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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    Contact us.    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 Google Group.

Recent news items    (15-Nov-2025) Thierry Arnoux added a new proof to the 100 theorem list, The Impossibility of Trisecting the Angle and Doubling the Cube, trisecnconstr and 2sqr3nconstr.

(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 8-Nov-2026 at 5:33 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
1-Nov-2026coires2 6283 Postcomposing with a restricted identity relation is the same as corestricting. (Contributed by BJ, 1-Nov-2026.)
(( I ↾ 𝐵) ∘ 𝐴) = (𝐴 ∩ (V × 𝐵))
 
1-Nov-2026cnvresid 6281 A restricted identity relation is equal to its converse. (Contributed by FL, 4-Mar-2007.) Remove dependency on ax-12 2216 and shorten. (Revised by BJ, 1-Nov-2026.)
◡( I ↾ 𝐴) = ( I ↾ 𝐴)
 
31-Oct-202613an22anass 1379 Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.) (Proof shortened by Wolf Lammen, 31-Oct-2026.)
((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
 
28-Oct-2026mirimanofff 36630 Mirimanoff's paradox mirimanoff 36628 for founded sets. A set 𝑥 is founded if every set containing 𝑥 has a minimal element under membership. Compare with zfreg 9617 the defining condition of 𝑀 in the hypothesis. (Contributed by Adrian Ducourtial, 28-Oct-2026.)
𝑀 = {𝑥 ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)}    ⇒   𝑀 ∉ V
 
28-Oct-2026mirimanofffr 36629 Relative version of mirimanofff 36630 (which corresponds to the case 𝑉 = V). (Contributed by Adrian Ducourtial, 28-Oct-2026.)
𝑀 = {𝑥 ∈ 𝑉 ∣ ∀𝑦(𝑥 ∈ 𝑦 → ∃𝑧 ∈ 𝑦 (𝑧 ∩ 𝑦) = ∅)}    ⇒   𝑀 ∉ 𝑉
 
27-Oct-2026cbvex2ivw 43455 One direction of cbvex2vw 2076 does not need a DV condition between 𝑥 and 𝑦. Idea from relopabVD 46074. (Contributed by SN, 27-Oct-2026.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 → 𝜓))    ⇒   (∃𝑥∃𝑦𝜑 → ∃𝑧∃𝑤𝜓)
 
26-Oct-2026elpsseu 36637 A class has a unique element just in case it has a unique proper subset. (Contributed by Adrian Ducourtial, 26-Oct-2026.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑦 𝑦 ⊊ 𝐴)
 
26-Oct-2026elpssmo 36636 Uniqueness of elements is equivalent to uniqueness of proper subsets. (Contributed by Adrian Ducourtial, 26-Oct-2026.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑦 𝑦 ⊊ 𝐴)
 
26-Oct-2026elpssex 36635 A class has an element just in case it has a proper subset. (Contributed by Adrian Ducourtial, 26-Oct-2026.)
(∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ⊊ 𝐴)
 
26-Oct-2026relopabi 5805 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5252, ax-nul 5263, ax-pr 5395. (Revised by KP, 25-Oct-2021.) Avoid ax-11 2197 and ax-12 2216. (Revised by SN, 26-Oct-2026.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   Rel 𝐴
 
25-Oct-2026resssres 38139 A restricted class is included in a class if and only if it is included in that class restricted to any class including the first restricting class. (Contributed by BJ, 25-Oct-2026.)
(𝑋 ⊆ 𝑌 → ((𝐴 ↾ 𝑋) ⊆ 𝐵 ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌)))
 
25-Oct-2026funcocnv2 6872 Precomposition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by BJ, 25-Oct-2026.)
(Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
 
25-Oct-2026coass 6284 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.) Remove dependency on ax-11 2197. (Revised by BJ, 25-Oct-2026.)
((𝐴 ∘ 𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵 ∘ 𝐶))
 
