 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝑂 = (compf‘𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    ⇒   𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 
1-Mar-2024homffval 16963 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝐹 = (Homf ‘𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
 
29-Feb-2024evls1pw 20492 Univariate polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑊 = (Poly1‘𝑈)    &   𝐺 = (mulGrp‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &    ↑ = (.g‘𝐺)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s 𝐾)))(𝑄‘𝑋)))
 
29-Feb-2024evlspw 20309 Polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    ↑ = (.g‘𝐺)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋)))
 
29-Feb-2024mpllvec 20236 The polynomial ring is a vector space. (Contributed by SN, 29-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ DivRing) → 𝑃 ∈ LVec)
 
27-Feb-2024pwmnd 18105 The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   𝑀 ∈ Mnd
 
27-Feb-2024pwmndid 18104 The identity of the monoid of the power set of a class 𝐴 under union is the empty set. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   (0g‘𝑀) = ∅
 
27-Feb-2024pwmndgplus 18103 The operation of the monoid of the power set of a class 𝐴 under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g‘𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥 ∪ 𝑦))    ⇒   ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(+g‘𝑀)𝑌) = (𝑋 ∪ 𝑌))
 
27-Feb-2024pwuncl 7495 Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)
((𝐴 ∈ 𝒫 𝑋 ∧ 𝐵 ∈ 𝒫 𝑋) → (𝐴 ∪ 𝐵) ∈ 𝒫 𝑋)
 
25-Feb-2024injsubmefmnd 18065 The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–1-1→𝐴} ∈ (SubMnd‘𝑀))
 
25-Feb-2024sursubmefmnd 18064 The set of surjective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → {ℎ ∣ ℎ:𝐴–onto→𝐴} ∈ (SubMnd‘𝑀))
 
25-Feb-2024insubm 17986 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))
 
25-Feb-2024nfsb 2564 If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Usage of this theorem is discouraged because it depends on ax-13 2389. For a version requiring more disjoint variables, but fewer axioms, see nfsbv 2348. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.) (New usage is discouraged.)
Ⅎ𝑧𝜑    ⇒   Ⅎ𝑧[𝑦 / 𝑥]𝜑
 
24-Feb-2024nfsumw 15050 Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. Version of nfsum 15051 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 11-Dec-2005.) (Revised by Gino Giotto, 24-Feb-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵
 
23-Feb-2024efmndtmd 22712 The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 22717. (Contributed by AV, 23-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → 𝑀 ∈ TopMnd)
 
22-Feb-2024selvcl 39144 Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐸 = (Base‘𝑇)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸)
 
22-Feb-2024selvval2lem5 39143 The fifth argument passed to evalSub is in the domain (a function 𝐼⟶𝐸). (Contributed by SN, 22-Feb-2024.)
𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐸 = (Base‘𝑇)    &   𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   (𝜑 → 𝐹 ∈ (𝐸 ↑m 𝐼))
 
21-Feb-2024remulcand 39256 Commuted version of remulcan2d 39162 without ax-mulcom 10604. (Contributed by SN, 21-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐶 ≠ 0)    ⇒   (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
21-Feb-2024readdcan2 39248 Commuted version of readdcan 10817 without ax-mulcom 10604. (Contributed by SN, 21-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
 
21-Feb-2024evlsvarpw 20310 Polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by SN, 21-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    ↑ = (.g‘𝐺)    &   𝑋 = ((𝐼 mVar 𝑈)‘𝑌)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑃 = (𝑆 ↑s (𝐵 ↑m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐵 = (Base‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝐼)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋)))
 
21-Feb-2024sb1 2502 One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2275) or a non-freeness hypothesis (sb5f 2537). Usage of this theorem is discouraged because it depends on ax-13 2389. Use the weaker sb1v 2094 when possible. (Contributed by NM, 13-May-1993.) Revise df-sb 2069. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
 
21-Feb-2024sb3 2501 One direction of a simplified definition of substitution when variables are distinct. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
 
21-Feb-2024sb4b 2498 Simplified definition of substitution when variables are distinct. Version of sb6 2092 with a distinctor. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 27-May-1997.) Revise df-sb 2069. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
 
19-Feb-2024grpsubfvalALT 18151 Shorter proof of grpsubfval 18150 using ax-rep 5193. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   𝐼 = (invg‘𝐺)    &    − = (-g‘𝐺)    ⇒    − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))
 
19-Feb-2024grpsubfval 18150 Group subtraction (division) operation. For a shorter proof using ax-rep 5193, see grpsubfvalALT 18151. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5193. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   𝐼 = (invg‘𝐺)    &    − = (-g‘𝐺)    ⇒    − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦)))
 
18-Feb-2024symgsubmefmnd 18529 The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → 𝐵 ∈ (SubMnd‘𝑀))
 
18-Feb-2024smndex2dlinvh 18085 The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g‘𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   𝑁 ∈ ℕ0    &   𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))    ⇒   (𝐻 ∘ 𝐷) = 0
 
18-Feb-2024smndex2hbas 18084 The halving functions 𝐻 are endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g‘𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   𝑁 ∈ ℕ0    &   𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))    ⇒   𝐻 ∈ 𝐵
 
18-Feb-2024smndex2dnrinv 18083 The doubling function 𝐷 has no right inverse in the monoid of endofunctions on ℕ0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g‘𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    ⇒   ∀𝑓 ∈ 𝐵 (𝐷 ∘ 𝑓) ≠ 0
 
18-Feb-2024smndex2dbas 18082 The doubling function 𝐷 is an endofunction on ℕ0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g‘𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    ⇒   𝐷 ∈ 𝐵
 
18-Feb-2024efmnd2hash 18062 The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼, 𝐽}    ⇒   ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 4)
 
17-Feb-2024nsmndex1 18081 The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set ℕ0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   𝐵 ∉ (SubMnd‘𝑀)
 
17-Feb-2024smndex1n0mnd 18080 The identity of the monoid 𝑀 of endofunctions on set ℕ0 is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   (0g‘𝑀) ∉ 𝐵
 
17-Feb-2024idresefmnd 18067 The structure with the singleton containing only the identity function restricted to a set 𝐴 as base set and the function composition as group operation, constructed by (structure) restricting the monoid of endofunctions on 𝐴 to that singleton, is a monoid whose base set is a subset of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 17-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐸 = (𝐺 ↾s {( I ↾ 𝐴)})    ⇒   (𝐴 ∈ 𝑉 → (𝐸 ∈ Mnd ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
 
17-Feb-2024idressubmefmnd 18066 The singleton containing only the identity function restricted to a set is a submonoid of the monoid of endofunctions on this set. (Contributed by AV, 17-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → {( I ↾ 𝐴)} ∈ (SubMnd‘𝐺))
 
17-Feb-2024submefmnd 18063 If the base set of a monoid is contained in the base set of the monoid of endofunctions on a set 𝐴, contains the identity function and has the function composition as group operation, then its base set is a submonoid of the monoid of endofunctions on set 𝐴. Analogous to pgrpsubgsymg 18540. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝑀)    &    0 = (0g‘𝑀)    &   𝐹 = (Base‘𝑆)    ⇒   (𝐴 ∈ 𝑉 → (((𝑆 ∈ Mnd ∧ 𝐹 ⊆ 𝐵 ∧ 0 ∈ 𝐹) ∧ (+g‘𝑆) = (𝑓 ∈ 𝐹, 𝑔 ∈ 𝐹 ↦ (𝑓 ∘ 𝑔))) → 𝐹 ∈ (SubMnd‘𝑀)))
 
17-Feb-20240subm 17985 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0g‘𝐺)    ⇒   (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 
17-Feb-2024resmndismnd 17976 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the other monoid restricted to the base set of the monoid is a monoid. Analogous to resgrpisgrp 18303. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝐺 ↾s 𝑆) ∈ Mnd))
 
17-Feb-2024mndissubm 17975 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. Analogous to grpissubg 18302. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
 
17-Feb-2024mgmsscl 17860 If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 18302. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    ⇒   (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)
 
16-Feb-2024smndex1id 18079 The modulo function 𝐼 is the identity of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   𝐼 = (0g‘𝑆)
 
16-Feb-2024smndex1mnd 18078 The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a monoid. (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   𝑆 ∈ Mnd
 
16-Feb-2024smndex1mndlem 18077 Lemma for smndex1mnd 18078 and smndex1id 18079. (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   (𝑋 ∈ 𝐵 → ((𝐼 ∘ 𝑋) = 𝑋 ∧ (𝑋 ∘ 𝐼) = 𝑋))
 
14-Feb-2024smndex1sgrp 18076 The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a semigroup. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   𝑆 ∈ Smgrp
 
14-Feb-2024smndex1mgm 18075 The monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a magma. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   𝑆 ∈ Mgm
 
14-Feb-2024smndex1igid 18072 The composition of the modulo function 𝐼 and a constant function (𝐺‘𝐾) results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝐾)) = (𝐺‘𝐾))
 
14-Feb-2024smndex1gid 18071 The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ0 results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾))
 
13-Feb-2024sn-ltp1 39253 ltp1 11483 without ax-mulcom 10604. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
13-Feb-2024sn-0lt1 39252 0lt1 11165 without ax-mulcom 10604. (Contributed by SN, 13-Feb-2024.)
0 < 1
 
13-Feb-2024relt0neg2 39251 Comparison of a real and its negative to zero. Compare lt0neg2 11150. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 −ℝ 𝐴) < 0))
 
13-Feb-2024relt0neg1 39250 Comparison of a real and its negative to zero. Compare lt0neg1 11149. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 −ℝ 𝐴)))
 
13-Feb-2024sn-ltaddpos 39249 ltaddpos 11133 without ax-mulcom 10604. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
 
13-Feb-2024renegneg 39247 A real number is equal to the negative of its negative. Compare negneg 10939. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
 
13-Feb-2024reltsubadd2 39223 'Less than' relationship between addition and subtraction. Compare ltsubadd2 11114. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
 
13-Feb-2024reltsub1 39222 Subtraction from both sides of 'less than'. Compare ltsub1 11139. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 −ℝ 𝐶) < (𝐵 −ℝ 𝐶)))
 
13-Feb-2024evlsgsummul 20308 Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    1 = (1r‘𝑊)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 1 )    ⇒   (𝜑 → (𝑄‘(𝐺 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝐻 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
 
13-Feb-2024evlsgsumadd 20307 Polynomial evaluation maps (additive) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &    0 = (0g‘𝑊)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑃 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥 ∈ 𝑁 ↦ 𝑌) finSupp 0 )    ⇒   (𝜑 → (𝑄‘(𝑊 Σg (𝑥 ∈ 𝑁 ↦ 𝑌))) = (𝑃 Σg (𝑥 ∈ 𝑁 ↦ (𝑄‘𝑌))))
 
12-Feb-2024smndex1bas 18074 The base set of the monoid of endofunctions on ℕ0 restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾). (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    &   𝑆 = (𝑀 ↾s 𝐵)    ⇒   (Base‘𝑆) = 𝐵
 
12-Feb-2024smndex1basss 18073 The modulo function 𝐼 and the constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    &   𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})    ⇒   𝐵 ⊆ (Base‘𝑀)
 
12-Feb-2024smndex1gbas 18070 The constant functions (𝐺‘𝐾) are endofunctions on ℕ0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))    ⇒   (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) ∈ (Base‘𝑀))
 
12-Feb-2024smndex1iidm 18069 The modulo function 𝐼 is idempotent. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    ⇒   (𝐼 ∘ 𝐼) = 𝐼
 
12-Feb-2024smndex1ibas 18068 The modulo function 𝐼 is an endofunction on ℕ0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    ⇒   𝐼 ∈ (Base‘𝑀)
 
10-Feb-2024cbvexdvaw 2045 Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 2430 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
10-Feb-2024cbvaldvaw 2044 Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 2429 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) (Revised by Gino Giotto, 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
5-Feb-2024remulid2 39255 Commuted version of ax-1rid 10610 and real number version of mulid2 10643 without ax-mulcom 10604. (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
 
5-Feb-2024remulinvcom 39254 A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 10604. (Contributed by SN, 5-Feb-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   (𝜑 → (𝐵 · 𝐴) = 1)
 
5-Feb-2024nnmulcom 39171 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
5-Feb-2024nnmul1com 39170 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 10604. Since (𝐴 · 1) is 𝐴 by ax-1rid 10610, this is equivalent to remulid2 39255 for natural numbers, but using fewer axioms (avoiding ax-resscn 10597, ax-addass 10605, ax-mulass 10606, ax-rnegex 10611, ax-pre-lttri 10614, ax-pre-lttrn 10615, ax-pre-ltadd 10616). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))
 
5-Feb-2024nnadddir 39169 Right-distributivity for natural numbers without ax-mulcom 10604. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
5-Feb-2024empty 1906 Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
(¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
 
3-Feb-2024sbequ2 2249 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2069. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)
(𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))
 
1-Feb-2024issubmndb 17973 The submonoid predicate. Analogous to issubg 18282. (Contributed by AV, 1-Feb-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    ⇒   (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ∈ Mnd ∧ (𝐺 ↾s 𝑆) ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆)))
 
31-Jan-2024efmnd1bas 18061 The monoid of endofunctions on a singleton consists of the identity only. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}    ⇒   (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}})
 
31-Jan-2024efmnd0nmnd 18058 Even the monoid of endofunctions on the empty set is actually a monoid. (Contributed by AV, 31-Jan-2024.)
(EndoFMnd‘∅) ∈ Mnd
 
31-Jan-2024efmndmnd 18057 The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd)
 
31-Jan-2024efmndtopn 18051 The topology of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝑋)    &   𝐵 = (Base‘𝐺)    ⇒   (𝑋 ∈ 𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
 
30-Jan-2024iotan0 6348 Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
 
30-Jan-2024iresn0n0 5926 The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.)
(𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)
 
29-Jan-2024sgrpidmnd 17919 A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 18105 and pwmndid 18104), but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
 
29-Jan-2024ccatw2s1ccatws2 14319 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 29-Jan-2024.)
(𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))
 
29-Jan-2024ccatw2s1p1 13998 Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁 ∧ 𝑋 ∈ 𝑉) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑁) = 𝑋)
 
29-Jan-2024sbiedvw 2103 Conversion of implicit substitution to explicit substitution (deduction version of sbievw 2102). Version of sbied 2544 and sbiedv 2545 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 29-Jan-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
 
29-Jan-2024sbrimvw 2101 Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2312 and sbrimv 2313 based on fewer axioms, but with more disjoint variable conditions. Based on an idea of Gino Giotto. (Contributed by Wolf Lammen, 29-Jan-2024.)
([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
29-Jan-2024sbrimvlem 2100 Common proof template for sbrimvw 2101 and sbrimv 2313. The hypothesis is an instance of 19.21 2206. (Contributed by Wolf Lammen, 29-Jan-2024.)
(∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))    ⇒   ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
28-Jan-2024symggrp 18531 The symmetric group on a set 𝐴 is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 28-Jan-2024.)
𝐺 = (SymGrp‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → 𝐺 ∈ Grp)
 
28-Jan-2024efmndsgrp 18054 The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   𝐺 ∈ Smgrp
 
28-Jan-2024efmndmgm 18053 The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   𝐺 ∈ Mgm
 
28-Jan-2024symggrplem 18052 Lemma for symggrp 18531 and efmndsgrp 18054. Conditions for an operation to be associative. Formerly part of proof for symggrp 18531. (Contributed by AV, 28-Jan-2024.)
((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑥 ∘ 𝑦))    ⇒   ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
28-Jan-2024ccat2s1fvwALT 14321 Alternate proof of ccat2s1fvw 14001 using words of length 2, see df-s2 14213. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 28-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))
 
28-Jan-2024ccat2s1fst 14003 The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 28-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ 0 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘0) = (𝑊‘0))
 
28-Jan-2024ccat2s1fvw 14001 Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊‘𝐼))
 
28-Jan-2024sbiedw 2331 Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2329). Version of sbied 2544 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2144. (Revised by Wolf Lammen, 28-Jan-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
 
28-Jan-2024sbrimv 2313 Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2312 not depending on ax-10 2144, but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024.)
Ⅎ𝑥𝜑    ⇒   ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
27-Jan-2024efmnd1hash 18060 The monoid of endofunctions on a singleton has cardinality 1. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}    ⇒   (𝐼 ∈ 𝑉 → (♯‘𝐵) = 1)
 
27-Jan-2024ielefmnd 18055 The identity function restricted to a set 𝐴 is an element of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
 
27-Jan-2024efmndcl 18050 The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
27-Jan-2024efmndov 18049 The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
 
27-Jan-2024efmndplusg 18048 The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒    + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
 
27-Jan-2024efmndfv 18046 The function value of an endofunction. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐴)
 
27-Jan-2024efmndbasfi 18045 The monoid of endofunctions on a finite set 𝐴 is finite. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ Fin → 𝐵 ∈ Fin)
 
27-Jan-2024efmndhash 18044 The monoid of endofunctions on 𝑛 objects has cardinality 𝑛↑𝑛. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ Fin → (♯‘𝐵) = ((♯‘𝐴)↑(♯‘𝐴)))
 
27-Jan-2024efmndbasf 18043 Elements in the monoid of endofunctions on 𝐴 are functions from 𝐴 into itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐹 ∈ 𝐵 → 𝐹:𝐴⟶𝐴)
 
27-Jan-2024elefmndbas 18041 Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐴 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐴))
 
27-Jan-2024mpo0v 7241 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) (Proof shortened by AV, 27-Jan-2024.)
(𝑥 ∈ ∅, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅
 
27-Jan-20240mpo0 7240 A mapping operation with empty domain is empty. Generalization of mpo0 7242. (Contributed by AV, 27-Jan-2024.)
((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∅)
 
26-Jan-2024nfixpw 8483 Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8484 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    ⇒   Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵
 
26-Jan-2024frsucmpt2w 8078 Version of frsucmpt2 8079 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   (𝑦 = 𝑥 → 𝐸 = 𝐶)    &   (𝑦 = (𝐹‘𝐵) → 𝐸 = 𝐷)    ⇒   ((𝐵 ∈ ω ∧ 𝐷 ∈ 𝑉) → (𝐹‘suc 𝐵) = 𝐷)
 
26-Jan-2024elovmporab1w 7395 Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. Version of elovmporab1 7396 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ ⦋𝑋 / 𝑚⦌𝑀))
 
26-Jan-2024eqoprab2bw 7227 Equivalence of ordered pair abstraction subclass and biconditional. Version of eqoprab2b 7228 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 4-Jan-2017.) (Revised by Gino Giotto, 26-Jan-2024.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦∀𝑧(𝜑 ↔ 𝜓))
 
26-Jan-2024oprabidw 7190 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of oprabid 7191 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 20-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
 
26-Jan-2024cbvriotavw 7127 Change bound variable in a restricted description binder. Version of cbvriotav 7131 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
 
26-Jan-2024cbvriotaw 7126 Change bound variable in a restricted description binder. Version of cbvriota 7130 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
 
26-Jan-2024nfriotadw 7125 Deduction version of nfriota 7129 with a disjoint variable condition, which contrary to nfriotad 7128 does not require ax-13 2389. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝐴)    ⇒   (𝜑 → Ⅎ𝑥(℩𝑦 ∈ 𝐴 𝜓))
 
26-Jan-2024rexrnmptw 6864 A restricted quantifier over an image set. Version of rexrnmpt 6866 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
 
26-Jan-2024ralrnmptw 6863 A restricted quantifier over an image set. Version of ralrnmpt 6865 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    ⇒   (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
 
26-Jan-2024elfvmptrab1w 6797 Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Version of elfvmptrab1 6798 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
 
26-Jan-2024cbviotavw 6325 Change bound variables in a description binder. Version of cbviotav 6327 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥𝜑) = (℩𝑦𝜓)
 
26-Jan-2024cbviotaw 6324 Change bound variables in a description binder. Version of cbviota 6326 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    ⇒   (℩𝑥𝜑) = (℩𝑦𝜓)
 
26-Jan-2024nfiotaw 6321 Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6323 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 23-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥(℩𝑦𝜑)
 
26-Jan-2024nfiotadw 6320 Deduction version of nfiotaw 6321. Version of nfiotad 6322 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥(℩𝑦𝜓))
 
26-Jan-2024eqopab2bw 5438 Equivalence of ordered pair abstraction equality and biconditional. Version of eqopab2b 5442 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 4-Jan-2017.) (Revised by Gino Giotto, 26-Jan-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓))
 
26-Jan-2024ssopab2bw 5437 Equivalence of ordered pair abstraction subclass and implication. Version of ssopab2b 5439 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 27-Dec-1996.) (Revised by Gino Giotto, 26-Jan-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓))
 
26-Jan-2024opabidw 5415 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 5416 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 14-Apr-1995.) (Revised by Gino Giotto, 26-Jan-2024.)
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
 
26-Jan-2024copsexgw 5384 Version of copsexg 5385 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
 
26-Jan-2024disjprgw 5064 Version of disjprg 5065 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴 → 𝐶 = 𝐷)    &   (𝑥 = 𝐵 → 𝐶 = 𝐸)    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷 ∩ 𝐸) = ∅))
 
26-Jan-2024invdisjrabw 5054 Version of invdisjrab 5055 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 26-Jan-2024.)
Disj 𝑦 ∈ 𝐴 {𝑥 ∈ 𝐵 ∣ 𝐶 = 𝑦}
 
26-Jan-2024nfdisjw 5046 Bound-variable hypothesis builder for disjoint collection. Version of nfdisj 5047 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    ⇒   Ⅎ𝑦Disj 𝑥 ∈ 𝐴 𝐵
 
26-Jan-2024sbcco3gw 4377 Composition of two substitutions. Version of sbcco3g 4382 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 27-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜑))
 
26-Jan-2024csbnestgw 4376 Nest the composition of two substitutions. Version of csbnestg 4381 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
 
26-Jan-2024sbcnestgw 4375 Nest the composition of two substitutions. Version of sbcnestg 4380 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 27-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
(𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
 
26-Jan-2024csbnestgfw 4374 Nest the composition of two substitutions. Version of csbnestgf 4379 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 26-Jan-2024.)
((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝐶) → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐶 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐶)
 
26-Jan-2024sbcnestgfw 4373 Nest the composition of two substitutions. Version of sbcnestgf 4378 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 11-Nov-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
((𝐴 ∈ 𝑉 ∧ ∀𝑦Ⅎ𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦]𝜑))
 
26-Jan-2024cbvrabcsfw 3927 Version of cbvrabcsf 3931 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Andrew Salmon, 13-Jul-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
Ⅎ𝑦𝐴    &   Ⅎ𝑥𝐵    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
 
25-Jan-2024efmndid 18056 The identity function restricted to a set 𝐴 is the identity element of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) = (0g‘𝐺))
 
25-Jan-2024efmndtset 18047 The topology of the monoid of endofunctions on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    ⇒   (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
 
25-Jan-2024efmndbas 18039 The base set of the monoid of endofunctions on class 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    ⇒   𝐵 = (𝐴 ↑m 𝐴)
 
25-Jan-2024efmnd 18038 The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (𝐴 ↑m 𝐴)    &    + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))    ⇒   (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 
25-Jan-2024df-efmnd 18037 Define the monoid of endofunctions on set 𝑥. We represent the monoid as the set of functions from 𝑥 to itself ((𝑥 ↑m 𝑥)) under function composition, and topologize it as a function space assuming the set is discrete. Analogous to the former definition of SymGrp, see df-symg 18499 and symgvalstruct 18528. (Contributed by AV, 25-Jan-2024.)
EndoFMnd = (𝑥 ∈ V ↦ ⦋(𝑥 ↑m 𝑥) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})
 
23-Jan-2024resubid1 39246 Real number version of subid1 10909, without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴)
 
23-Jan-2024readdid1 39245 Real number version of addid1 10823, without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
 
23-Jan-2024resubid 39244 Subtraction of a real number from itself (compare subid 10908). (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0)
 
23-Jan-2024remul01 39243 Real number version of mul01 10822 proven without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 · 0) = 0)
 
23-Jan-2024sn-0ne2 39242 0ne2 11847 without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
0 ≠ 2
 
23-Jan-2024remul02 39241 Real number version of mul02 10821 proven without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 · 𝐴) = 0)
 
23-Jan-2024sn-addid2 39240 addid2 10826 without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
23-Jan-2024readdid2 39239 Real number version of addid2 10826. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
 
23-Jan-2024re0m0e0 39238 Real number version of 0m0e0 11760 proven without ax-mulcom 10604. (Contributed by SN, 23-Jan-2024.)
(0 −ℝ 0) = 0
 
23-Jan-2024nelb 3271 A definition of ¬ 𝐴 ∈ 𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.)
(¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
 
22-Jan-20243cubes 39293 Every rational number is a sum of three rational cubes. (S. Ryley, The Ladies' Diary 122 (1825), 35) (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝐴 ∈ ℚ ↔ ∃𝑎 ∈ ℚ ∃𝑏 ∈ ℚ ∃𝑐 ∈ ℚ 𝐴 = (((𝑎↑3) + (𝑏↑3)) + (𝑐↑3)))
 
22-Jan-20243cubeslem4 39292 Lemma for 3cubes 39293. This is Ryley's explicit formula for decomposing a rational 𝐴 into a sum of three rational cubes. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → 𝐴 = (((((((3↑3) · (𝐴↑3)) − 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3) + ((((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)) + (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)))
 
22-Jan-20243cubeslem3 39291 Lemma for 3cubes 39293. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)))
 
22-Jan-20243cubeslem3r 39290 Lemma for 3cubes 39293. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
22-Jan-20243cubeslem3l 39289 Lemma for 3cubes 39293. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
22-Jan-20243cubeslem2 39288 Lemma for 3cubes 39293. Used to show that the denominators in 3cubeslem4 39292 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0)
 
22-Jan-20243cubeslem1 39287 Lemma for 3cubes 39293. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑 → 𝐴 ∈ ℚ)    ⇒   (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))
 
21-Jan-2024rexlimdv3d 39286 An extended version of rexlimdvv 3296 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝜓 → 𝜒)))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))
 
21-Jan-2024negexpidd 39285 The sum of a real number to the power of N and the negative of the number to the power of N equals zero if N is a nonnegative odd integer. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ¬ 2 ∥ 𝑁)    ⇒   (𝜑 → ((𝐴↑𝑁) + (-𝐴↑𝑁)) = 0)
 
21-Jan-2024sqnegd 39284 The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (-𝐴↑2) = (𝐴↑2))
 
21-Jan-2024df-rspec 30989 Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))
 
21-Jan-2024df-idlsrg 30987 Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
IDLsrg = (𝑟 ∈ Ring ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((+𝑓‘𝑟) “ (𝑖 × 𝑗)))⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘((+𝑓‘(mulGrp‘𝑟)) “ (𝑖 × 𝑗))))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ (𝑏 ∖ {𝑗 ∈ (PrmIdeal‘𝑟) ∣ 𝑗 ⊆ 𝑖}))⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}))
 
21-Jan-2024mxidlprm 30981 Every maximal ideal is prime. Statement in [Lang] p. 92 (Contributed by Thierry Arnoux, 21-Jan-2024.)
× = (LSSum‘(mulGrp‘𝑅))    ⇒   ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
 
21-Jan-2024lsmidl 30955 The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    ⊕ = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    ⇒   (𝜑 → (𝐼 ⊕ 𝐽) ∈ (LIdeal‘𝑅))
 
21-Jan-2024lsmidllsp 30954 The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    ⊕ = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    ⇒   (𝜑 → (𝐼 ⊕ 𝐽) = (𝐾‘(𝐼 ∪ 𝐽)))
 
21-Jan-2024lsmsnidl 30953 The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐵 × {𝑋}) ∈ (LPIdeal‘𝑅))
 
21-Jan-2024lsmsnpridl 30952 The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐵 × {𝑋}) = (𝐾‘{𝑋}))
 
21-Jan-2024lsmsnorb 30949 The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18440. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    ⊕ = (LSSum‘𝐺)    &    ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑔 + 𝑥) = 𝑦)}    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐴 ⊕ {𝑋}) = [𝑋] ∼ )
 
21-Jan-2024elgrplsmsn 30948 Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    ⊕ = (LSSum‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑍 ∈ (𝐴 ⊕ {𝑋}) ↔ ∃𝑥 ∈ 𝐴 𝑍 = (𝑥 + 𝑋)))
 
21-Jan-2024fmptssfisupp 30431 The restriction of a mapping function has finite support if that function has finite support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
(𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) finSupp 𝑍)    &   (𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝑍 ∈ 𝑉)    ⇒   (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐵) finSupp 𝑍)
 
21-Jan-2024fvdifsupp 30430 Function value is zero outside of its support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑍 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ (𝐴 ∖ (𝐹 supp 𝑍)))    ⇒   (𝜑 → (𝐹‘𝑋) = 𝑍)
 
21-Jan-2024emptyex 1907 On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
(¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
 
20-Jan-2024nfiund 44784 Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2389. See nfiundg 44785 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → Ⅎ𝑦𝐴)    &   (𝜑 → Ⅎ𝑦𝐵)    ⇒   (𝜑 → Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵)
 
20-Jan-2024bj-elsn0 34451 If the intersection of two classes is a set, then these classes are equal if and only if one is an element of the singleton formed on the other. Stronger form of elsng 4584 and elsn2g 4606 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
20-Jan-2024bnj1441 32116 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2389. See bnj1441g 32117 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) (New usage is discouraged.)
(𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴)    &   (𝜑 → ∀𝑦𝜑)    ⇒   (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑})
 
20-Jan-2024ringlsmss 30951 Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐸 ⊆ 𝐵)    &   (𝜑 → 𝐹 ⊆ 𝐵)    ⇒   (𝜑 → (𝐸 × 𝐹) ⊆ 𝐵)
 
20-Jan-2024elringlsm 30950 Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐸 ⊆ 𝐵)    &   (𝜑 → 𝐹 ⊆ 𝐵)    ⇒   (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦)))
 
20-Jan-2024cygabl 19013 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 
20-Jan-2024cycsubmcmn 19011 The set of nonnegative integer powers of an element 𝐴 of a monoid forms a commutative monoid. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    ⇒   ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ CMnd)
 
20-Jan-2024cycsubmcom 18350 The operation of a monoid is commutative over the set of nonnegative integer powers of an element 𝐴 of the monoid. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    &    + = (+g‘𝐺)    ⇒   (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
20-Jan-2024cyccom 18349 Condition for an operation to be commutative. Lemma for cycsubmcom 18350 and cygabl 19013. Formerly part of proof for cygabl 19013. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 20-Jan-2024.)
(𝜑 → ∀𝑐 ∈ 𝐶 ∃𝑥 ∈ 𝑍 𝑐 = (𝑥 · 𝐴))    &   (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑛 ∈ 𝑍 ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴)))    &   (𝜑 → 𝑋 ∈ 𝐶)    &   (𝜑 → 𝑌 ∈ 𝐶)    &   (𝜑 → 𝑍 ⊆ ℂ)    ⇒   (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
20-Jan-2024mndinvmod 17944 Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃*𝑤 ∈ 𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
 
20-Jan-2024ccat2s1p2 13989 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.)
(𝑌 ∈ 𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌)
 
20-Jan-2024ccat2s1p1 13988 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.)
(𝑋 ∈ 𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘0) = 𝑋)
 
20-Jan-2024ccats1val1 13984 Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by JJ, 20-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (𝑊‘𝐼))
 
20-Jan-2024cbviinv 4969 Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2389. See cbviinvg 4971 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
 
20-Jan-2024cbviunv 4968 Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.) Add disjoint variable condition to avoid ax-13 2389. See cbviunvg 4970 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
 
20-Jan-2024cbviin 4965 Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2389. See cbviing 4967 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐵    &   Ⅎ𝑥𝐶    &   (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
 
20-Jan-2024cbviun 4964 Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) Add disjoint variable condition to avoid ax-13 2389. See cbviung 4966 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐵    &   Ⅎ𝑥𝐶    &   (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
 
20-Jan-2024nfiin 4953 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2389. See nfiing 4955 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    ⇒   Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵
 
20-Jan-2024nfiun 4952 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2389. See nfiung 4954 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    ⇒   Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵
 
20-Jan-2024nfrex 3312 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) Add disjoint variable condition to avoid ax-13 2389. See nfrexg 3313 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑
 
20-Jan-2024nfrexd 3310 Deduction version of nfrex 3312. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2389. See nfrexdg 3311 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)
 
20-Jan-2024nfaba1 2989 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2389. See nfaba1g 2990 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}
 
20-Jan-2024nfab 2987 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2389. See nfabg 2988 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥{𝑦 ∣ 𝜑}
 
20-Jan-2024hblem 2946 Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) Add disjoint variable condition to avoid ax-13 2389. See hblemg 2947 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
 
20-Jan-2024nfsab 2815 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2389. See nfsabg 2816 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
 
20-Jan-2024hbab 2813 Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2389. See hbabg 2814 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝜑 → ∀𝑥𝜑)    ⇒   (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
 
20-Jan-2024sb1v 2094 One direction of sb5 2275, provable from fewer axioms. Version of sb1 2502 with a disjoint variable condition using fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Wolf Lammen, 20-Jan-2024.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
 
19-Jan-2024mxidlnzr 30980 A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)
 
19-Jan-2024mxidln1 30979 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1 ∈ 𝑀)
 
19-Jan-2024mxidlmax 30978 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    ⇒   (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
 
19-Jan-2024mxidlnr 30977 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
 
19-Jan-2024mxidlidl 30976 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
 
19-Jan-2024ismxidl 30975 The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
 
19-Jan-2024mxidlval 30974 The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))})
 
19-Jan-2024df-mxidl 30973 Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
 
18-Jan-2024ccatval1 13933 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆‘𝐼))
 
18-Jan-2024ccat0 13932 The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) = ∅ ↔ (𝑆 = ∅ ∧ 𝑇 = ∅)))
 
17-Jan-2024cbvmptv 5172 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 ax-13 2389. See cbvmptvg 5173 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
 
17-Jan-2024cbvmpt 5170 Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) Add disjoint variable condition to avoid ax-13 2389. See cbvmptg 5171 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
Ⅎ𝑦𝐵    &   Ⅎ𝑥𝐶    &   (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
 
17-Jan-2024cbvmptf 5168 Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2389. See cbvmptfg 5169 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    &   Ⅎ𝑥𝐶    &   (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)
 
17-Jan-2024cbvopab1 5142 Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2389. See cbvopab1g 5143 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
Ⅎ𝑧𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
16-Jan-2024qsidom 30971 An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))
 
16-Jan-2024qsidomlem2 30970 A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)
 
16-Jan-2024qsidomlem1 30969 If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
 
16-Jan-2024qusxpid 30932 The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵))
 
16-Jan-2024qsxpid 30931 The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
(𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
 
15-Jan-2024rspsnid 30941 A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (𝐾‘{𝐺}))
 
15-Jan-2024rspsnel 30940 Membership in a principal ideal. Analogous to lspsnel 19778. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝐾 = (RSpan‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥 ∈ 𝐵 𝐼 = (𝑥 · 𝑋)))
 
15-Jan-2024qustrivr 30934 Converse of qustriv 30933. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐻))    ⇒   ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → 𝐻 = 𝐵)
 
15-Jan-2024qustriv 30933 The quotient of a group 𝐺 by itself is trivial. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐵))    ⇒   (𝐺 ∈ Grp → (Base‘𝑄) = {𝐵})
 
15-Jan-2024eqg0el 30930 Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
∼ = (𝐺 ~QG 𝐻)    ⇒   ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))
 
15-Jan-2024ecxpid 30929 The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
(𝑋 ∈ 𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
 
14-Jan-2024cringm4 30966 Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))
 
14-Jan-2024lidlnsg 30965 An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
 
14-Jan-2024df-prmidl 30957 Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵 ⊆ 𝐼 for ideals 𝐴 and 𝐵, either 𝐴 ⊆ 𝐼 or 𝐵 ⊆ 𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 30962 and isprmidlc 30967. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
 
14-Jan-2024wlklenvclwlk 27439 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)
(𝑊 ∈ Word (Vtx‘𝐺) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊)))
 
14-Jan-2024ccat2s1len 13980 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 14-Jan-2024.)
(♯‘(⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)) = 2
 
13-Jan-2024dvdemo2 5278 Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑧 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑦).

That theorem bundles the theorems (∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (∃𝑥(𝑥 = 𝑥 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint) and (∃𝑥(𝑥 = 𝑧 → 𝑧 ∈ 𝑥) with 𝑥, 𝑧 disjoint).

Compare with dvdemo1 5277, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5277 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5278 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑦 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ax-10 2144, ax-11 2160, ax-12 2176, ax-13 2389. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
 
13-Jan-2024dvdemo1 5277 Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑦 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑧).

That theorem bundles the theorems (∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (∃𝑥(𝑥 = 𝑦 → 𝑥 ∈ 𝑥) with 𝑥, 𝑦 disjoint) and (∃𝑥(𝑥 = 𝑦 → 𝑦 ∈ 𝑥) with 𝑥, 𝑦 disjoint).

Compare with dvdemo2 5278, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5278 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.)