25-Oct-2026idresrnsscocnv2 6276 A class precomposed with its converse includes the identity relation restricted to its range. (Contributed by BJ, 25-Oct-2026.)
( I ↾ ran 𝐴) ⊆ (𝐴 ∘ ◡𝐴)
 
25-Oct-2026idresdmsscocnv1 6275 A class postcomposed with its converse includes the identity relation restricted to its domain. (Contributed by BJ, 25-Oct-2026.)
( I ↾ dom 𝐴) ⊆ (◡𝐴 ∘ 𝐴)
 
25-Oct-2026rncocnv1 5974 The range of a class postcomposed with its converse is equal to its domain. (Contributed by BJ, 25-Oct-2026.)
ran (◡𝐴 ∘ 𝐴) = dom 𝐴
 
25-Oct-2026rncocnv2 5973 The range of a class precomposed with its converse is equal to its range. (Contributed by BJ, 25-Oct-2026.)
ran (𝐴 ∘ ◡𝐴) = ran 𝐴
 
25-Oct-2026dmcocnv2 5971 The domain of a class precomposed with its converse is equal to its range. (Contributed by BJ, 25-Oct-2026.)
dom (𝐴 ∘ ◡𝐴) = ran 𝐴
 
25-Oct-2026dmcocnv1 5970 The domain of a class postcomposed with its converse is equal to its domain. (Contributed by BJ, 25-Oct-2026.)
dom (◡𝐴 ∘ 𝐴) = dom 𝐴
 
25-Oct-2026brcoi 5853 Ordered pair membership in a composition. (Contributed by BJ, 25-Oct-2026.)
𝑋 ∈ V    &   𝑌 ∈ V    &   𝑍 ∈ V    ⇒   ((𝑋𝑅𝑌 ∧ 𝑌𝑆𝑍) → 𝑋(𝑆 ∘ 𝑅)𝑍)
 
24-Oct-2026df-finfld 46236 Define a function taking a prime power 𝑞 to the finite field of order 𝑞. When 𝑞 is an integer, there is a field of order 𝑞 (that is, its base set has 𝑞 elements) if and only if 𝑞 is a prime power. Additionally, any two finite fields of the same order are isomorphic. Thus, we usually refer to "the" finite field of order 𝑞 — which is often denoted 𝔽 𝑞 — without specifying a particular one. Usually, this causes no problems, but in Metamath the notation (𝔽‘𝑞) must unambigously denote a specific set. Since HFStructs can be well-ordered, we can use the smallest candidate according to the ordering. This depends on the fact that any finite field is isomorphic to an HFStruct. Indeed, this isomorphism result holds for most abstract mathematical structures with a finite base set. (For groups, see fingrphf 46231.) Thus, the technique used here could apply in other cases in which we want a particular representative of some class of finite structures. (Contributed by Eric Schmidt, 24-Oct-2026.)
𝔽 = (𝑞 ∈ (↑ “ (ℙ × ℕ)) ↦ inf((HFStruct ∩ {𝑘 ∈ Field ∣ (♯‘(Base‘𝑘)) = 𝑞}), HFStruct, <HFStruct ))
 
24-Oct-2026hfstructwe 46234 The relation <HFStruct is a well ordering of the class of HFStructs. (Contributed by Eric Schmidt, 24-Oct-2026.)
<HFStruct We HFStruct
 
24-Oct-2026df-lthfstruct 46233 Define a well ordering on HFStructs. The exact definition is not important. Any explicit definition of a well ordering would do. The definition we have chosen makes proving hfstructwe 46234 easier. (Contributed by Eric Schmidt, 24-Oct-2026.)
<HFStruct = {⟨𝑓, 𝑔⟩ ∣ ((ℎ ∈ HFStruct ↦ (ackBij‘(ℎ ∘ (♯ ↾ ω))))‘𝑓) E ((ℎ ∈ HFStruct ↦ (ackBij‘(ℎ ∘ (♯ ↾ ω))))‘𝑔)}
 
24-Oct-2026fingrphf 46231 Every finite group is isomorphic to a group with hereditarily finite components. (Contributed by Eric Schmidt, 24-Oct-2026.)
𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    ⇒   (𝜑 → ∃ℎ ∈ HFStruct 𝐺 ≃𝑔 ℎ)
 
24-Oct-2026grpcanonhfstruct 46230 A group defined on a hereditarily finite set is an HFStruct. (Contributed by Eric Schmidt, 24-Oct-2026.)
𝐵 = (Base‘𝐺)    &    ⨣ = (+𝑓‘𝐺)    &   𝐻 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⨣ ⟩}    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ HF )    ⇒   (𝜑 → 𝐻 ∈ HFStruct)
 
24-Oct-2026ackbij3 46220 The Ackermann bijection is a bijection. This is just a restatement of ackbij2 10347. (Contributed by Eric Schmidt, 24-Oct-2026.)
ackBij: HF ⟶1-1ontoω
 
24-Oct-2026df-ackbij 46219 Define the Ackermann bijection from hereditarily finite sets to ω. This definition is derived from ackbij2 10347. (Contributed by Eric Schmidt, 24-Oct-2026.)
ackBij = ∪ (rec((𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ ((𝑧 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑤 ∈ 𝑧 ({𝑤} × 𝒫 𝑤)))‘(𝑥 “ 𝑦)))), ∅) “ ω)
 
24-Oct-2026grpgimcanon 46123 The result of "canonicalizing" a group by discarding unneeded structural components and ensuring that the group operation is a binary operation is isomorphic to the original group. (Contributed by Eric Schmidt, 24-Oct-2026.)
𝐵 = (Base‘𝐺)    &    ⨣ = (+𝑓‘𝐺)    &   𝐻 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ⨣ ⟩}    ⇒   (𝐺 ∈ Grp → ( I ↾ 𝐵) ∈ (𝐺 GrpIso 𝐻))
 
24-Oct-2026imasgimw 46122 A relabeling of the elements of a group induces an isomorphism to the relabeled group. This theorem is the same as imasgim 19513 but with an extra, unused, hypothesis 𝐵 ∈ V. We do this in anticipation of a revision to theorems about “s in which we use a sethood hypothesis to avoid ax-rep 5235. When this happens, this theorem can be deleted and all usages replaced with usages of imasgim 19513. (Contributed by Eric Schmidt, 24-Oct-2026.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &   (𝜑 → 𝐹:𝑉⟶1-1onto𝐵)    &   (𝜑 → 𝑅 ∈ Grp)    &   𝐵 ∈ V    ⇒   (𝜑 → 𝐹 ∈ (𝑅 GrpIso 𝑈))
 
24-Oct-2026mpoexwd 46118 Deduction version of mpoexw 8116. (Contributed by Eric Schmidt, 24-Oct-2026.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐷 ∈ V    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)    ⇒   (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
 
24-Oct-2026relpropg 46113 A pair of ordered pairs is a relation. (Contributed by Eric Schmidt, 24-Oct-2026.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → Rel {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
 
24-Oct-2026fineqvhfbregs 35906 All sets are finite iff all sets are hereditarily finite. (Contributed by BTernaryTau, 30-Dec-2025.) Use HF. (Revised by BTernaryTau, 24-Oct-2026.)
(Fin = V ↔ HF = V)
 
24-Oct-2026eqhfregs 35905 The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. This version of eqhf 35820 replaces setinds2 9779 with setinds2regs 35899 and trssfihf 35819 with trssfihfregs 35904. (Contributed by BTernaryTau, 21-Jan-2026.) Use HF. (Revised by BTernaryTau, 24-Oct-2026.)
(𝐻 = HF ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
 
24-Oct-2026trssfihfregs 35904 If every element in a transitive class is finite, then every element is also hereditarily finite. This version of trssfihf 35819 replaces setinds2 9779 with setinds2regs 35899. (Contributed by BTernaryTau, 20-Jan-2026.) Use HF. (Revised by BTernaryTau, 24-Oct-2026.)
((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ HF )
 