Note that dvdemo1 5277 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑧 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require ax-11 2160 nor ax-13 2389. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

∃𝑥(𝑥 = 𝑦 → 𝑧 ∈ 𝑥)
 
12-Jan-2024prmidlc 30968 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃))
 
12-Jan-2024isprmidlc 30967 The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))
 
12-Jan-2024prmidlidl 30964 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
 
12-Jan-2024prmidl2 30962 A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 35352 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
 
12-Jan-2024prmidl 30961 The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
 
12-Jan-2024prmidlnr 30960 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ≠ 𝐵)
 
12-Jan-2024isprmidl 30959 The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
 
12-Jan-2024prmidlval 30958 The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
 
10-Jan-2024nfcsbw 3912 Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3913 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵
 
10-Jan-2024csbcow 3901 Composition law for chained substitutions into a class. Version of csbco 3902 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵
 
10-Jan-2024cbvcsbw 3896 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on 𝐴. Version of cbvcsb 3897 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝐶    &   Ⅎ𝑥𝐷    &   (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑦⦌𝐷
 
10-Jan-2024cbvsbcvw 3808 Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3810 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 30-Sep-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvsbcw 3807 Change bound variables in a wff substitution. Version of cbvsbc 3809 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Jeff Hankins, 19-Sep-2009.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
 
10-Jan-2024sbccow 3798 A composition law for class substitution. Version of sbcco 3801 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
 
10-Jan-2024nfsbcw 3797 Bound-variable hypothesis builder for class substitution. Version of nfsbc 3800 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 7-Sep-2014.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥[𝐴 / 𝑦]𝜑
 
10-Jan-2024nfsbcdw 3796 Deduction version of nfsbcw 3797. Version of nfsbcd 3799 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
10-Jan-2024euxfrw 3715 Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr 3717 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
10-Jan-2024euxfr2w 3714 Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Version of euxfr2 3716 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 14-Nov-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
𝐴 ∈ V    &   ∃*𝑦 𝑥 = 𝐴    ⇒   (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦𝜑)
 
10-Jan-2024cbvrabw 3492 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3493 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
 
10-Jan-2024cbvrexsvw 3471 Change bound variable by using a substitution. Version of cbvrexsv 3473 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 2-Mar-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
(∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
 
10-Jan-2024cbvralsvw 3470 Change bound variable by using a substitution. Version of cbvralsv 3472 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 20-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
(∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
 
10-Jan-2024cbvral3vw 3466 Change bound variables of triple restricted universal quantification, using implicit substitution. Version of cbvral3v 3469 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-May-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑤 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑣 → (𝜒 ↔ 𝜃))    &   (𝑧 = 𝑢 → (𝜃 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ∀𝑢 ∈ 𝐶 𝜓)
 
10-Jan-2024cbvrex2vw 3465 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 3468 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvral2vw 3464 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 3467 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜒))    &   (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐵 𝜓)
 
10-Jan-2024cbvreuvw 3454 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3457 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrexvw 3453 Change the bound variable of a restricted existential quantifier using implicit substitution. Version of cbvrexv 3456 with a disjoint variable condition, which does not require ax-10 2144, ax-11 2160, ax-12 2176, ax-13 2389. (Contributed by NM, 2-Jun-1998.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvralvw 3452 Change the bound variable of a restricted universal quantifier using implicit substitution. Version of cbvralv 3455 with a disjoint variable condition, which does not require ax-10 2144, ax-11 2160, ax-12 2176, ax-13 2389. (Contributed by NM, 28-Jan-1997.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrmow 3447 Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3451 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvreuw 3446 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3450 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrexw 3445 Rule used to change bound variables, using implicit substitution. Version of cbvrex 3449 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvralw 3444 Rule used to change bound variables, using implicit substitution. Version of cbvral 3448 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvrexfw 3441 Rule used to change bound variables, using implicit substitution. Version of cbvrexf 3443 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024cbvralfw 3440 Rule used to change bound variables, using implicit substitution. Version of cbvralf 3442 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfrabw 3388 A variable not free in a wff remains so in a restricted class abstraction. Version of nfrab 3389 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝐴    ⇒   Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑}
 
10-Jan-2024nfrmow 3378 Bound-variable hypothesis builder for restricted uniqueness. Version of nfrmo 3380 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑
 
10-Jan-2024nfreuw 3377 Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3379 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑
 
10-Jan-2024nfra2w 3230 Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 41200. Version of nfra2 3231 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
 
10-Jan-2024nfralw 3228 Bound-variable hypothesis builder for restricted quantification. Version of nfral 3229 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
 
10-Jan-2024nfraldw 3226 Deduction version of nfralw 3228. Version of nfrald 3227 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
 
10-Jan-2024nfabdw 3003 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 3004 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
 
10-Jan-2024clelsb3fw 2984 Substitution applied to an atomic wff (class version of elsb3 2121). Version of clelsb3f 2985 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝐴    ⇒   ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
 
10-Jan-2024cbvabw 2893 Rule used to change bound variables, using implicit substitution. Version of cbvab 2894 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
 
10-Jan-2024cbveuw 2689 Version of cbveu 2690 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
10-Jan-2024cbvmow 2687 Rule used to change bound variables, using implicit substitution. Version of cbvmo 2688 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
10-Jan-2024nfeuw 2678 Bound-variable hypothesis builder for the unique existential quantifier. Version of nfeu 2679 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 8-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃!𝑦𝜑
 
10-Jan-2024nfeudw 2676 Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2679. Version of nfeud 2677 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
10-Jan-2024cbval2v 2362 Rule used to change bound variables, using implicit substitution. Version of cbval2 2431 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑧𝜑    &   Ⅎ𝑤𝜑    &   Ⅎ𝑥𝜓    &   Ⅎ𝑦𝜓    &   ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
 
10-Jan-2024cbvexdw 2358 Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2428 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 2-Jan-2002.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
10-Jan-2024cbvaldw 2357 Deduction used to change bound variables, using implicit substitution. Version of cbvald 2427 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 2-Jan-2002.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
10-Jan-2024cbv2w 2356 Rule used to change bound variables, using implicit substitution. Version of cbv2 2422 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
10-Jan-2024hbsbw 2350 If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2566 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.)
(𝜑 → ∀𝑧𝜑)    ⇒   ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 
10-Jan-2024sbiedwOLD 2332 Obsolete version of sbiedw 2331 as of 28-Jan-2024. (Contributed by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Ⅎ𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
 
10-Jan-2024nfnaew 2152 All variables are effectively bound in a distinct variable specifier. Version of nfnae 2455 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
10-Jan-20242sbievw 2104 Conversion of double implicit substitution to explicit substitution. Version of 2sbiev 2546 with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))    ⇒   ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)
 
10-Jan-2024cbvex4vw 2048 Rule used to change bound variables, using implicit substitution. Version of cbvex4v 2436 with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))    &   ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒))    ⇒   (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒)
 
10-Jan-2024cbvex2vw 2047 Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2435 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)
 
10-Jan-2024cbval2vw 2046 Rule used to change bound variables, using implicit substitution. Version of cbval2vv 2434 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)
 
7-Jan-2024bj-fvimacnv0 34572 Variant of fvimacnv 6826 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with definition df-afv 43326. (Contributed by BJ, 7-Jan-2024.)
((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
 
6-Jan-2024bj-rveccvec 34590 Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec)
 
6-Jan-2024bj-rvecsscvec 34589 Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ ℂVec
 
6-Jan-2024bj-rvecsscmod 34588 Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ ℂMod
 
6-Jan-2024bj-rveccmod 34587 Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod)
 
6-Jan-2024bj-rvecssvec 34586 Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ LVec
 
6-Jan-2024bj-isrvec2 34585 The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
(𝜑 → (Scalar‘𝑉) = 𝐾)    ⇒   (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LVec ∧ 𝐾 = ℝfld)))
 
6-Jan-2024bj-rvecvec 34584 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec)
 
6-Jan-2024bj-isrvecd 34583 The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
(𝜑 → (Scalar‘𝑉) = 𝐾)    ⇒   (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ 𝐾 = ℝfld)))
 
6-Jan-2024bj-rvecrr 34582 The field of scalars of a real vector space is the field of real numbers. (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → (Scalar‘𝑉) = ℝfld)
 
6-Jan-2024bj-rvecssmod 34581 Real vector spaces are modules. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ LMod
 
6-Jan-2024bj-rvecmod 34580 Real vector spaces are modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod)
 
6-Jan-2024bj-isrvec 34579 The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) = ℝfld))
 
6-Jan-2024bj-isclm 34576 The predicate "is a subcomplex module". (Contributed by BJ, 6-Jan-2024.)
(𝜑 → 𝐹 = (Scalar‘𝑊))    &   (𝜑 → 𝐾 = (Base‘𝐹))    ⇒   (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
 
6-Jan-2024bj-rrdrg 34575 The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.)
ℝfld ∈ DivRing
 
6-Jan-2024bj-flddrng 34574 Fields are division rings. (Contributed by BJ, 6-Jan-2024.)
Field ⊆ DivRing
 
6-Jan-2024bj-isvec 34573 The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.)
(𝜑 → 𝐾 = (Scalar‘𝑉))    ⇒   (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
 
5-Jan-2024bj-grpssmndel 34561 Groups are monoids (elemental version). Shorter proof of grpmnd 18113. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
(𝐴 ∈ Grp → 𝐴 ∈ Mnd)
 
5-Jan-2024bj-grpssmnd 34560 Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
Grp ⊆ Mnd
 
4-Jan-2024cycpmco2lem7 30778 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑 → 𝐾 ∈ ran 𝑊)    &   (𝜑 → 𝐾 ≠ 𝐽)    &   (𝜑 → (◡𝑈‘𝐾) ∈ (0..^𝐸))    ⇒   (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))
 
4-Jan-2024cycpmco2lem6 30777 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑 → 𝐾 ∈ ran 𝑊)    &   (𝜑 → 𝐾 ≠ 𝐼)    &   (𝜑 → (◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))    ⇒   (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))
 
4-Jan-2024cycpmco2lem5 30776 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑 → 𝐾 ∈ ran 𝑊)    &   (𝜑 → (◡𝑈‘𝐾) = ((♯‘𝑈) − 1))    ⇒   (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾))
 
4-Jan-2024cycpmco2lem4 30775 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼))
 
4-Jan-2024cycpmco2lem3 30774 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
 
4-Jan-2024cycpmco2lem2 30773 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → (𝑈‘𝐸) = 𝐼)
 
4-Jan-2024cycpmco2lem1 30772 Lemma for cycpmco2 30779. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑊)‘𝐽))
 
4-Jan-2024cycpmco2rn 30771 The orbit of the composition of a cyclic permutation and a well-chosen transposition is one element more than the orbit of the original permutation. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
 
4-Jan-2024cycpmco2f1 30770 The word U used in cycpmco2 30779 is injective, so it can represent a cycle and form a cyclic permutation (𝑀‘𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → 𝑈:dom 𝑈–1-1→𝐷)
 
4-Jan-2024offsplitfpar 7818 Express the function operation map ∘f by the functions defined in fsplit 7815 and fpar 7814. (Contributed by AV, 4-Jan-2024.)
𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = (◡(1st ↾ I ) ↾ 𝐴)    ⇒   (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))
 
3-Jan-2024simpcntrab 43134 The center of a simple group is trivial or the group is abelian. (Contributed by SS, 3-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   𝑍 = (Cntr‘𝐺)    &   (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel))
 
3-Jan-2024ex-fpar 28244 Formalized example provided in the comment for fpar 7814. (Contributed by AV, 3-Jan-2024.)
𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝐴 = (0[,)+∞)    &   𝐵 = ℝ    &   𝐹 = (√ ↾ 𝐴)    &   𝐺 = (sin ↾ 𝐵)    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
 
3-Jan-2024isfrgr 28042 The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
 
3-Jan-2024df-frgr 28041 Define the class of all friendship graphs: a simple graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. This condition is called the friendship condition , see definition in [MertziosUnger] p. 152. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.)
FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
 
3-Jan-2024fsplitfpar 7817 Merge two functions with a common argument in parallel. Combination of fsplit 7815 and fpar 7814. (Contributed by AV, 3-Jan-2024.)
𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = (◡(1st ↾ I ) ↾ 𝐴)    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
 
2-Jan-2024cycpmco2 30779 The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ dom 𝑀)    &   (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑 → 𝐽 ∈ ran 𝑊)    &   𝐸 = ((◡𝑊‘𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    ⇒   (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀‘𝑈))
 
1-Jan-2024fzom1ne1 30527 Elementhood in a half-open interval, except the lower bound, shifted by one. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1)))
 
1-Jan-2024fzone1 30526 Elementhood in a half-open interval, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)..^𝑁))
 
1-Jan-2024fzm1ne1 30515 Elementhood of an integer and its predecessor in finite intervals of integers. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → (𝐾 − 1) ∈ (𝑀...(𝑁 − 1)))
 
1-Jan-2024fzne1 30514 Elementhood in a finite set of sequential integers, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ≠ 𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁))
 
1-Jan-2024ccatlen 13930 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.)
((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
 
31-Dec-2023bj-elpwg 34349 If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4545 and elpw2g 5250 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
 
31-Dec-2023bj-bixor 33929 Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023.)
((𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ⊻ (𝜓 ↔ 𝜒)))
 
31-Dec-2023rinvmod 18932 Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovmo 7388. (Contributed by AV, 31-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃*𝑤 ∈ 𝐵 (𝐴 + 𝑤) = 0 )
 
31-Dec-2023fsplit 7815 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7814 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5466 with df-id 5463. (Revised by BJ, 31-Dec-2023.)
◡(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
 
31-Dec-2023pwidb 4565 A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023.)
(𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)
 
31-Dec-2023elpw 4546 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) (Proof shortened by BJ, 31-Dec-2023.)
𝐴 ∈ V    ⇒   (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
 
31-Dec-2023elpwg 4545 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5250. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
 
30-Dec-2023eqwrd 13912 Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)
((𝑈 ∈ Word 𝑆 ∧ 𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈‘𝑖) = (𝑊‘𝑖))))
 
29-Dec-2023mndbn0 17930 The base set of a monoid is not empty. Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Mnd → 𝐵 ≠ ∅)
 
29-Dec-2023ncanth 7115 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5222). Specifically, the identity function maps the universe onto its power class. Compare canth 7114 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3774): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4838). See also the remark in ru 3774 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.)

I :V–onto→𝒫 V
 
28-Dec-2023bj-opabssvv 34446 A variant of relopabiv 5696 (which could be proved from it, similarly to relxp 5576 from xpss 5574). (Contributed by BJ, 28-Dec-2023.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
 
28-Dec-2023cycsubm 18348 The set of nonnegative integer powers of an element 𝐴 of a monoid forms a submonoid containing 𝐴 (see cycsubmcl 18347), called the cyclic monoid generated by the element 𝐴. This corresponds to the statement in [Lang] p. 6. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    ⇒   ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ (SubMnd‘𝐺))
 
28-Dec-2023cycsubmcl 18347 The set of nonnegative integer powers of an element 𝐴 contains 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    ⇒   (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶)
 
28-Dec-2023cycsubmel 18346 Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    ⇒   (𝑋 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
 
28-Dec-2023mulgnn0gsum 18237 Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
28-Dec-2023mulgnngsum 18236 Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)    ⇒   ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
28-Dec-2023cnvrescnv 6055 Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)
◡(◡𝑅 ↾ 𝐵) = (𝑅 ∩ (V × 𝐵))
 
28-Dec-2023alcomiw 2049 Weak version of alcom 2162. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.)
(𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑)
 
27-Dec-2023gsumfsupp 44096 A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   𝐼 = (𝐹 supp 0 )    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐹 finSupp 0 )    ⇒   (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹))
 
27-Dec-2023nn0mnd 44093 The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂfld ↾s ℕ0). (Contributed by AV, 27-Dec-2023.)
𝑀 = {⟨(Base‘ndx), ℕ0⟩, ⟨(+g‘ndx), + ⟩}    ⇒   𝑀 ∈ Mnd
 
27-Dec-2023bj-idres 34456 Alternate expression for the restricted identity relation. The advantage of that expression is to expose it as a "bounded" class, being included in the Cartesian square of the restricting class. (Contributed by BJ, 27-Dec-2023.)

This is an alternate of idinxpresid 5918 (see idinxpres 5917). See also elrid 5916 and elidinxp 5914. (Proof modification is discouraged.)

( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
 
27-Dec-2023bj-ideqgALT 34454 Alternate proof of bj-ideqg 34453 from brabga 5424 instead of bj-opelid 34452 itself proved from bj-opelidb 34448. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
 
27-Dec-2023bj-inexeqex 34450 Lemma for bj-opelid 34452 (but not specific to the identity relation): if the intersection of two classes is a set and the two classes are equal, then both are sets (all three classes are equal, so they all belong to 𝑉, but it is more convenient to have V in the consequent for theorems using it). (Contributed by BJ, 27-Dec-2023.)
(((𝐴 ∩ 𝐵) ∈ 𝑉 ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
27-Dec-2023bj-opelidb1 34449 Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 34448 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
27-Dec-2023bj-opelidb 34448 Characterization of the ordered pair elements of the identity relation.

Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than ⊤ which already appears in the proof. Here for instance this could be the definition I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.)

(⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
 
27-Dec-2023bj-funidres 34447 The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.)

TODO: relabel funi 6390 to funid.