24-Oct-2026eqhf 35820 The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. (Contributed by BTernaryTau, 24-Jan-2026.) Use HF. (Revised by BTernaryTau, 24-Oct-2026.)
(𝐻 = HF ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))
 
24-Oct-2026trssfihf 35819 If every element in a transitive class is finite, then every element is also hereditarily finite. (Contributed by BTernaryTau, 24-Jan-2026.) Use HF. (Revised by BTernaryTau, 24-Oct-2026.)
((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ HF )
 
24-Oct-2026fundif 6610 A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.) Shorten proof and remove dependency on ax-sep 5252 and ax-pr 5395. (Revised by Eric Schmidt, 24-Oct-2026.)
(Fun 𝐹 → Fun (𝐹 ∖ 𝐴))
 
24-Oct-2026fununfun 6608 If the union of classes is a function, the classes themselves are functions. (Contributed by AV, 18-Jul-2019.) Shorten proof and remove dependency on ax-sep 5252 and ax-pr 5395. (Revised by Eric Schmidt, 24-Oct-2026.)
(Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))
 
23-Oct-2026nf5di 2184 Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.) Reduce axiom dependencies. (Revised by SN and Sky Winter, 23-Oct-2026.)
(𝜑 → Ⅎ𝑥𝜑)    ⇒   Ⅎ𝑥𝜑
 
23-Oct-2026hb3an 1907 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) Reduce axiom dependencies. (Revised by SN and Sky Winter, 23-Oct-2026.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜒)    ⇒   ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
 
22-Oct-2026reldm 8079 An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) (Proof shortened by Sky Winter, 22-Oct-2026.)
(Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))
 
22-Oct-2026elpreima 7080 Membership in the preimage of a class under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Sky Winter, 22-Oct-2026.)
(𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶)))
 
22-Oct-2026abbib 2832 Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2149 and df-clel 2839 (by avoiding use of cleqh 2893). (Revised by BJ, 23-Jun-2019.) Reverse the biconditional. (Revised by Wolf Lammen, 23-Feb-2025.) (Proof shortened by SN, 22-Oct-2026.)
({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
 
22-Oct-2026hban 1906 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) Reduce axiom dependencies. (Revised by SN and Sky Winter, 22-Oct-2026.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    ⇒   ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
 
20-Oct-2026ecscottd 35844 Applying the Scott operation to equivalence classes preserves their equality property. (Contributed by BTernaryTau, 20-Oct-2026.)
(𝜑 → 𝑅 Er 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ Scott [𝐴]𝑅 = Scott [𝐵]𝑅))
 
20-Oct-2026erscottd 35843 Applying the Scott operation to equivalence classes preserves their equality property. (Contributed by BTernaryTau, 20-Oct-2026.)
(𝜑 → 𝑅 Er 𝑉)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ Scott {𝑥 ∣ 𝑥𝑅𝐴} = Scott {𝑥 ∣ 𝑥𝑅𝐵}))
 
20-Oct-2026trscott 35842 The Scott's trick set of a transitive class is also transitive. (Contributed by BTernaryTau, 20-Oct-2026.)
(Tr 𝐴 → Tr Scott 𝐴)
 
19-Oct-2026gdlopcfv8 35979 The value of the combined Gödel operations function when its first argument is 8. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨8o, 𝐴, 𝐵⟩) = (𝐴 ∩ (Cnv3‘𝐵)))
 
19-Oct-2026gdlopcfv7 35978 The value of the combined Gödel operations function when its first argument is 7. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨7o, 𝐴, 𝐵⟩) = (𝐴 ∩ (Cnv2‘𝐵)))
 
19-Oct-2026gdlopcfv6 35977 The value of the combined Gödel operations function when its first argument is 6. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨6o, 𝐴, 𝐵⟩) = (𝐴 ∩ ◡𝐵))
 
19-Oct-2026gdlopcfv5 35976 The value of the combined Gödel operations function when its first argument is 5. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨5o, 𝐴, 𝐵⟩) = (𝐴 ∩ dom 𝐵))
 