Fun ( I ↾ 𝑉)
 
27-Dec-2023bj-opelrelex 34440 The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5607, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
27-Dec-2023copsex2b 34436 Biconditional form of copsex2d 34435. TODO: prove a relative version, that is, with ∃𝑥 ∈ 𝑉∃𝑦 ∈ 𝑊...(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊). (Contributed by BJ, 27-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 
27-Dec-2023bj-reabeq 34343 Relative form of abeq2 2948. (Contributed by BJ, 27-Dec-2023.)
((𝑉 ∩ 𝐴) = {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝜑))
 
27-Dec-2023bj-rcleq 34342 Relative version of dfcleq 2818. (Contributed by BJ, 27-Dec-2023.)
((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
 
27-Dec-2023bj-rcleqf 34341 Relative version of cleqf 3013. (Contributed by BJ, 27-Dec-2023.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    &   Ⅎ𝑥𝑉    ⇒   ((𝑉 ∩ 𝐴) = (𝑉 ∩ 𝐵) ↔ ∀𝑥 ∈ 𝑉 (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
 
27-Dec-2023gsumxp2 19103 Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑊)    &   (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑 → 𝐹 finSupp 0 )    ⇒   (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝐴 ↦ (𝐺 Σg (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))))
 
27-Dec-2023mndlsmidm 18799 Subgroup sum is idempotent for monoids. This corresponds to the observation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
⊕ = (LSSum‘𝐺)    &   𝐵 = (Base‘𝐺)    ⇒   (𝐺 ∈ Mnd → (𝐵 ⊕ 𝐵) = 𝐵)
 
27-Dec-2023lsmidm 18791 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.)
⊕ = (LSSum‘𝐺)    ⇒   (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈)
 
27-Dec-2023smndlsmidm 18784 The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.)
⊕ = (LSSum‘𝐺)    ⇒   (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈)
 
27-Dec-2023fnresi 6479 The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004.) (Proof shortened by BJ, 27-Dec-2023.)
( I ↾ 𝐴) Fn 𝐴
 
26-Dec-2023bj-ififc 33919 A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.)
(𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋 ∈ 𝐴, 𝑋 ∈ 𝐵))
 
26-Dec-2023bj-dfif 33918 Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑 ∧ 𝑥 ∈ 𝐴) ∨ (¬ 𝜑 ∧ 𝑥 ∈ 𝐵))}
 
26-Dec-2023gsumreidx 19040 Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵)    &   (𝜑 → 𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    ⇒   (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ 𝐻)))
 
26-Dec-2023gsumccat 18009 Homomorphic property of composites. Second formula in [Lang] p. 4. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
26-Dec-2023gsumsgrpccat 18007 Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 18009. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   ((𝐺 ∈ Smgrp ∧ (𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵) ∧ (𝑊 ≠ ∅ ∧ 𝑋 ≠ ∅)) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
26-Dec-2023gsumsplit1r 17900 Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵)    ⇒   (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
26-Dec-2023lidrididd 17883 If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 17882) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
(𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    &   𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    ⇒   (𝜑 → 𝐿 = 0 )
 
26-Dec-2023lidrideqd 17882 If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
(𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    ⇒   (𝜑 → 𝐿 = 𝑅)
 
26-Dec-2023rnep 5800 The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.)
ran E = (V ∖ {∅})
 
26-Dec-2023dmep 5796 The domain of the membership relation is the universal class. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof shortened by BJ, 26-Dec-2023.)
dom E = V
 
25-Dec-2023sn-00id 39237 00id 10818 proven without ax-mulcom 10604 but using ax-1ne0 10609. (Though note that the current version of 00id 10818 can be changed to avoid ax-icn 10599, ax-addcl 10600, ax-mulcl 10602, ax-i2m1 10608, ax-cnre 10613. Most of this is by using 0cnALT3 39159 instead of 0cn 10636). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
(0 + 0) = 0
 
25-Dec-2023sn-00idlem3 39236 Lemma for sn-00id 39237. (Contributed by SN, 25-Dec-2023.)
((0 −ℝ 0) = 1 → (0 + 0) = 0)
 
25-Dec-2023sn-00idlem2 39235 Lemma for sn-00id 39237. (Contributed by SN, 25-Dec-2023.)
((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1)
 
25-Dec-2023sn-00idlem1 39234 Lemma for sn-00id 39237. (Contributed by SN, 25-Dec-2023.)
(𝐴 ∈ ℝ → (𝐴 · (0 −ℝ 0)) = (𝐴 −ℝ 𝐴))
 
25-Dec-2023re1m1e0m0 39233 Equality of two left-additive identities. See resubidaddid1 39231. Uses ax-i2m1 10608. (Contributed by SN, 25-Dec-2023.)
(1 −ℝ 1) = (0 −ℝ 0)
 
25-Dec-2023bj-ideqg1ALT 34461 Alternate proof of bj-ideqg1 using brabga 5424 instead of the "unbounded" version bj-brab2a1 34445 or brab2a 5647. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 34453 is in the main section.

((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
 
25-Dec-2023bj-idreseq 34458 Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 34453 with V substituted for 𝑉 is a direct consequence of bj-idreseq 34458. This is a strengthening of resieq 5867 which should be proved from it (note that currently, resieq 5867 relies on ideq 5726). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → .... (Contributed by BJ, 25-Dec-2023.)
((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵 ↔ 𝐴 = 𝐵))
 
25-Dec-2023bj-brab2a1 34445 "Unbounded" version of brab2a 5647. (Contributed by BJ, 25-Dec-2023.)
((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))
 
25-Dec-2023bj-brresdm 34442 If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5863 and brrelex1 5608.

Remark: there are many pairs like bj-opelresdm 34441 / bj-brresdm 34442, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34441 / brrelex12 5607 or the opelopabg 5428 / brabg 5429 family). They are straightforwardly equivalent by df-br 5070. The latter is indeed a very direct definition, introducing a "shorthand", and barely necessary, were it not for the frequency of the expression 𝐴𝑅𝐵. Therefore, in the spirit of "definitions are here to be used", most theorems, apart from the most elementary ones, should only have the "br" version, not the "opel" one. (Contributed by BJ, 25-Dec-2023.)

(𝐴(𝑅 ↾ 𝑋)𝐵 → 𝐴 ∈ 𝑋)
 
25-Dec-2023bj-opelresdm 34441 If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5862. (Contributed by BJ, 25-Dec-2023.)
(⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)
 
25-Dec-2023copsex2d 34435 Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
 
25-Dec-2023bj-nfexd 34434 Variant of nfexd 2347. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∃𝑦𝜓)
 
25-Dec-2023bj-nfald 34433 Variant of nfald 2346. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦𝜓)
 
25-Dec-2023bj-exlimd 33962 A slightly more general exlimd 2217. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2217. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (∃𝑥𝜃 → 𝜏))    &   (𝜓 → (𝜒 → 𝜃))    ⇒   (𝜑 → (∃𝑥𝜒 → 𝜏))
 
25-Dec-2023bj-sylge 33961 Dual statement of sylg 1822 (the final "e" in the label stands for "existential (version of sylg 1822)". Variant of exlimih 2296. (Contributed by BJ, 25-Dec-2023.)
(∃𝑥𝜑 → 𝜓)    &   (𝜒 → 𝜑)    ⇒   (∃𝑥𝜒 → 𝜓)
 
25-Dec-2023bj-alrimd 33957 A slightly more general alrimd 2214. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2214. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (𝜒 → ∀𝑥𝜃))    &   (𝜓 → (𝜃 → 𝜏))    ⇒   (𝜑 → (𝜒 → ∀𝑥𝜏))
 
24-Dec-2023bj-imdirid 34479 Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)
(𝜑 → 𝐴 ∈ 𝑈)    ⇒   (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
 
24-Dec-2023bj-ideqg1 34460 For sets, the identity relation is the same thing as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ, 24-Dec-2023.)

TODO: delete once bj-ideqg 34453 is in the main section.

((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
 
24-Dec-2023bj-idreseqb 34459 Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)
(𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵))
 
24-Dec-2023bj-ideqb 34455 Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
(𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
24-Dec-2023bj-ideqg 34453 Characterization of the classes related by the identity relation when their intersection is a set. Note that the antecedent is more general than either class being a set. (Contributed by NM, 30-Apr-2004.) Weaken the antecedent to sethood of the intersection. (Revised by BJ, 24-Dec-2023.)

TODO: replace ideqg 5725, or at least prove ideqg 5725 from it.

((𝐴 ∩ 𝐵) ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
 
23-Dec-2023df-invdir 34481 Definition of the functionalized inverse image, which maps a binary relation between two given sets to its associated inverse image relation. (Contributed by BJ, 23-Dec-2023.)
𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ 𝑥 = (◡𝑟 “ 𝑦))}))
 
23-Dec-2023idfn 6478 The identity relation is a function on the universal class. See also funi 6390. (Contributed by BJ, 23-Dec-2023.)
I Fn V
 
23-Dec-2023idinxpresid 5918 The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.)
( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
 
23-Dec-2023idinxpres 5917 The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 5918) to cartesian product. (Revised by BJ, 23-Dec-2023.)
( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴 ∩ 𝐵))
 
21-Dec-2023pwvabrel 5606 The powerclass of the cartesian square of the universal class is the class of all sets which are binary relations. (Contributed by BJ, 21-Dec-2023.)
𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}
 
18-Dec-2023norasslem3 1531 This lemma specializes biorf 933 suitably for the proof of norass 1532. (Contributed by Wolf Lammen, 18-Dec-2023.)
(¬ 𝜑 → ((𝜓 → 𝜒) ↔ ((𝜑 ∨ 𝜓) → 𝜒)))
 
18-Dec-2023norasslem2 1530 This lemma specializes biimt 363 suitably for the proof of norass 1532. (Contributed by Wolf Lammen, 18-Dec-2023.)
(𝜑 → (𝜓 ↔ ((𝜑 ∨ 𝜒) → 𝜓)))
 
18-Dec-2023norasslem1 1529 This lemma shows the equivalence of two expressions, used in norass 1532. (Contributed by Wolf Lammen, 18-Dec-2023.)
(((𝜑 ∨ 𝜓) → 𝜒) ↔ ((𝜑 ⊽ 𝜓) ∨ 𝜒))
 
18-Dec-2023impimprbi 826 An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 477, but ↔ is a weaker operator than ∧. Note that an implication and its reverse can never be simultaneously false, because of pm2.521 178. (Contributed by Wolf Lammen, 18-Dec-2023.)
((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑)))
 
17-Dec-2023brabd 34444 Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))
 
17-Dec-2023brabd0 34443 Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))
 
17-Dec-2023opelopabbv 34439 Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 
17-Dec-2023opelopabb 34438 Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 
17-Dec-2023opelopabd 34437 Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
 
17-Dec-2023subne0nn 30541 A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.)
(𝜑 → 𝑀 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℂ)    &   (𝜑 → (𝑀 − 𝑁) ∈ ℕ0)    &   (𝜑 → 𝑀 ≠ 𝑁)    ⇒   (𝜑 → (𝑀 − 𝑁) ∈ ℕ)
 
17-Dec-2023hadcoma 1598 Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
 
17-Dec-2023falnorfal 1590 A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊥ ⊽ ⊥) ↔ ⊤)
 
17-Dec-2023trunorfal 1587 A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊤ ⊽ ⊥) ↔ ⊥)
 
17-Dec-2023norass 1532 A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass 1500. Remark: Like alternative denial, joint denial is also commutative, see norcom 1522. (Contributed by RP, 29-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊽ 𝜓) ⊽ 𝜒) ↔ (𝜑 ⊽ (𝜓 ⊽ 𝜒))))
 
16-Dec-2023bj-imdirval3 34478 Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
(𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))    ⇒   (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) ∧ (𝑅 “ 𝑋) = 𝑌)))
 
16-Dec-2023bj-imdirval2 34477 Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
(𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝑅 ⊆ (𝐴 × 𝐵))    ⇒   (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑅 “ 𝑥) = 𝑦)})
 
16-Dec-2023bj-imdirval 34476 Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
(𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑦 ⊆ 𝐵) ∧ (𝑟 “ 𝑥) = 𝑦)}))
 
16-Dec-2023df-imdir 34475 Definition of the functionalized direct image, which maps a binary relation between two given sets to its associated direct image relation. (Contributed by BJ, 16-Dec-2023.)
𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ⊆ 𝑎 ∧ 𝑦 ⊆ 𝑏) ∧ (𝑟 “ 𝑥) = 𝑦)}))
 
16-Dec-2023bj-pwvrelb 34218 Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023.)
(𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))
 
16-Dec-2023pwvrel 5605 A set is a binary relation if and only if it belongs to the powerclass of the cartesian square of the universal class. (Contributed by Peter Mazsa, 14-Jun-2018.) (Revised by BJ, 16-Dec-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
 
15-Dec-2023selvval2lem3 39140 The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐽 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))
 
15-Dec-2023selvval2lem2 39139 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐽 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇))
 
15-Dec-2023selvval2lem1 39138 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼 ∖ 𝐽) and we have 𝐽 ∈ 𝑊 instead of 𝐽 ⊆ 𝐼. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐽 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ CRing)    ⇒   (𝜑 → 𝑇 ∈ AssAlg)
 
15-Dec-2023equs5av 2278 A property related to substitution that replaces the distinctor from equs5 2482 to a disjoint variable condition. Version of equs5a 2479 with a disjoint variable condition, which does not require ax-13 2389. See also sb56 2276. (Contributed by NM, 2-Feb-2007.) (Revised by Gino Giotto, 15-Dec-2023.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
 
15-Dec-2023sb4av 2243 Version of sb4a 2508 with a disjoint variable condition, which does not require ax-13 2389. The distinctor antecedent from sb4b 2498 is replaced by a disjoint variable condition in this theorem. (Contributed by NM, 2-Feb-2007.) (Revised by BJ, 15-Dec-2023.)
([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))
 
14-Dec-2023cu3addd 39283 Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    ⇒   (𝜑 → (((𝐴 + 𝐵) + 𝐶)↑3) = (((((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))) + (((3 · ((𝐴↑2) · 𝐶)) + (((3 · 2) · (𝐴 · 𝐵)) · 𝐶)) + (3 · ((𝐵↑2) · 𝐶)))) + (((3 · (𝐴 · (𝐶↑2))) + (3 · (𝐵 · (𝐶↑2)))) + (𝐶↑3))))
 
14-Dec-2023binom2d 39282 Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
 
14-Dec-2023splfv3 30636 Symbols to the right of a splice are unaffected. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝜑 → 𝑆 ∈ Word 𝐴)    &   (𝜑 → 𝐹 ∈ (0...𝑇))    &   (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑 → 𝑅 ∈ Word 𝐴)    &   (𝜑 → 𝑋 ∈ (0..^((♯‘𝑆) − 𝑇)))    &   (𝜑 → 𝐾 = (𝐹 + (♯‘𝑅)))    ⇒   (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝑋 + 𝐾)) = (𝑆‘(𝑋 + 𝑇)))
 
14-Dec-2023elunsn 30276 Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐶)))
 
14-Dec-2023elfzom1p1elfzo 13120 Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Proof shortened by Thierry Arnoux, 14-Dec-2023.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁))
 
13-Dec-2023sn-1ne2 39164 A proof of 1ne2 11848 without using ax-mulcom 10604, ax-mulass 10606, ax-pre-mulgt0 10617. Based on mul02lem2 10820. (Contributed by SN, 13-Dec-2023.)
1 ≠ 2
 
13-Dec-2023swrdrndisj 30635 Condition for the range of two subwords of an injective word to be disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑀 ∈ (0...𝑁))    &   (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → 𝑂 ∈ (𝑁...𝑃))    &   (𝜑 → 𝑃 ∈ (𝑁...(♯‘𝑊)))    ⇒   (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅)
 
13-Dec-2023swrdrn3 30633 Express the range of a subword. Stronger version of swrdrn2 30632. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁)))
 
13-Dec-2023pfxf1 30622 Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑 → 𝑊 ∈ Word 𝑆)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝑆)    &   (𝜑 → 𝐿 ∈ (0...(♯‘𝑊)))    ⇒   (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆)
 
13-Dec-2023pfxrn3 30621 Express the range of a prefix of a word. Stronger version of pfxrn2 30620. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
 
13-Dec-2023nelun 30277 Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)))
 
12-Dec-2023swrdf1 30634 Condition for a subword to be injective. (Contributed by Thierry Arnoux, 12-Dec-2023.)
(𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑀 ∈ (0...𝑁))    &   (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    ⇒   (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1→𝐷)
 
12-Dec-2023swrdrn2 30632 The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14017. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ ran 𝑊)
 
12-Dec-2023pfxrn2 30620 The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14050. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
 
12-Dec-2023incom 4181 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) (Proof shortened by SN, 12-Dec-2023.)
(𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
 
12-Dec-2023rspcev 3626 Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2144, ax-11 2160, ax-12 2176. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
 
12-Dec-2023rspcv 3621 Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2144, ax-11 2160, ax-12 2176. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))
 
11-Dec-2023ccatf1 30629 Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ Word 𝑆)    &   (𝜑 → 𝐵 ∈ Word 𝑆)    &   (𝜑 → 𝐴:dom 𝐴–1-1→𝑆)    &   (𝜑 → 𝐵:dom 𝐵–1-1→𝑆)    &   (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)    ⇒   (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆)
 
11-Dec-2023s1f1 30623 Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑 → 𝐼 ∈ 𝐷)    ⇒   (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
 
9-Dec-2023bj-wnfnf 34072 When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 34079, bj-nnfe1 34093 and bj-nnfa1 34092. (Contributed by BJ, 9-Dec-2023.)
Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
 
9-Dec-2023bj-wnfenf 34058 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "exists" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓))
 
9-Dec-2023bj-wnfanf 34057 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "forall" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
 
9-Dec-2023bj-wnf2 34056 When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
(∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
 
9-Dec-2023bj-wnf1 34055 When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
 
9-Dec-2023bj-eximcom 33980 A commuted form of exim 1833 which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.)
(∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
 
9-Dec-2023bj-aleximiALT 33979 Alternate proof of aleximi 1831 from exim 1833, which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
9-Dec-2023bj-eximALT 33978 Alternate proof of exim 1833 directly from alim 1810 by using df-ex 1780 (using duality of ∀ and ∃. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
 
9-Dec-2023bj-nfimexal 33963 A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1838) and the converse implication is the join of instances of bj-alrimg 33956 and bj-exlimg 33960 (see 19.38a 1839 and 19.38b 1840). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.)
(((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓)))
 
9-Dec-2023bj-exlimg 33960 The general form of the *exlim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg 33956. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓)))
 
9-Dec-2023bj-alrimg 33956 The general form of the *alrim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg 33960. (Contributed by BJ, 9-Dec-2023.)
((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓 → 𝜒) → (𝜑 → ∀𝑥𝜒)))
 
8-Dec-2023noror 1527 ∨ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑 ∨ 𝜓) ↔ ((𝜑 ⊽ 𝜓) ⊽ (𝜑 ⊽ 𝜓)))
 
8-Dec-2023noran 1525 ∧ is expressible via ⊽. (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))
 
8-Dec-2023nornot 1523 ¬ is expressible via ⊽. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
(¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
 
7-Dec-2023swrdwlk 32377 Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.)
((𝐹(Walks‘𝐺)𝑃 ∧ 𝐵 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 substr ⟨𝐵, 𝐿⟩)(Walks‘𝐺)(𝑃 substr ⟨𝐵, (𝐿 + 1)⟩))
 
7-Dec-2023trunortru 1585 A ⊽ identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
((⊤ ⊽ ⊤) ↔ ⊥)
 
4-Dec-2023bj-nnclav 33888 When ⊥ is substituted for 𝜓, this formula is the Clavius law with a doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
(((𝜑 → 𝜓) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
 
4-Dec-2023revwlkb 32376 Two words represent a walk if and only if their reverses also represent a walk. (Contributed by BTernaryTau, 4-Dec-2023.)
((𝐹 ∈ Word 𝑊 ∧ 𝑃 ∈ Word 𝑈) → (𝐹(Walks‘𝐺)𝑃 ↔ (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃)))
 
3-Dec-2023swrdrevpfx 32367 A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
((𝑊 ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿 − 𝐹))))
 
3-Dec-2023rabbidva 3481 Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.) (Proof shortened by SN, 3-Dec-2023.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})
 
2-Dec-2023xlimlimsupleliminf 42150 A sequence of extended reals converges if and only if its superior limit is smaller than or equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Dec-2023.)
(𝜑 → 𝑀 ∈ ℤ)    &   𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝐹:𝑍⟶ℝ*)    ⇒   (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
 
2-Dec-2023bj-19.41t 34107 Closed form of 19.41 2236 from the same axioms as 19.41v 1949. The same is doable with 19.27 2228, 19.28 2229, 19.31 2235, 19.32 2234, 19.44 2238, 19.45 2239. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)))
 
2-Dec-2023bj-19.42t 34106 Closed form of 19.42 2237 from the same axioms as 19.42v 1953. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
 
2-Dec-2023bj-19.37im 34105 One direction of 19.37 2233 from the same axioms as 19.37imv 1947. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))
 
2-Dec-2023bj-19.36im 34104 One direction of 19.36 2231 from the same axioms as 19.36imv 1945. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
 
2-Dec-2023bj-19.23t 34103 Statement 19.23t 2209 proved from modalK (obsoleting 19.23v 1942). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
 
2-Dec-2023bj-19.21t 34102 Statement 19.21t 2205 proved from modalK (obsoleting 19.21v 1939). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
2-Dec-2023bj-stdpc5t 34101 Alias of bj-nnf-alrim 34088 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2207 proved from modalK (obsoleting stdpc5v 1938). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 34088 instead. (New usaged is discouraged.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
 
2-Dec-2023bj-nnf-exlim 34089 Proof of the closed form of exlimi 2216 from modalK (compare exlimiv 1930). See also bj-sylget2 33959. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓)))
 
2-Dec-2023bj-nnfbid 34086 Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)    ⇒   (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒))
 
2-Dec-2023bj-nnfbit 34085 Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓))
 
2-Dec-2023bj-nnford 34084 Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 34083 and bj-nnfand 34082. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)    ⇒   (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒))
 
2-Dec-2023bj-nnfimd 34080 Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)    ⇒   (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒))
 
2-Dec-2023pfxwlk 32374 A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝐹(Walks‘𝐺)𝑃 ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))
 
2-Dec-2023revpfxsfxrev 32366 The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝑊 ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘(𝑊 prefix 𝐿)) = ((reverse‘𝑊) substr ⟨((♯‘𝑊) − 𝐿), (♯‘𝑊)⟩))
 
1-Dec-2023mpteq12dv 5154 An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Drop ax-10 2144 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023.)
(𝜑 → 𝐴 = 𝐶)    &   (𝜑 → 𝐵 = 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
 
1-Dec-2023sbcbidv 3830 Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) Drop ax-12 2176. (Revised by Gino Giotto, 1-Dec-2023.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))
 
1-Dec-2023rexab2 3694 Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2115. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒))
 
1-Dec-2023ralab2 3691 Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2115. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   (∀𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∀𝑦(𝜑 → 𝜒))
 
1-Dec-2023ceqsexgv 3650 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2144 and ax-12 2176. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))
 
30-Nov-2023bj-stabpeirce 33893 Over minimal implicational calculus, Peirce's law is implied by the (classical refutation equivalent of) the double negation of the stability of any proposition. (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)
((((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → (((𝜓 → 𝜑) → 𝜓) → 𝜓))
 
30-Nov-2023bj-peircestab 33892 Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any proposition (that is the interpretation when ⊥ is substitued for 𝜓). (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)
(((((𝜑 → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
 
30-Nov-2023revwlk 32375 The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)
(𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))
 
29-Nov-2023disjdifr 30278 A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
((𝐵 ∖ 𝐴) ∩ 𝐴) = ∅
 
27-Nov-2023nrmo 33762 "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(𝑥 ∈ 𝐴 → ¬ 𝜑)    ⇒   ∃*𝑥 ∈ 𝐴 𝜑
 
27-Nov-2023tocyccntz 30790 All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑍 = (Cntz‘𝑆)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥)    &   (𝜑 → 𝐴 ⊆ dom 𝑀)    ⇒   (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴)))
 
27-Nov-2023cntzsnid 30700 The centralizer of the identity element is the whole base set. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)    &    0 = (0g‘𝑀)    ⇒   (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵)
 
27-Nov-2023cntzun 30699 The centralizer of a union is the intersection of the centralizers. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)    ⇒   ((𝑋 ⊆ 𝐵 ∧ 𝑌 ⊆ 𝐵) → (𝑍‘(𝑋 ∪ 𝑌)) = ((𝑍‘𝑋) ∩ (𝑍‘𝑌)))
 
27-Nov-2023disjxun0 30327 Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = ∅)    ⇒   (𝜑 → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ Disj 𝑥 ∈ 𝐴 𝐶))
 
27-Nov-2023rmounid 30262 Case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝜓)    ⇒   (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓))
 
27-Nov-2023rmoun 30261 "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))
 
23-Nov-2023ich2ex 43636 Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)
[𝑥⇄𝑦]∃𝑥∃𝑦𝜑
 
23-Nov-2023ich2al 43635 Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.)
[𝑥⇄𝑦]∀𝑥∀𝑦𝜑
 
23-Nov-2023ichf 43617 Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    ⇒   [𝑥⇄𝑦]𝜑
 
23-Nov-2023ichv 43616 Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
[𝑥⇄𝑦]𝜑
 
23-Nov-2023nf5r 2192 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1784 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
21-Nov-2023tocyc01 30764 Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.)
𝐶 = (toCyc‘𝐷)    ⇒   ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ (dom 𝐶 ∩ (◡♯ “ {0, 1}))) → (𝐶‘𝑊) = ( I ↾ 𝐷))
 
21-Nov-20231cshid 30637 Cyclically shifting a single letter word keeps it unchanged. (Contributed by Thierry Arnoux, 21-Nov-2023.)
((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ (♯‘𝑊) = 1) → (𝑊 cyclShift 𝑁) = 𝑊)
 
21-Nov-2023hashgt1 30533 Restate "set contains at least two elements" in terms of elementhood. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ (◡♯ “ {0, 1}) ↔ 1 < (♯‘𝐴)))
 
21-Nov-2023xnn01gt 30498 An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than 1. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ {0, 1} ↔ 1 < 𝑁))
 
21-Nov-2023undifr 30287 Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴 ⊆ 𝐵 ↔ ((𝐵 ∖ 𝐴) ∪ 𝐴) = 𝐵)
 
20-Nov-2023cycpmrn 30789 The range of the word used to build a cycle is the cycle's orbit, i.e. the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑀 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → 1 < (♯‘𝑊))    ⇒   (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I ))
 
20-Nov-2023symgcntz 30733 All elements of a (finite) set of permutations commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (Cntz‘𝑆)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → Disj 𝑥 ∈ 𝐴 dom (𝑥 ∖ I ))    ⇒   (𝜑 → 𝐴 ⊆ (𝑍‘𝐴))
 
20-Nov-2023gsumzresunsn 30695 Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝑌 = (𝐹‘𝑋)    &   (𝜑 → 𝐹:𝐶⟶𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐶)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑋 ∈ 𝐶)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))))    ⇒   (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌))
 
20-Nov-2023eldmne0 30376 A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)
 
20-Nov-20230res 30357 Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(∅ ↾ 𝐴) = ∅
 
20-Nov-2023iunxunpr 30322 Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋 → 𝐵 = 𝐶)    &   (𝑥 = 𝑌 → 𝐵 = 𝐷)    ⇒   ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ (𝐶 ∪ 𝐷)))
 
20-Nov-2023iunxunsn 30321 Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋 → 𝐵 = 𝐶)    ⇒   (𝑋 ∈ 𝑉 → ∪ 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = (∪ 𝑥 ∈ 𝐴 𝐵 ∪ 𝐶))
 
20-Nov-2023neldifpr2 30297 The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
20-Nov-2023neldifpr1 30296 The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
20-Nov-2023inpr0 30295 Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴))
 
20-Nov-2023nelpr 30294 A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝐴 ∈ 𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶)))
 
20-Nov-2023nelbOLD 30235 Obsolete version of nelb 3271 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
(¬ 𝐴 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
 
19-Nov-2023bj-sbft 34108 Version of sbft 2269 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
(Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))
 
19-Nov-2023bj-nnfor 34083 Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 34079, bj-nnfnt 34073 and bj-nnfbi 34061, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓))
 
19-Nov-2023bj-nnfand 34082 Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 34081, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 34081 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 34082 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)    ⇒   (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒))
 
19-Nov-2023bj-nnfan 34081 Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 34079, bj-nnfnt 34073 and bj-nnfbi 34061, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓))
 
19-Nov-2023prv1n 32682 No wff encoded as a Godel-set of membership is true in a model with only one element. (Contributed by AV, 19-Nov-2023.)
((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋 ∈ 𝑉) → ¬ {𝑋}⊧(𝐼∈𝑔𝐽))
 
19-Nov-2023prv0 32681 Every wff encoded as 𝑈 is true in an "empty model" (𝑀 = ∅). Since ⊧ is defined in terms of the interpretations making the given formula true, it is not defined on the "empty model", since there are no interpretations. In particular, the empty set on the LHS of ⊧ should not be interpreted as the empty model, because ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by AV, 19-Nov-2023.)
(𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈)
 
19-Nov-2023ex-sategoelel12 32678 Example of a valuation of a simplified satisfaction predicate over a proper pair (of ordinal numbers) as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 1o ∈ 2o = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o∈𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.)
𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o))    ⇒   𝑆 ∈ ({1o, 2o} Sat∈ (2o∈𝑔1o))
 
19-Nov-2023ex-sategoelelomsuc 32677 Example of a valuation of a simplified satisfaction predicate over the ordinal numbers as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 𝑍 ∈ suc 𝑍 = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o∈𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.)
𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))    ⇒   (𝑍 ∈ ω → 𝑆 ∈ (ω Sat∈ (2o∈𝑔1o)))
 
19-Nov-2023fmlan0 32642 The empty set is not a Godel formula. (Contributed by AV, 19-Nov-2023.)
∅ ∉ (Fmla‘ω)
 
19-Nov-2023pm2.61i 184 Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2023.)
(𝜑 → 𝜓)    &   (¬ 𝜑 → 𝜓)    ⇒   𝜓
 
18-Nov-2023alephiso3 39924 ℵ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
ℵ Isom E , ≺ (On, (ran card ∖ ω))
 
18-Nov-2023alephiso2 39923 ℵ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
 
18-Nov-2023aleph1min 39922 (ℵ‘1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.)
(ℵ‘1o) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥}
 
18-Nov-2023bj-elsnb 34358 Biconditional version of elsng 4584. (Contributed by BJ, 18-Nov-2023.)
(𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
18-Nov-2023bj-elsn12g 34357 Join of elsng 4584 and elsn2g 4606. (Contributed by BJ, 18-Nov-2023.)
((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
18-Nov-2023bj-nnfbii 34063 If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 34061. (Contributed by BJ, 18-Nov-2023.)
(𝜑 ↔ 𝜓)    ⇒   (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
 
18-Nov-2023rnrhmsubrg 19570 The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
(𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
 
18-Nov-2023notzfaus 5265 In the Separation Scheme zfauscl 5208, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction if that requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥 ∈ 𝑦)    ⇒    ¬ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
 
17-Nov-2023satefvfmla1 32676 The simplified satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))    ⇒   ((𝑀 ∈ 𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat∈ 𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝐼) ∈ (𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾) ∈ (𝑎‘𝐿))})
 
17-Nov-20232goelgoanfmla1 32675 Two Godel-sets of membership combined with a Godel-set for NAND is a Godel formula of height 1. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))    ⇒   (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
 
17-Nov-2023satfv1fvfmla1 32674 The value of the satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼∈𝑔𝐽)⊼𝑔(𝐾∈𝑔𝐿))    ⇒   (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝐼)𝐸(𝑎‘𝐽) ∨ ¬ (𝑎‘𝐾)𝐸(𝑎‘𝐿))})
 
17-Nov-2023symgcom2 30732 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 17-Nov-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅)    ⇒   (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋))
 
17-Nov-2023nfpconfp 30380 The set of fixed points of 𝐹 is the complement of the set of points moved by 𝐹. (Contributed by Thierry Arnoux, 17-Nov-2023.)
(𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))
 
17-Nov-2023pm2.18 128 Clavius law, or "consequentia mirabilis" ("admirable consequence"). If a formula is implied by its negation, then it is true. Can be used in proofs by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. See also the weak Clavius law pm2.01 191. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 17-Nov-2023.)
((¬ 𝜑 → 𝜑) → 𝜑)
 
17-Nov-2023pm2.18d 127 Deduction form of the Clavius law pm2.18 128. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Revised to shorten pm2.18 128. (Revised by Wolf Lammen, 17-Nov-2023.)
(𝜑 → (¬ 𝜓 → 𝜓))    ⇒   (𝜑 → 𝜓)
 
16-Nov-2023ensucne0 39901 A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.)
(𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅)
 
16-Nov-2023pmtrcnelor 30739 Composing a permutation 𝐹 with a transposition which results in moving one or two less points. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹‘𝐼)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I ))    &   𝐸 = dom (𝐹 ∖ I )    &   𝐴 = dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )    ⇒   (𝜑 → (𝐴 = (𝐸 ∖ {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 ∖ {𝐼})))
 
16-Nov-2023pmtrcnel2 30738 Variation on pmtrcnel 30737. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹‘𝐼)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I ))    ⇒   (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))
 
16-Nov-2023pmtrcnel 30737 Composing a permutation 𝐹 with a transposition which results in moving at least one less point. Here the set of points moved by a permutation 𝐹 is expressed as dom (𝐹 ∖ I ). (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹‘𝐼)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I ))    ⇒   (𝜑 → dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∖ {𝐼}))
 
16-Nov-2023eqdif 30284 If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
(((𝐴 ∖ 𝐵) = ∅ ∧ (𝐵 ∖ 𝐴) = ∅) → 𝐴 = 𝐵)
 
16-Nov-2023rnasclassa 20127 The scalar multiples of the unit vector form a subalgebra of the vectors. (Contributed by SN, 16-Nov-2023.)
𝐴 = (algSc‘𝑊)    &   𝑈 = (𝑊 ↾s ran 𝐴)    &   (𝜑 → 𝑊 ∈ AssAlg)    ⇒   (𝜑 → 𝑈 ∈ AssAlg)
 
14-Nov-2023nsyl2 143 A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 14-Nov-2023.)
(𝜑 → ¬ 𝜓)    &   (¬ 𝜒 → 𝜓)    ⇒   (𝜑 → 𝜒)
 
13-Nov-2023pnpcan 10928 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by SN, 13-Nov-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 − 𝐶))
 
13-Nov-2023ralrexbid 3325 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.)
(𝜑 → (𝜓 ↔ 𝜃))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))
 
11-Nov-2023ensucne0OLD 39902 A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅)
 
11-Nov-2023dfsucon 39895 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)
((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
 
10-Nov-2023adh-minimp-pm2.43 43267 Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minimp-ax1 43261, adh-minimp-ax2 43265, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) Polish prefix notation: CCpCpqCpq . (Contributed by BJ, 31-May-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
 
10-Nov-2023adh-minimp-idALT 43266 Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minimp-ax1 43261, adh-minimp-ax2 43265, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝜑)
 
10-Nov-2023adh-minimp-ax2 43265 Derivation of ax-2 7 from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
 
10-Nov-2023adh-minimp-ax2-lem4 43264 Fourth lemma for the derivation of ax-2 7 from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → (𝜑 → 𝜒)) → (𝜓 → 𝜒)))
 
10-Nov-2023adh-minimp-ax2c 43263 Derivation of a commuted form of ax-2 7 from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
 
10-Nov-2023adh-minimp-imim1 43262 Derivation of imim1 83 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
 
10-Nov-2023adh-minimp-ax1 43261 Derivation of ax-1 6 from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → 𝜑))
 
10-Nov-2023adh-minimp-sylsimp 43260 Derivation of jarr 106 (also called "syll-simp") from minimp 1621 and ax-mp 5. Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
 
10-Nov-2023adh-minimp-jarr-ax2c-lem3 43259 Third lemma for the derivation of jarr 106 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃))) → 𝜏) → 𝜏)
 
10-Nov-2023adh-minimp-jarr-lem2 43258 Second lemma for the derivation of jarr 106, and indirectly ax-1 6, a commuted form of ax-2 7, and ax-2 7 proper, from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → 𝜓) → (((𝜒 → 𝜃) → (((𝜏 → 𝜒) → (𝜃 → 𝜂)) → (𝜒 → 𝜂))) → 𝜁)) → (𝜓 → 𝜁))
 
10-Nov-2023adh-minimp-jarr-imim1-ax2c-lem1 43257 First lemma for the derivation of jarr 106, imim1 83, and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7, from adh-minimp 43256 and ax-mp 5. Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → 𝜓) → (((𝜒 → 𝜑) → (𝜓 → 𝜃)) → (𝜑 → 𝜃)))
 
10-Nov-2023adh-minimp 43256 Another single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. Among single axioms of this length, it is the one with simplest antecedents (i.e., in the corresponding ordering of binary trees which first compares left subtrees, it is the first one). Known as "HI-2" on Dolph Edward "Ted" Ulrich's web page. In the next 4 lemmas and 5 theorems, ax-1 6 and ax-2 7 are derived from this other single axiom in 20 detachments (instances of ax-mp 5) in total. Polish prefix notation: CpCCqrCCCsqCrtCqt ; or CtCCpqCCCspCqrCpr in Carew Arthur Meredith and Arthur Norman Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, volume IV, number 3, July 1963, pages 171--187, on page 180. (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.)
(𝜑 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏))))
 
10-Nov-2023adh-minim-pm2.43 43255 Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 43249, adh-minim-ax2 43253, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))
 
10-Nov-2023adh-minim-idALT 43254 Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 43249, adh-minim-ax2 43253, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝜑)
 
10-Nov-2023adh-minim-ax2 43253 Derivation of ax-2 7 from adh-minim 43244 and ax-mp 5. Carew Arthur Meredith derived ax-2 7 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
 
10-Nov-2023adh-minim-ax2c 43252 Derivation of a commuted form of ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
 
10-Nov-2023adh-minim-ax2-lem6 43251 Sixth lemma for the derivation of ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ((((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))) → 𝜂)) → (𝜑 → 𝜂))
 
10-Nov-2023adh-minim-ax2-lem5 43250 Fifth lemma for the derivation of ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓 → 𝜒) → 𝜃) → ((𝜒 → (𝜃 → 𝜏)) → (𝜒 → 𝜏))))
 
10-Nov-2023adh-minim-ax1 43249 Derivation of ax-1 6 from adh-minim 43244 and ax-mp 5. Carew Arthur Meredith derived ax-1 6 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CpCqp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → 𝜑))
 
10-Nov-2023adh-minim-ax1-ax2-lem4 43248 Fourth lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → 𝜓) → 𝜒) → ((𝜓 → (𝜒 → 𝜃)) → (𝜓 → 𝜃)))
 
10-Nov-2023adh-minim-ax1-ax2-lem3 43247 Third lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CCpCqrCqCsCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜃 → (𝜑 → 𝜒))))
 