19-Oct-2026gdlopcfv4 35975 The value of the combined Gödel operations function when its first argument is 4. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨4o, 𝐴, 𝐵⟩) = (𝐴 ↾ 𝐵))
 
19-Oct-2026gdlopcfv3 35974 The value of the combined Gödel operations function when its first argument is 3. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨3o, 𝐴, 𝐵⟩) = (𝐴 ∖ 𝐵))
 
19-Oct-2026gdlopcfv2 35973 The value of the combined Gödel operations function when its first argument is 2. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨2o, 𝐴, 𝐵⟩) = (𝐴 ∩ E ))
 
19-Oct-2026gdlopcfv1 35972 The value of the combined Gödel operations function when its first argument is 1. (Contributed by BTernaryTau, 19-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ ‘⟨1o, 𝐴, 𝐵⟩) = {𝐴, 𝐵})
 
19-Oct-2026gdlopcfvlem 35971 Lemma for gdlopcfv2 35973 through gdlopcfv8 35979. (Contributed by BTernaryTau, 19-Oct-2026.)
𝐴 ∈ On    &   𝐵 ∈ On    &   𝐴 ∈ 𝐵    ⇒   ((𝜑 ∧ 𝐵 = 𝐶) → if(𝐶 = 𝐴, 𝐷, 𝐸) = 𝐸)
 
19-Oct-2026suceldifsn0 35791 A successor that is a member of a set is also a member of that set with the empty set removed. (Contributed by BTernaryTau, 19-Oct-2026.)
𝐴 ∈ 𝐵    &   𝐴 = suc 𝐶    ⇒   𝐴 ∈ (𝐵 ∖ {∅})
 
19-Oct-2026eqsucidi 35790 A set belongs to its successor. Inference form of sucidg 6464 with an equality hypothesis. (Contributed by BTernaryTau, 19-Oct-2026.)
𝐴 ∈ 𝑉    &   𝐵 = suc 𝐴    ⇒   𝐴 ∈ 𝐵
 
19-Oct-2026ontr1i 35789 Transitive law for ordinal numbers. Inference form of ontr1 6428. (Contributed by BTernaryTau, 19-Oct-2026.)
𝐶 ∈ On    &   𝐴 ∈ 𝐵    &   𝐵 ∈ 𝐶    ⇒   𝐴 ∈ 𝐶
 
18-Oct-2026aiotaexim 48373 Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is a set (that means that "the unique element such that 𝜑 " exists). Analog to iotan0 6547 which requires 𝐴 to be nonempty instead of being a set. That gives a more natural statement, closer to statements like spcgv 3554. (Proposed by BJ, 18-Oct-2026.) (Contributed by AV, 18-Oct-2026.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐴 = (℩'𝑥𝜑)) → 𝜓)
 
18-Oct-2026gdlop8fv 35970 The value of the eighth Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ8‘⟨𝐴, 𝐵⟩) = (𝐴 ∩ (Cnv3‘𝐵)))
 
18-Oct-2026gdlop7fv 35969 The value of the seventh Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ7‘⟨𝐴, 𝐵⟩) = (𝐴 ∩ (Cnv2‘𝐵)))
 
18-Oct-2026gdlop6fv 35968 The value of the sixth Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ6‘⟨𝐴, 𝐵⟩) = (𝐴 ∩ ◡𝐵))
 
18-Oct-2026gdlop5fv 35967 The value of the fifth Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ5‘⟨𝐴, 𝐵⟩) = (𝐴 ∩ dom 𝐵))
 
18-Oct-2026gdlop4fv 35966 The value of the fourth Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ4‘⟨𝐴, 𝐵⟩) = (𝐴 ↾ 𝐵))
 
18-Oct-2026gdlop3fv 35965 The value of the third Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ3‘⟨𝐴, 𝐵⟩) = (𝐴 ∖ 𝐵))
 
18-Oct-2026gdlop2fv 35964 The value of the second Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ2‘⟨𝐴, 𝐵⟩) = (𝐴 ∩ E ))
 