10-Nov-2023adh-minim-ax1-ax2-lem2 43246 Second lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CCpCCqCCrCpsCrstCpt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ((𝜓 → ((𝜒 → (𝜑 → 𝜃)) → (𝜒 → 𝜃))) → 𝜏)) → (𝜑 → 𝜏))
 
10-Nov-2023adh-minim-ax1-ax2-lem1 43245 First lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43244 and ax-mp 5. Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓 → 𝜏)) → (𝜃 → 𝜏))) → 𝜂)) → (𝜓 → 𝜂)))
 
10-Nov-2023adh-minim 43244 A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. This is the axiom from Carew Arthur Meredith, A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. A two-line review by Alonzo Church of this article can be found in The Journal of Symbolic Logic, volume 19, issue 2, June 1954, page 144, https://doi.org/10.2307/2268914. Known as "HI-1" on Dolph Edward "Ted" Ulrich's web page. In the next 6 lemmas and 3 theorems, ax-1 6 and ax-2 7 are derived from this single axiom in 16 detachments (instances of ax-mp 5) in total. Polish prefix notation: CCCpqrCsCCqCrtCqt . (Contributed by ADH, 10-Nov-2023.)
(((𝜑 → 𝜓) → 𝜒) → (𝜃 → ((𝜓 → (𝜒 → 𝜏)) → (𝜓 → 𝜏))))
 
10-Nov-2023adh-jarrsc 43243 Replacement of a nested antecedent with an outer antecedent. Commuted simplificated form of elimination of a nested antecedent. Also holds intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.)
(((𝜑 → 𝜓) → 𝜒) → (𝜃 → (𝜓 → 𝜒)))
 
9-Nov-2023satfv1 32614 The value of the satisfaction predicate as function over wff codes of height 1. (Contributed by AV, 9-Nov-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘1o) = ((𝑆‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (¬ (𝑎‘𝑖)𝐸(𝑎‘𝑗) ∨ ¬ (𝑎‘𝑘)𝐸(𝑎‘𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝑗)), if-(𝑗 = 𝑛, (𝑎‘𝑖)𝐸𝑧, (𝑎‘𝑖)𝐸(𝑎‘𝑗)))}))}))
 
9-Nov-2023satfv1lem 32613 Lemma for satfv1 32614. (Contributed by AV, 9-Nov-2023.)
((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀 ↑m ω) ∣ (𝑏‘𝐼)𝐸(𝑏‘𝐽)}} = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎‘𝐽)), if-(𝐽 = 𝑁, (𝑎‘𝐼)𝐸𝑧, (𝑎‘𝐼)𝐸(𝑎‘𝐽)))})
 
9-Nov-20232ex2rexrot 3253 Rotate two existential quantifiers and two restricted existential quantifiers. (Contributed by AV, 9-Nov-2023.)
(∃𝑥∃𝑦∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∃𝑥∃𝑦𝜑)
 
8-Nov-2023iscard5 39907 Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴))
 
8-Nov-2023iscard4 39906 Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
 
8-Nov-2023rexopabb 5418 Restricted existential quantification over an ordered-pair class abstraction. (Contributed by AV, 8-Nov-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜒))    ⇒   (∃𝑜 ∈ 𝑂 𝜓 ↔ ∃𝑥∃𝑦(𝜑 ∧ 𝜒))
 
5-Nov-2023harsucnn 39909 The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.)
(𝐴 ∈ ω → (har‘𝐴) = suc 𝐴)
 
5-Nov-2023nndomog 39903 Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 8715 when both are natural numbers. (Originally by NM, 17-Jun-1998.) (Contributed by RP, 5-Nov-2023.)
((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ 𝐴 ⊆ 𝐵))
 
5-Nov-2023selvval2lem4 39142 The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑆 = (𝑇 ↾s ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑋 = (Base‘𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋)
 
5-Nov-2023selvval2lemn 39141 A lemma to illustrate the purpose of selvval2lem3 39140 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.)
𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑆 = (𝑇 ↾s ran 𝐷)    &   𝑋 = (𝑇 ↑s (𝐵 ↑m 𝐼))    &   𝐵 = (Base‘𝑇)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑋))
 
5-Nov-2023elnanelprv 32680 The wff (𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴) encoded as ((𝐴∈𝑔𝐵) ⊼𝑔(𝐵∈𝑔𝐴)) is true in any model 𝑀. This is the model theoretic proof of elnanel 9073. (Contributed by AV, 5-Nov-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝑀⊧((𝐴∈𝑔𝐵)⊼𝑔(𝐵∈𝑔𝐴)))
 
5-Nov-2023prv 32679 The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀 ↑m ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by AV, 5-Nov-2023.)
((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀⊧𝑈 ↔ (𝑀 Sat∈ 𝑈) = (𝑀 ↑m ω)))
 
5-Nov-2023ex-sategoel 32673 Instance of sategoelfv 32671 for the example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    &   𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))    ⇒   (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → (𝑆‘𝐴) ∈ (𝑆‘𝐵))
 
5-Nov-2023ex-sategoelel 32672 Example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    &   𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))    ⇒   (((𝑀 ∈ WUni ∧ 𝑍 ∈ 𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴 ≠ 𝐵)) → 𝑆 ∈ 𝐸)
 
5-Nov-2023sategoelfv 32671 Condition of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership: The sets in model 𝑀 corresponding to the variables 𝐴 and 𝐵 under the assignment of 𝑆 are in a membership relation in 𝑀. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    ⇒   ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑆 ∈ 𝐸) → (𝑆‘𝐴) ∈ (𝑆‘𝐵))
 
5-Nov-2023sategoelfvb 32670 Characterization of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat∈ (𝐴∈𝑔𝐵))    ⇒   ((𝑀 ∈ 𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆 ∈ 𝐸 ↔ (𝑆 ∈ (𝑀 ↑m ω) ∧ (𝑆‘𝐴) ∈ (𝑆‘𝐵))))
 
5-Nov-2023sate0 32666 The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.)
(𝑈 ∈ 𝑉 → (∅ Sat∈ 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
 
5-Nov-2023rnasclmulcl 20126 (Vector) multiplication is closed for scalar multiples of the unit vector. (Contributed by SN, 5-Nov-2023.)
𝐶 = (algSc‘𝑊)    &    × = (.r‘𝑊)    &   (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ((𝜑 ∧ (𝑋 ∈ ran 𝐶 ∧ 𝑌 ∈ ran 𝐶)) → (𝑋 × 𝑌) ∈ ran 𝐶)
 
5-Nov-2023rnasclsubrg 20125 The scalar multiples of the unit vector form a subring of the vectors. (Contributed by SN, 5-Nov-2023.)
𝐶 = (algSc‘𝑊)    &   (𝜑 → 𝑊 ∈ AssAlg)    ⇒   (𝜑 → ran 𝐶 ∈ (SubRing‘𝑊))
 
5-Nov-2023ascldimul 20119 The algebra scalars function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof shortened by SN, 5-Nov-2023.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r‘𝑊)    &    · = (.r‘𝐹)    ⇒   ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴‘𝑅) × (𝐴‘𝑆)))
 
5-Nov-2023elnanel 9073 Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9072 and serves as an example in the context of Godel codes, see elnanelprv 32680. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.)
(𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴)
 
5-Nov-2023f1iun 7648 The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.) (Proof shortened by AV, 5-Nov-2023.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    &   𝐵 ∈ V    ⇒   (∀𝑥 ∈ 𝐴 (𝐵:𝐷–1-1→𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷–1-1→𝑆)
 
4-Nov-2023harval3on 39911 For any ordinal number 𝐴 let (har‘𝐴) denote the least cardinal that is greater than 𝐴; (Contributed by RP, 4-Nov-2023.)
(𝐴 ∈ On → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
 
4-Nov-2023harval3 39910 (har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
(𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
 
4-Nov-2023satefvfmla0 32669 The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.)
((𝑀 ∈ 𝑉 ∧ 𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat∈ 𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋))) ∈ (𝑎‘(2nd ‘(2nd ‘𝑋)))})
 
4-Nov-2023selvval 20334 Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
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 ∈ 𝑊)    &   (𝜑 → 𝐽 ⊆ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
 
4-Nov-2023selvfval 20333 Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
(𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝐽) mPoly 𝑅) / 𝑢⦌⦋(𝐽 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥)))))))
 
4-Nov-2023selvffval 20332 Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
(𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 ∖ 𝑗) mPoly 𝑅) / 𝑢⦌⦋(𝑗 mPoly 𝑢) / 𝑡⦌⦋(algSc‘𝑡) / 𝑐⦌⦋(𝑐 ∘ (algSc‘𝑢)) / 𝑑⦌((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑 ∘ 𝑓))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼 ∖ 𝑗) mVar 𝑅)‘𝑥))))))))
 
4-Nov-2023fviunfun 7649 The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023.)
𝑈 = ∪ 𝑖 ∈ 𝐼 (𝐹‘𝑖)    ⇒   ((Fun 𝑈 ∧ 𝐽 ∈ 𝐼 ∧ 𝑋 ∈ dom (𝐹‘𝐽)) → (𝑈‘𝑋) = ((𝐹‘𝐽)‘𝑋))
 
3-Nov-2023spALT 40560 sp 2181 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2181 instead. (New usage is discouraged.)
(∀𝑥𝜑 → 𝜑)
 
3-Nov-2023csbeq12dv 3895 Formula-building inference for class substitution. (Contributed by SN, 3-Nov-2023.)
(𝜑 → 𝐴 = 𝐶)    &   (𝜑 → 𝐵 = 𝐷)    ⇒   (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐷)
 
2-Nov-2023satfv0fvfmla0 32664 The value of the satisfaction predicate as function over a wff code at ∅. (Contributed by AV, 2-Nov-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))})
 
1-Nov-2023bj-mpgs 33947 From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the inference 𝜑 ⇒ 𝜓. Strong necessity is stronger than necessity, and equivalent to it when sp 2181 (modal T) is available. Therefore, this theorem is stronger than mpg 1797 when sp 2181 is not available. (Contributed by BJ, 1-Nov-2023.)
𝜑    &   ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)    ⇒   𝜓
 
1-Nov-2023evpmsubg 30793 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐴 = (pmEven‘𝐷)    ⇒   (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆))
 
1-Nov-2023cnmsgn0g 30792 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})    ⇒   1 = (0g‘𝑈)
 
1-Nov-2023evpmval 30791 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEven‘𝐷)    ⇒   (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1}))
 
31-Oct-2023sate0fv0 32668 A simplified satisfaction predicate as function over wff codes over an empty model is an empty set. (Contributed by AV, 31-Oct-2023.)
(𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (∅ Sat∈ 𝑈) → 𝑆 = ∅))
 
31-Oct-2023ablsimpgfindlem1 19232 Lemma for ablsimpgfind 19235. An element of an abelian finite simple group which doesn't square to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    · = (.g‘𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑 → 𝐺 ∈ Abel)    &   (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (2 · 𝑥) ≠ 0 ) → (𝑂‘𝑥) ≠ 0)
 
31-Oct-2023ablsimpgcygd 19231 An abelian simple group is cyclic. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
(𝜑 → 𝐺 ∈ Abel)    &   (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   (𝜑 → 𝐺 ∈ CycGrp)
 
30-Oct-2023satef 32667 The simplified satisfaction predicate as function over wff codes over an empty model. (Contributed by AV, 30-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (𝑀 Sat∈ 𝑈)) → 𝑆:ω⟶𝑀)
 
30-Oct-2023satefv 32665 The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
 
29-Oct-2023tr3dom 39900 An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.)
{𝐴, 𝐵, 𝐶} ≼ 3o
 
29-Oct-2023pr2dom 39899 An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.)
{𝐴, 𝐵} ≼ 2o
 
29-Oct-2023sn1dom 39898 A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.)
{𝐴} ≼ 1o
 
29-Oct-2023satfvel 32663 An element of the value of the satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at the code 𝑈 for a wff using ∈ , ⊼ , ∀ is a valuation 𝑆:ω⟶𝑀 of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1o), etc.) so that 𝑈 is true under the assignment 𝑆. (Contributed by AV, 29-Oct-2023.)
(((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)
 
29-Oct-2023satfun 32662 The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 29-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))
 
29-Oct-2023norassOLD 1533 Obsolete version of norass 1532 as of 17-Dec-2023. (Contributed by RP, 29-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ 𝜒) ↔ (((𝜑 ⊽ 𝜓) ⊽ 𝜒) ↔ (𝜑 ⊽ (𝜓 ⊽ 𝜒))))
 
28-Oct-2023satff 32661 The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 28-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀 ↑m ω))
 
28-Oct-2023satffun 32660 The value of the satisfaction predicate as function over wff codes at a natural number is a function. (Contributed by AV, 28-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))
 
28-Oct-2023satffunlem2 32659 Lemma 2 for satffun 32660: induction step. (Contributed by AV, 28-Oct-2023.)
((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)))
 
28-Oct-2023satffunlem1 32658 Lemma 1 for satffun 32660: induction basis. (Contributed by AV, 28-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
 
28-Oct-2023satffunlem2lem1 32655 Lemma 1 for satffunlem2 32659. (Contributed by AV, 28-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   ((Fun (𝑆‘suc 𝑁) ∧ (𝑆‘𝑁) ⊆ (𝑆‘suc 𝑁)) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))})
 
28-Oct-2023pm2.521g 176 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → (𝜓 → 𝜒))
 
28-Oct-2023conax1k 173 Weakening of conax1 172. General instance of pm2.51 174 and of pm2.52 175. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → (𝜒 → ¬ 𝜓))
 
28-Oct-2023conax1 172 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑 → 𝜓) → ¬ 𝜓)
 
27-Oct-2023satffunlem2lem2 32657 Lemma 2 for satffunlem2 32659. (Contributed by AV, 27-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))}) = ∅)
 
27-Oct-2023satffunlem 32652 Lemma for satffunlem1lem1 32653 and satffunlem2lem1 32655. (Contributed by AV, 27-Oct-2023.)
(((Fun 𝑍 ∧ (𝑠 ∈ 𝑍 ∧ 𝑟 ∈ 𝑍) ∧ (𝑢 ∈ 𝑍 ∧ 𝑣 ∈ 𝑍)) ∧ (𝑥 = ((1st ‘𝑠)⊼𝑔(1st ‘𝑟)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑠) ∩ (2nd ‘𝑟)))) ∧ (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑤 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))))) → 𝑦 = 𝑤)
 
27-Oct-2023funeldmdif 7750 Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.)
((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
 
27-Oct-2023funelss 7749 If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.)
((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐴) → ((1st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵))
 
26-Oct-2023releldmdifi 7747 One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.)
((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶))
 
26-Oct-2023nororOLD 1528 Obsolete version of noror 1527 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∨ 𝜓) ↔ ((𝜑 ⊽ 𝜓) ⊽ (𝜑 ⊽ 𝜓)))
 
26-Oct-2023noranOLD 1526 Obsolete version of noran 1525 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ 𝜓) ↔ ((𝜑 ⊽ 𝜑) ⊽ (𝜓 ⊽ 𝜓)))
 
25-Oct-2023dmopab3rexdif 32656 The domain of an ordered pair class abstraction with three nested restricted existential quantifiers with differences. (Contributed by AV, 25-Oct-2023.)
((∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑈 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) ∧ 𝑆 ⊆ 𝑈) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷)) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (∃𝑢 ∈ (𝑈 ∖ 𝑆)(∃𝑣 ∈ 𝑈 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶) ∨ ∃𝑢 ∈ 𝑆 ∃𝑣 ∈ (𝑈 ∖ 𝑆)𝑥 = 𝐴)})
 
25-Oct-2023rexdifi 4125 Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)
 
25-Oct-2023falnorfalOLD 1591 Obsolete version of falnorfal 1590 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊥ ⊽ ⊥) ↔ ⊤)
 
25-Oct-2023falnortru 1589 A ⊽ identity. (Contributed by Remi, 25-Oct-2023.)
((⊥ ⊽ ⊤) ↔ ⊥)
 
25-Oct-2023trunorfalOLD 1588 Obsolete version of trunorfal 1587 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊽ ⊥) ↔ ⊥)
 
25-Oct-2023trunortruOLD 1586 Obsolete version of trunortru 1585 as of 7-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊽ ⊤) ↔ ⊥)
 
25-Oct-2023nornotOLD 1524 Obsolete version of nornot 1523 as of 8-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ 𝜑 ↔ (𝜑 ⊽ 𝜑))
 
25-Oct-2023norcom 1522 The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.)
((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))
 
25-Oct-2023df-nor 1521 Define joint denial ("not-or" or "nor"). After we define the constant true ⊤ (df-tru 1539) and the constant false ⊥ (df-fal 1549), we will be able to prove these truth table values: ((⊤ ⊽ ⊤) ↔ ⊥) (trunortru 1585), ((⊤ ⊽ ⊥) ↔ ⊥) (trunorfal 1587), ((⊥ ⊽ ⊤) ↔ ⊥) (falnortru 1589), and ((⊥ ⊽ ⊥) ↔ ⊤) (falnorfal 1590). Contrast with ∧ (df-an 399), ∨ (df-or 844), → (wi 4), ⊼ (df-nan 1482), and ⊻ (df-xor 1502). (Contributed by Remi, 25-Oct-2023.)
((𝜑 ⊽ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓))
 
24-Oct-2023dff15 32357 A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023.)
(𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)))
 
23-Oct-2023satffunlem1lem2 32654 Lemma 2 for satffunlem1 32658. (Contributed by AV, 23-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) = ∅)
 
23-Oct-2023acycgrsubgr 32409 The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ AcyclicGraph)
 
23-Oct-2023subgrcycl 32386 If a cycle exists in a subgraph of a graph 𝐺, then that cycle also exists in 𝐺. (Contributed by BTernaryTau, 23-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Cycles‘𝑆)𝑃 → 𝐹(Cycles‘𝐺)𝑃))
 
23-Oct-2023dmopab2rex 5789 The domain of an ordered pair class abstraction with two nested restricted existential quantifiers. (Contributed by AV, 23-Oct-2023.)
(∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑈 (∃𝑣 ∈ 𝑉 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))} = {𝑥 ∣ ∃𝑢 ∈ 𝑈 (∃𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶)})
 
23-Oct-2023equsexvw 2010 Version of equsexv 2268 with a disjoint variable condition, and of equsex 2439 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2009. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓)
 
22-Oct-2023goalr 32648 If the "Godel-set of universal quantification" applied to a class is a Godel formula, the class is also a Godel formula. Remark: The reverse is not valid for 𝐴 being of the same height as the "Godel-set of universal quantification". (Contributed by AV, 22-Oct-2023.)
((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))
 
22-Oct-2023goalrlem 32647 Lemma for goalr 32648 (induction step). (Contributed by AV, 22-Oct-2023.)
(𝑁 ∈ ω → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
 
22-Oct-2023goaln0 32644 The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.)
(∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
 
22-Oct-2023subgrpth 32385 If a path exists in a subgraph of a graph 𝐺, then that path also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Paths‘𝑆)𝑃 → 𝐹(Paths‘𝐺)𝑃))
 
22-Oct-2023subgrtrl 32384 If a trail exists in a subgraph of a graph 𝐺, then that trail also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Trails‘𝑆)𝑃 → 𝐹(Trails‘𝐺)𝑃))
 
22-Oct-2023subgrwlk 32383 If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃 → 𝐹(Walks‘𝐺)𝑃))
 
22-Oct-2023speiv 1975 Inference from existential specialization. (Contributed by NM, 19-Aug-1993.) (Revised by Wolf Lammen, 22-Oct-2023.)
(𝑥 = 𝑦 → (𝜓 → 𝜑))    &   𝜓    ⇒   ∃𝑥𝜑
 
22-Oct-2023spimew 1973 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 22-Oct-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   (𝜑 → ∃𝑥𝜓)
 
21-Oct-2023pren2d 39921 A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ≠ 𝐵)    ⇒   (𝜑 → {𝐴, 𝐵} ≈ 2o)
 
21-Oct-2023pr2eldif2 39920 If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴}))
 
21-Oct-2023pr2eldif1 39919 If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵}))
 
21-Oct-2023pr2cv2 39917 If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ V)
 
21-Oct-2023pr2el2 39916 If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ {𝐴, 𝐵})
 
21-Oct-2023pr2cv1 39915 If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ V)
 
21-Oct-2023pr2el1 39914 If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ {𝐴, 𝐵})
 
21-Oct-2023gonar 32646 If the "Godel-set of NAND" applied to classes is a Godel formula, the classes are also Godel formulas. Remark: The reverse is not valid for 𝐴 or 𝐵 being of the same height as the "Godel-set of NAND". (Contributed by AV, 21-Oct-2023.)
((𝑁 ∈ ω ∧ (𝑎⊼𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))
 
21-Oct-2023gonarlem 32645 Lemma for gonar 32646 (induction step). (Contributed by AV, 21-Oct-2023.)
(𝑁 ∈ ω → (((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎⊼𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
 
21-Oct-2023gonan0 32643 The "Godel-set of NAND" is a Godel formula of at least height 1. (Contributed by AV, 21-Oct-2023.)
((𝐴⊼𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
 
21-Oct-2023fmlaomn0 32641 The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)
(𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
 
21-Oct-2023satfvsucsuc 32616 The satisfaction predicate as function over wff codes of height (𝑁 + 1), expressed by the minimally necessary satisfaction predicates as function over wff codes of height 𝑁. (Contributed by AV, 21-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆‘𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = 𝐴))}))
 
21-Oct-2023gonanegoal 32603 The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.)
(𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢
 
21-Oct-2023pthacycspth 32408 A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, 21-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Paths‘𝐺)𝑃) → 𝐹(SPaths‘𝐺)𝑃)
 
21-Oct-2023elneeldif 3953 The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.)
((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑋 ≠ 𝑌)
 
21-Oct-2023ralrexbidOLD 3326 Obsolete version of ralrexbid 3325 as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ↔ 𝜃))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜃))
 
20-Oct-2023fmlasucdisj 32650 The valid Godel formulas of height (𝑁 + 1) is disjoint with the difference ((Fmla‘suc suc 𝑁) ∖ (Fmla‘suc 𝑁)), expressed by formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification based on the valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 20-Oct-2023.)
(𝑁 ∈ ω → ((Fmla‘suc 𝑁) ∩ {𝑥 ∣ (∃𝑢 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑣 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ∨ ∃𝑢 ∈ (Fmla‘𝑁)∃𝑣 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑢⊼𝑔𝑣))}) = ∅)
 
20-Oct-2023fmla0disjsuc 32649 The set of valid Godel formulas of height 0 is disjoint with the formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification. (Contributed by AV, 20-Oct-2023.)
((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
 
20-Oct-2023fmlasssuc 32640 The Godel formulas of height 𝑁 are a subset of the Godel formulas of height 𝑁 + 1. (Contributed by AV, 20-Oct-2023.)
(𝑁 ∈ ω → (Fmla‘𝑁) ⊆ (Fmla‘suc 𝑁))
 
20-Oct-2023cusgracyclt3v 32407 A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3))
 
20-Oct-2023pthisspthorcycl 32379 A path is either a simple path or a cycle (or both). (Contributed by BTernaryTau, 20-Oct-2023.)
(𝐹(Paths‘𝐺)𝑃 → (𝐹(SPaths‘𝐺)𝑃 ∨ 𝐹(Cycles‘𝐺)𝑃))
 
20-Oct-202319.3v 1985 Version of 19.3 2201 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1987. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2014. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑 ↔ 𝜑)
 
20-Oct-2023spvw 1984 Version of sp 2181 when 𝑥 does not occur in 𝜑. Converse of ax-5 1910. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1985. (Revised by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑 → 𝜑)
 
20-Oct-2023exgen 1977 Rule of existential generalization, similar to universal generalization ax-gen 1795, but valid only if an individual exists. Its proof requires ax-6 1969 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen 1795, ax-4 1809 and this theorem alone, not requiring ax-7 2014 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
𝜑    ⇒   ∃𝑥𝜑
 
19-Oct-2023dmopabelb 5788 A set is an element of the domain of a ordered pair class abstraction iff there is a second set so that both sets fulfil the wff of the class abstraction. (Contributed by AV, 19-Oct-2023.)
(𝑥 = 𝑋 → (𝜑 ↔ 𝜓))    ⇒   (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))
 
19-Oct-2023vtocl2d 3560 Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.) (Revised by BTernaryTau, 19-Oct-2023.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝜓)    ⇒   (𝜑 → 𝜒)
 
18-Oct-2023mo4 2649 At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2650, but this proof is based on fewer axioms.

By the way, swapping 𝑥, 𝑦 and 𝜑, 𝜓 leads to an expression for ∃*𝑦𝜓, which is equivalent to ∃*𝑥𝜑 (is a proof line), so the right hand side is a rare instance of an expression where swapping the quantifiers can be done without ax-11 2160. (Contributed by NM, 26-Jul-1995.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2023.)

(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
 
17-Oct-2023satffunlem1lem1 32653 Lemma for satffunlem1 32658. (Contributed by AV, 17-Oct-2023.)
(Fun ((𝑀 Sat 𝐸)‘𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑘 ∈ 𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})
 
17-Oct-2023upgracycusgr 32406 An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
 
17-Oct-2023umgracycusgr 32405 An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
 
17-Oct-2023umgr2cycl 32392 A multigraph with two distinct edges that connect the same vertices has a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐼 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ UMGraph ∧ ∃𝑗 ∈ dom 𝐼∃𝑘 ∈ dom 𝐼((𝐼‘𝑗) = (𝐼‘𝑘) ∧ 𝑗 ≠ 𝑘)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2))
 
17-Oct-2023umgr2cycllem 32391 Lemma for umgr2cycl 32392. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐹 = ⟨“𝐽𝐾”⟩    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → 𝐺 ∈ UMGraph)    &   (𝜑 → 𝐽 ∈ dom 𝐼)    &   (𝜑 → 𝐽 ≠ 𝐾)    &   (𝜑 → (𝐼‘𝐽) = (𝐼‘𝐾))    ⇒   (𝜑 → ∃𝑝 𝐹(Cycles‘𝐺)𝑝)
 
17-Oct-20231one2o 8272 Ordinal one is not ordinal two. Analogous to 1ne2 11848. (Contributed by AV, 17-Oct-2023.)
1o ≠ 2o
 
17-Oct-2023funfv1st2nd 7748 The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.)
((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋))
 
17-Oct-2023omsucne 7601 A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
(𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)
 
17-Oct-2023relcnvtr 6123 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.)
(Rel 𝑅 → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅))
 
17-Oct-2023relcnvtrg 6122 General form of relcnvtr 6123. (Contributed by Peter Mazsa, 17-Oct-2023.)
((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇))
 
17-Oct-20233anidm 1100 Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.)
((𝜑 ∧ 𝜑 ∧ 𝜑) ↔ 𝜑)
 
16-Oct-2023gonafv 32601 The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
 
16-Oct-20232cycl2d 32390 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = ⟨“𝐴𝐵𝐴”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼‘𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → 𝐽 ≠ 𝐾)    ⇒   (𝜑 → 𝐹(Cycles‘𝐺)𝑃)
 
16-Oct-20232cycld 32389 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → 𝐽 ≠ 𝐾)    &   (𝜑 → 𝐴 = 𝐶)    ⇒   (𝜑 → 𝐹(Cycles‘𝐺)𝑃)
 
15-Oct-2023satfv0fun 32622 The value of the satisfaction predicate as function over wff codes at ∅ is a function. (Contributed by AV, 15-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
 
15-Oct-2023satfsschain 32615 The binary relation of a satisfaction predicate as function over wff codes is an increasing chain (with respect to inclusion). (Contributed by AV, 15-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵 ⊆ 𝐴 → (𝑆‘𝐵) ⊆ (𝑆‘𝐴)))
 
15-Oct-2023upgracycumgr 32404 An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph)
 
15-Oct-2023acycgrislfgr 32403 An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
 
15-Oct-2023lfuhgr3 32370 A hypergraph is loop-free if and only if none of its edges connect to only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ¬ ∃𝑎{𝑎} ∈ (Edg‘𝐺)))
 
15-Oct-2023lfuhgr2 32369 A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1))
 
15-Oct-2023lfuhgr 32368 A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
 
15-Oct-2023cyc3conja 30803 All 3-cycles are conjugate in the alternating group An for n>= 5. Property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (𝑀 “ (◡♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g‘𝑆)    &    − = (-g‘𝑆)    &   (𝜑 → 5 ≤ 𝑁)    &   (𝜑 → 𝐷 ∈ Fin)    &   (𝜑 → 𝑄 ∈ 𝐶)    &   (𝜑 → 𝑇 ∈ 𝐶)    ⇒   (𝜑 → ∃𝑝 ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) − 𝑝))
 
15-Oct-2023tocycfvres2 30757 A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    ⇒   (𝜑 → ((𝐶‘𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊)))
 
15-Oct-2023tocycfvres1 30756 A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    ⇒   (𝜑 → ((𝐶‘𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊))
 
15-Oct-2023symgsubg 30735 The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    − = (-g‘𝐺)    ⇒   ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 ∘ ◡𝑌))
 
15-Oct-2023odpmco 30734 The composition of two odd permutations is even. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (pmEven‘𝐷)    ⇒   ((𝐷 ∈ Fin ∧ 𝑋 ∈ (𝐵 ∖ 𝐴) ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → (𝑋 ∘ 𝑌) ∈ 𝐴)
 
15-Oct-2023symgcom 30731 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑋 ↾ 𝐸) = ( I ↾ 𝐸))    &   (𝜑 → (𝑌 ↾ 𝐹) = ( I ↾ 𝐹))    &   (𝜑 → (𝐸 ∩ 𝐹) = ∅)    &   (𝜑 → (𝐸 ∪ 𝐹) = 𝐴)    ⇒   (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋))
 
14-Oct-2023isfmlasuc 32639 The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)
((𝑁 ∈ ω ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))))
 
14-Oct-2023cycpmconjs 30802 All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (◡♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g‘𝑆)    &    − = (-g‘𝑆)    &   (𝜑 → 𝑃 ∈ (0...𝑁))    &   (𝜑 → 𝐷 ∈ Fin)    &   (𝜑 → 𝑄 ∈ 𝐶)    &   (𝜑 → 𝑇 ∈ 𝐶)    ⇒   (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝))
 
14-Oct-2023cycpmconjslem2 30801 Lemma for cycpmconjs 30802 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (◡♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g‘𝑆)    &    − = (-g‘𝑆)    &   (𝜑 → 𝑃 ∈ (0...𝑁))    &   (𝜑 → 𝐷 ∈ Fin)    &   (𝜑 → 𝑄 ∈ 𝐶)    ⇒   (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ((◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
 
14-Oct-2023cycpmconjslem1 30800 Lemma for cycpmconjs 30802 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (◡♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → (♯‘𝑊) = 𝑃)    ⇒   (𝜑 → ((◡𝑊 ∘ (𝑀‘𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1))
 
13-Oct-2023satfdmfmla 32651 The domain of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is the set of valid Godel formulas of height 𝑁. (Contributed by AV, 13-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
 
13-Oct-2023satfrnmapom 32621 The range of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is a subset of the power set of all mappings from the natural numbers into the model 𝑀. (Contributed by AV, 13-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω))
 
13-Oct-2023satfdm 32620 The domain of the satisfaction predicate as function over wff codes does not depend on the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 13-Oct-2023.)
(((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑁 ∈ 𝑋 ∧ 𝐹 ∈ 𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
 
13-Oct-2023satfbrsuc 32617 The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝑃 = (𝑆‘𝑁)    ⇒   (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})))))
 
13-Oct-2023loop1cycl 32388 A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)
(𝐺 ∈ UHGraph → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))
 
13-Oct-2023cycpmgcl 30799 Cyclic permutations are permutations, similar to cycpmcl 30762, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐶 = (𝑀 “ (◡♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    ⇒   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵)
 
12-Oct-2023satfdmlem 32619 Lemma for satfdm 32620. (Contributed by AV, 12-Oct-2023.)
(((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st ‘𝑎)⊼𝑔(1st ‘𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑎))))
 
12-Oct-2023satfrel 32618 The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
 
12-Oct-2023acycgr2v 32401 A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → 𝐺 ∈ AcyclicGraph)
 
12-Oct-2023acycgr1v 32400 A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → 𝐺 ∈ AcyclicGraph)
 
12-Oct-2023acycgrcycl 32398 Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 = ∅)
 
12-Oct-2023fnunres2 30427 Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.)
((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
 
11-Oct-2023en2pr 39912 A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.)
(𝐴 ≈ 2o ↔ ∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦))
 
11-Oct-2023prclisacycgr 32402 A proper class (representing a null graph, see vtxvalprc 26833) has the property of an acyclic graph (see also acycgr0v 32399). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)    ⇒   (¬ 𝐺 ∈ V → ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅))
 
11-Oct-2023acycgr0v 32399 A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ 𝑊 ∧ 𝑉 = ∅) → 𝐺 ∈ AcyclicGraph)
 
11-Oct-2023isacycgr1 32397 The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓∀𝑝(𝑓(Cycles‘𝐺)𝑝 → 𝑓 = ∅)))
 
11-Oct-2023isacycgr 32396 The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺 ∈ 𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ 𝑓 ≠ ∅)))
 
11-Oct-2023dfacycgr1 32395 An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ ∀𝑓∀𝑝(𝑓(Cycles‘𝑔)𝑝 → 𝑓 = ∅)}
 
11-Oct-2023df-acycgr 32394 Define the class of all acyclic graphs. A graph is called acyclic if it has no (non-trivial) cycles. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓∃𝑝(𝑓(Cycles‘𝑔)𝑝 ∧ 𝑓 ≠ ∅)}
 
10-Oct-2023satfvsuc 32612 The value of the satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 10-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))
 
9-Oct-2023cycpmconjv 30788 A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g‘𝑆)    &    − = (-g‘𝑆)    &   𝐵 = (Base‘𝑆)    ⇒   ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (𝑀‘(𝐺 ∘ 𝑊)))
 
9-Oct-2023cycpmconjvlem 30787 Lemma for cycpmconjv 30788 (Contributed by Thierry Arnoux, 9-Oct-2023.)
(𝜑 → 𝐹:𝐷–1-1-onto→𝐷)    &   (𝜑 → 𝐵 ⊆ 𝐷)    ⇒   (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵))))
 
9-Oct-2023reldisjun 30356 Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.)
((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)))
 
9-Oct-2023rabelpw 5256 A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023.)
(𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ 𝒫 𝐴)
 
9-Oct-2023difelpw 5255 A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.)
(𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐴)
 
8-Oct-2023pren2 39918 An unordered pair is equinumerous to ordinal two iff both parts are sets not equal to each other. (Contributed by RP, 8-Oct-2023.)
({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵))
 
8-Oct-2023pr2cv 39913 If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
8-Oct-2023snen1el 39897 A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.)
({𝐴} ≈ 1o ↔ 𝐴 ∈ {𝐴})
 
8-Oct-2023snen1g 39896 A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.)
({𝐴} ≈ 1o ↔ 𝐴 ∈ V)
 
8-Oct-2023satfvsuclem2 32611 Lemma 2 for satfvsuc 32612. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} ∈ V)
 
8-Oct-2023satfvsuclem1 32610 Lemma 1 for satfvsuc 32612. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀 ↑m ω))} ∈ V)
 
8-Oct-2023satfv0 32609 The value of the satisfaction predicate as function over wff codes at ∅. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
 
8-Oct-2023goeleq12bg 32600 Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)
(((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁)))
 
8-Oct-2023spthcycl 32380 A walk is a trivial path if and only if it is both a simple path and a cycle. (Contributed by BTernaryTau, 8-Oct-2023.)
((𝐹(Paths‘𝐺)𝑃 ∧ 𝐹 = ∅) ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ 𝐹(Cycles‘𝐺)𝑃))
 
8-Oct-2023funen1cnv 32361 If a function is equinumerous to ordinal 1, then its converse is also a function. (Contributed by BTernaryTau, 8-Oct-2023.)
((Fun 𝐹 ∧ 𝐹 ≈ 1o) → Fun ◡𝐹)
 
6-Oct-2023satom 32607 The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at omega (ω). (Contributed by AV, 6-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑛 ∈ ω ((𝑀 Sat 𝐸)‘𝑛))
 
6-Oct-2023satfn 32606 The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 is a function over suc ω. (Contributed by AV, 6-Oct-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) Fn suc ω)
 
6-Oct-2023fiun 7647 The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7648. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    &   𝐵 ∈ V    ⇒   (∀𝑥 ∈ 𝐴 (𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷⟶𝑆)
 
6-Oct-2023fiunlem 7646 Lemma for fiun 7647 and f1iun 7648. Formerly part of f1iun 7648. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   (((𝐵:𝐷⟶𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
 
6-Oct-2023vtoclgft 3556 Closed theorem form of vtoclgf 3568. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) Avoid ax-13 2389. (Revised by Gino Giotto, 6-Oct-2023.)
(((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴 ∈ 𝑉) → 𝜓)
 
5-Oct-2023finorwe 34667 If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.)
(¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴))
 
5-Oct-2023usgrcyclgt2v 32382 A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝑉))
 
5-Oct-2023pthhashvtx 32378 A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉))
 
4-Oct-2023cusgr3cyclex 32387 Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
 
4-Oct-2023cusgredgex2 32373 Any two distinct vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ 𝐸))
 
4-Oct-2023f1resfz0f1d 32365 If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.)
(𝜑 → 𝐾 ∈ ℕ0)    &   (𝜑 → 𝐹:(0...𝐾)⟶𝑉)    &   (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1→𝑉)    &   (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)    ⇒   (𝜑 → 𝐹:(0...𝐾)–1-1→𝑉)
 
4-Oct-2023cshf1o 30640 Condition for the cyclic shift to be a bijection. (Contributed by Thierry Arnoux, 4-Oct-2023.)
((𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):dom 𝑊–1-1-onto→ran 𝑊)
 
3-Oct-2023cusgredgex 32372 Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝐺 ∈ ComplUSGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))
 
3-Oct-2023fisshasheq 32356 A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.)
((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 = 𝐵)
 
3-Oct-2023dfeumo 2618 An elementary proof showing the reverse direction of dfmoeu 2617. Here the characterizing expression of existential uniqueness (eu6 2658) is derived from that of uniqueness (df-mo 2621). (Contributed by Wolf Lammen, 3-Oct-2023.)
((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
 
3-Oct-20232ax6e 2493 We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2492 with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.) (New usage is discouraged.)
∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)
 
2-Oct-2023cplgredgex 32371 Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝐺 ∈ ComplGraph → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒))
 
2-Oct-2023fzodif1 30519 Set difference of two half-open range of sequential integers sharing the same starting value. (Contributed by Thierry Arnoux, 2-Oct-2023.)
(𝐾 ∈ (𝑀...𝑁) → ((𝑀..^𝑁) ∖ (𝑀..^𝐾)) = (𝐾..^𝑁))
 
2-Oct-20232eu5 2741 An alternate definition of double existential uniqueness (see 2eu4 2740). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦". (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining ∀𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published (∃* means "exists at most one"). (Contributed by NM, 26-Oct-2003.) Avoid ax-13 2389. (Revised by Wolf Lammen, 2-Oct-2023.)
((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
 
2-Oct-20232eu1v 2736 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Version of 2eu1 2734 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2389. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
 
2-Oct-2023moexex 2722 "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2389. Use the version moexexvw 2712 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2712. (Revised by Wolf Lammen, 2-Oct-2023.) (New usage is discouraged.)
Ⅎ𝑦𝜑    ⇒   ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
 
2-Oct-20232exeuv 2716 Double existential uniqueness implies double unique existential quantification. Version of 2exeu 2730 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2389. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
 
2-Oct-20232euexv 2715 Double quantification with existential uniqueness. Version of 2euex 2725 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2389. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
(∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
 
2-Oct-20232moswapv 2713 A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap 2728 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2712. (Revised by Wolf Lammen, 2-Oct-2023.)
(∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
 
2-Oct-2023moexexvw 2712 "At most one" double quantification. Version of moexexv 2723 with an additional disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 26-Jan-1997.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2722. (Revised by Wolf Lammen, 2-Oct-2023.)
((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
 
2-Oct-2023moexexlem 2710 Factor out the proof skeleton of moexex 2722 and moexexvw 2712. (Contributed by Wolf Lammen, 2-Oct-2023.)
Ⅎ𝑦𝜑    &   Ⅎ𝑦∃*𝑥𝜑    &   Ⅎ𝑥∃*𝑦∃𝑥(𝜑 ∧ 𝜓)    ⇒   ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓))
 
2-Oct-2023nfmov 2643 Bound-variable hypothesis builder for the at-most-one quantifier. See nfmo 2645 for a version without disjoint variable conditions but requiring ax-13 2389. (Contributed by NM, 9-Mar-1995.) (Revised by Wolf Lammen, 2-Oct-2023.)
Ⅎ𝑥𝜑    ⇒   Ⅎ𝑥∃*𝑦𝜑
 
1-Oct-2023elrncard 39908 Let us define a cardinal number to be an element 𝐴 ∈ On such that 𝐴 is not equipotent with any 𝑥 ∈ 𝐴. (Contributed by RP, 1-Oct-2023.)
(𝐴 ∈ ran card ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴))
 
1-Oct-2023eu0 39892 There is only one empty set. (Contributed by RP, 1-Oct-2023.)
(∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
 
1-Oct-2023hashf1dmcdm 32360 The size of the domain of a one-to-one set function is less than or equal to the size of its codomain, if it exists. (Contributed by BTernaryTau, 1-Oct-2023.)
((𝐹 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐴) ≤ (♯‘𝐵))
 
1-Oct-2023hashf1dmrn 32359 The size of the domain of a one-to-one set function is equal to the size of its range. (Contributed by BTernaryTau, 1-Oct-2023.)
((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐴) = (♯‘ran 𝐹))
 
1-Oct-2023euim 2700 Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof shortened by Wolf Lammen, 1-Oct-2023.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
 
30-Sep-2023hashfundm 32358 The size of a set function is equal to the size of its domain. (Contributed by BTernaryTau, 30-Sep-2023.)
((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → (♯‘𝐹) = (♯‘dom 𝐹))
 
30-Sep-20230nn0m1nnn0 32355 A number is zero if and only if it's a nonnegative integer that becomes negative after subtracting 1. (Contributed by BTernaryTau, 30-Sep-2023.)
(𝑁 = 0 ↔ (𝑁 ∈ ℕ0 ∧ ¬ (𝑁 − 1) ∈ ℕ0))
 
29-Sep-2023syl2anc2 587 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
(𝜑 → 𝜓)    &   (𝜓 → 𝜒)    &   ((𝜓 ∧ 𝜒) → 𝜃)    ⇒   (𝜑 → 𝜃)
 
28-Sep-2023f1resrcmplf1d 32364 If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)
(𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)    &   (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵)    &   (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)    ⇒   (𝜑 → 𝐹:𝐴–1-1→𝐵)
 
27-Sep-2023infordmin 39905 ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.)
∀𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
 
27-Sep-2023dfom6 39904 Let ω be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023.)
ω = ∪ (On ∩ Fin)
 
27-Sep-2023ontric3g 39894 For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)
∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))
 
27-Sep-2023epelon2 39893 Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴 ∈ 𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7500. This is a weak form of epelg 5469 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
 
27-Sep-2023f1resrcmplf1dlem 32363 Lemma for f1resrcmplf1d 32364. (Contributed by BTernaryTau, 27-Sep-2023.)
(𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝐷 ⊆ 𝐴)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)    ⇒   (𝜑 → ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐷) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))
 
27-Sep-2023f1resveqaeq 32362 If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)
(((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
 
27-Sep-2023prsrcmpltd 32351 If a statement is true for all pairs of elements of a class, all pairs of elements of its complement relative to a second class, and all pairs with one element in each, then it is true for all pairs of elements of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
(𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → 𝜓))    &   (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))    &   (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓))    &   (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))    ⇒   (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓))
 
27-Sep-2023srcmpltd 32350 If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
(𝜑 → (𝐶 ∈ 𝐴 → 𝜓))    &   (𝜑 → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))    ⇒   (𝜑 → (𝐶 ∈ 𝐵 → 𝜓))
 
27-Sep-2023cyc3genpm 30798 The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐶 = (𝑀 “ (◡♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    ⇒   (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
 
27-Sep-2023s3clhash 30628 Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.)
⟨“𝐼𝐽𝐾”⟩ ∈ (◡♯ “ {3})
 
27-Sep-2023fnpr2ob 16834 Biconditional version of fnpr2o 16833. (Contributed by Jim Kingdon, 27-Sep-2023.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 
27-Sep-2023elunant 4157 A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
((𝐶 ∈ (𝐴 ∪ 𝐵) → 𝜑) ↔ ((𝐶 ∈ 𝐴 → 𝜑) ∧ (𝐶 ∈ 𝐵 → 𝜑)))
 
27-Sep-2023eubii 2669 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) Avoid ax-5 1910. (Revised by Wolf Lammen, 27-Sep-2023.)
(𝜑 ↔ 𝜓)    ⇒   (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
 
26-Sep-2023wrdt2ind 30631 Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.)
(𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑖𝑗”⟩) → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵) → (𝜒 → 𝜃))    ⇒   ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏)
 
26-Sep-2023pfxlsw2ccat 30630 Reconstruct a word from its prefix and its last two symbols. (Contributed by Thierry Arnoux, 26-Sep-2023.)
𝑁 = (♯‘𝑊)    ⇒   ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ 𝑁) → 𝑊 = ((𝑊 prefix (𝑁 − 2)) ++ ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩))
 
25-Sep-2023trsp2cyc 30769 Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐶 = (toCyc‘𝐷)    ⇒   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
 
25-Sep-2023tocycf 30763 The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    ⇒   (𝐷 ∈ 𝑉 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵)
 
25-Sep-2023mhplss 20345 Homogeneous polynomials form a linear subspace of the polynomials. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (𝐻‘𝑁) ∈ (LSubSp‘𝑃))
 
25-Sep-2023mhpvscacl 20344 Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    · = ( ·𝑠 ‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐾)    &   (𝜑 → 𝐹 ∈ (𝐻‘𝑁))    ⇒   (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁))
 
25-Sep-2023mhpsubg 20343 Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (𝐻‘𝑁) ∈ (SubGrp‘𝑃))
 
25-Sep-2023xpsrnbas 16847 The indexed structure product that appears in xpsval 16846 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    ⇒   (𝜑 → ran 𝐹 = (Base‘𝑈))
 
25-Sep-2023xpsval 16846 Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ 𝑊)    &   𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})    ⇒   (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))
 
25-Sep-2023fvpr1o 16836 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐵 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
 
25-Sep-2023fvpr0o 16835 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐴 ∈ 𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
 
25-Sep-2023fnpr2o 16833 Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 
25-Sep-2023df-xps 16786 Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
 
25-Sep-2023axsepgfromrep 5204 A more general version axsepg 5207 of the axiom scheme of separation ax-sep 5206 derived from the axiom scheme of replacement ax-rep 5193 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2123 to ax-13 2389. (Revised by SN, 25-Sep-2023.) Use ax-sep 5206 instead (or axsepg 5207 if the extra generality is needed). (New usage is discouraged.)
∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
 
24-Sep-2023usgrgt2cycl 32381 A non-trivial cycle in a simple graph has a length greater than 2. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ 𝐹 ≠ ∅) → 2 < (♯‘𝐹))
 
24-Sep-2023nn0ltp1ne 32354 Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵 ∧ 𝐵 ≠ (𝐴 + 1))))
 
24-Sep-2023nnltp1ne 32353 Positive integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵 ∧ 𝐵 ≠ (𝐴 + 1))))
 
24-Sep-2023zltp1ne 32352 Integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵 ∧ 𝐵 ≠ (𝐴 + 1))))
 
24-Sep-2023cyc3genpmlem 30797 Lemma for cyc3genpm 30798. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (𝑀 “ (◡♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    · = (+g‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    &   (𝜑 → 𝐾 ∈ 𝐷)    &   (𝜑 → 𝐿 ∈ 𝐷)    &   (𝜑 → 𝐸 = (𝑀‘⟨“𝐼𝐽”⟩))    &   (𝜑 → 𝐹 = (𝑀‘⟨“𝐾𝐿”⟩))    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐼 ≠ 𝐽)    &   (𝜑 → 𝐾 ≠ 𝐿)    ⇒   (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐))
 
24-Sep-2023cyc3evpm 30796 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    ⇒   (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)
 
24-Sep-2023cyc2fv2 30768 Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    &   (𝜑 → 𝐼 ≠ 𝐽)    &   𝑆 = (SymGrp‘𝐷)    ⇒   (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
 
24-Sep-2023cyc2fv1 30767 Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    &   (𝜑 → 𝐼 ≠ 𝐽)    &   𝑆 = (SymGrp‘𝐷)    ⇒   (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
 
24-Sep-2023cycpm2cl 30766 Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    &   (𝜑 → 𝐼 ≠ 𝐽)    &   𝑆 = (SymGrp‘𝐷)    ⇒   (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
 
24-Sep-2023cycpm2tr 30765 A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    &   (𝜑 → 𝐼 ≠ 𝐽)    &   𝑇 = (pmTrsp‘𝐷)    ⇒   (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) = (𝑇‘{𝐼, 𝐽}))
 
24-Sep-2023cycpmcl 30762 Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   𝑆 = (SymGrp‘𝐷)    ⇒   (𝜑 → (𝐶‘𝑊) ∈ (Base‘𝑆))
 
24-Sep-2023coprprop 30438 Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → 𝐴 ≠ 𝐶)    &   (𝜑 → 𝐸 ∈ 𝑋)    &   (𝜑 → 𝐹 ∈ 𝑋)    &   (𝜑 → 𝐸 ≠ 𝐹)    ⇒   (𝜑 → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∘ {⟨𝐸, 𝐴⟩, ⟨𝐹, 𝐶⟩}) = {⟨𝐸, 𝐵⟩, ⟨𝐹, 𝐷⟩})
 
24-Sep-2023mptprop 30437 Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    &   (𝜑 → 𝐴 ≠ 𝐶)    ⇒   (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
 
24-Sep-2023brprop 30436 Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑊)    ⇒   (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))
 
24-Sep-2023cnvprop 30435 Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})
 
24-Sep-2023cosnop 30434 Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    ⇒   (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})
 
24-Sep-2023cosnopne 30433 Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐴 ≠ 𝐷)    ⇒   (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
 
24-Sep-2023mobii 2630 Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) Avoid ax-5 1910. (Revised by Wolf Lammen, 24-Sep-2023.)
(𝜓 ↔ 𝜒)    ⇒   (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)
 
23-Sep-2023ichnfb 43632 If 𝑥 and 𝑦 are interchangeable in 𝜑, they are both free or both not free in 𝜑. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
([𝑥⇄𝑦]𝜑 → (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑦Ⅎ𝑥𝜑))
 
23-Sep-2023ichnfim 43631 If in an interchangeability context 𝑥 is not free in 𝜑, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)
 
23-Sep-2023currysetALT 34265 Alternate proof of curryset 34261, or more precisely alternate exposal of the same proof. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) (New usage is discouraged.)
¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
 
23-Sep-2023currysetlem3 34264 Lemma for currysetALT 34265. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒    ¬ 𝑋 ∈ 𝑉
 
23-Sep-2023currysetlem2 34263 Lemma for currysetALT 34265. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))
 
23-Sep-2023currysetlem1 34262 Lemma for currysetALT 34265. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}    ⇒   (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))
 
23-Sep-2023curryset 34261 Curry's paradox in set theory. This can be seen as a generalization of Russell's paradox, which corresponds to the case where 𝜑 is ⊥. See alternate exposal of basically the same proof currysetALT 34265. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉
 
23-Sep-2023currysetlem 34260 Lemma for currysetlem 34260, where it is used with (𝑥 ∈ 𝑥 → 𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑)))
 
23-Sep-2023bj-currypara 33899 Curry's paradox. Note that the proof is intuitionistic (use ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.)
((𝜑 ↔ (𝜑 → 𝜓)) → 𝜓)
 
23-Sep-2023bj-animbi 33898 Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)
((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓)))
 
23-Sep-2023brsnop 30432 Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴 ∧ 𝑌 = 𝐵)))
 
23-Sep-2023sbal2 2572 Move quantifier in and out of substitution. Usage of this theorem is discouraged because it depends on ax-13 2389. Check out sbal 2165 for a version replacing the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.) (Proof shortened by Wolf Lammen, 23-Sep-2023.) (New usage is discouraged.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
22-Sep-2023sn-axprlem3 39115 axprlem3 5329 using only Tarski's FOL axiom schemes and ax-rep 5193. (Contributed by SN, 22-Sep-2023.)
∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
 
22-Sep-2023satfvsucom 32608 The satisfaction predicate as function over wff codes at a successor of ω. (Contributed by AV, 22-Sep-2023.)
𝑆 = (𝑀 Sat 𝐸)    ⇒   ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → (𝑆‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
 
22-Sep-2023satfsucom 32605 The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at an element of the successor of ω. (Contributed by AV, 22-Sep-2023.)
((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘𝑁))
 
22-Sep-2023cycpmfv3 30761 Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → 𝑋 ∈ 𝐷)    &   (𝜑 → ¬ 𝑋 ∈ ran 𝑊)    ⇒   (𝜑 → ((𝐶‘𝑊)‘𝑋) = 𝑋)
 
22-Sep-2023cycpmfv2 30760 Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → 0 < (♯‘𝑊))    &   (𝜑 → 𝑁 = ((♯‘𝑊) − 1))    ⇒   (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘0))
 
22-Sep-2023cycpmfv1 30759 Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → 𝑁 ∈ (0..^((♯‘𝑊) − 1)))    ⇒   (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘(𝑁 + 1)))
 
22-Sep-2023cycpmfvlem 30758 Lemma for cycpmfv1 30759 and cycpmfv2 30760. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊)))    ⇒   (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁)))
 
22-Sep-2023tocycval 30754 Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    ⇒   (𝐷 ∈ 𝑉 → 𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤))))
 
21-Sep-2023sn-dtru 39117 dtru 5274 without ax-8 2115 or ax-12 2176. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦
 
21-Sep-2023sn-axrep5v 39114 A condensed form of axrep5 5199. (Contributed by SN, 21-Sep-2023.)
(∀𝑤 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑥 𝜑))
 
21-Sep-2023prmsimpcyc 30860 A group of prime order is cyclic if and only if it is simple. This is the first family of finite simple groups. (Contributed by Thierry Arnoux, 21-Sep-2023.)
𝐵 = (Base‘𝐺)    ⇒   ((♯‘𝐵) ∈ ℙ → (𝐺 ∈ SimpGrp ↔ 𝐺 ∈ CycGrp))
 
21-Sep-2023hashgt23el 13788 A set with more than two elements has at least three different elements. (Contributed by BTernaryTau, 21-Sep-2023.)
((𝑉 ∈ 𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))
 
21-Sep-2023eu6 2658 Alternate definition of the unique existential quantifier df-eu 2653 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2653 was then proved as dfeu 2680. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2160. (Revised by SN, 21-Sep-2023.)
(∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
 
20-Sep-2023fmla1 32638 The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎⊼𝑔𝑏) and ∀𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥 ∈ 𝑦. (Contributed by AV, 20-Sep-2023.)
(Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖∈𝑔𝑗)⊼𝑔(𝑘∈𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖∈𝑔𝑗))})
 
20-Sep-2023fmlasuc 32637 The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)
(𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢⊼𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
 
20-Sep-2023fmla 32632 The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)
(Fmla‘ω) = ∪ 𝑛 ∈ ω (Fmla‘𝑛)
 
20-Sep-2023ineq1 4184 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.) (Proof shortened by SN, 20-Sep-2023.)
(𝐴 = 𝐵 → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))

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 http://us2.metamath.org:88/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: http://us2.metamath.org:88/ocat/mmj2/mmj2.zip http://us2.metamath.org:88/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 us2.metamath.org:8888 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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