18-Oct-2026gdlop1fv 35963 The value of the first Gödel operation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (ℱ1‘⟨𝐴, 𝐵⟩) = {𝐴, 𝐵})
 
18-Oct-2026mpofvop2d 35799 Value of a two-argument function (usually an operation) in maps-to notation at an ordered pair. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝑅 ∈ 𝐴)    &   (𝜑 → 𝑆 ∈ 𝐵)    &   (𝜑 → 𝐿 ∈ 𝑉)    &   ((𝜑 ∧ 𝑅 = 𝑥) → 𝐾 = 𝐶)    &   ((𝜑 ∧ 𝑆 = 𝑦) → 𝐿 = 𝐾)    &   𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   (𝜑 → (𝐹‘⟨𝑅, 𝑆⟩) = 𝐿)
 
18-Oct-2026mpofvopd 35798 Value of a two-argument function (usually an operation) in maps-to notation at an ordered pair. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝑅 ∈ 𝐴)    &   (𝜑 → 𝑆 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝑉)    &   ((𝜑 ∧ ⟨𝑅, 𝑆⟩ = ⟨𝑥, 𝑦⟩) → 𝑌 = 𝐶)    &   𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   (𝜑 → (𝐹‘⟨𝑅, 𝑆⟩) = 𝑌)
 
18-Oct-2026mpofvd 35797 Value of a two-argument function (usually an operation) in maps-to notation. (Contributed by BTernaryTau, 18-Oct-2026.)
(𝜑 → 𝑋 ∈ (𝐴 × 𝐵))    &   (𝜑 → 𝑌 ∈ 𝑉)    &   ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦⟩) → 𝑌 = 𝐶)    &   𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   (𝜑 → (𝐹‘𝑋) = 𝑌)
 
17-Oct-2026fnlfvk3n0 36028 The value of 𝐹𝐿 when the value of 𝐾3 is nonzero. Part of Proposition 15.14 of [TakeutiZaring] p. 158. (Contributed by BTernaryTau, 17-Oct-2026.)
((𝐴 ∈ On ∧ (𝐾3‘𝐴) ≠ ∅) → (𝐹𝐿‘𝐴) = (ℱ ‘⟨(𝐾3‘𝐴), (𝐹𝐿‘(𝐾1‘𝐴)), (𝐹𝐿‘(𝐾2‘𝐴))⟩))
 
17-Oct-2026fnlfvk30 36027 The value of 𝐹𝐿 when the value of 𝐾3 is zero. Part of Proposition 15.14 of [TakeutiZaring] p. 158. (Contributed by BTernaryTau, 17-Oct-2026.)
((𝐴 ∈ On ∧ (𝐾3‘𝐴) = ∅) → (𝐹𝐿‘𝐴) = (𝐹𝐿 “ 𝐴))
 
17-Oct-20261olt9o 35880 Ordinal 1 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
1o ∈ 9o
 
17-Oct-20262olt9o 35879 Ordinal 2 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
2o ∈ 9o
 
17-Oct-20263olt9o 35878 Ordinal 3 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
3o ∈ 9o
 
17-Oct-20264olt9o 35877 Ordinal 4 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
4o ∈ 9o
 
17-Oct-20265olt9o 35876 Ordinal 5 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
5o ∈ 9o
 
17-Oct-20266olt9o 35875 Ordinal 6 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
6o ∈ 9o
 
17-Oct-20267olt9o 35874 Ordinal 7 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
7o ∈ 9o
 
17-Oct-20268olt9o 35873 Ordinal 8 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026.)
8o ∈ 9o
 
17-Oct-2026iotan0 6547 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.) Remove sethood antecedent. (Revised by Eric Schmidt, 17-Oct-2026.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
 
16-Oct-2026bj-coi1 38152 The identity relation is a right-unit for composition. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) Strengthen to biconditional form and shorten (Revised by BJ, 16-Oct-2026.)
(Rel 𝐴 ↔ (𝐴 ∘ I ) = 𝐴)
 
16-Oct-2026bj-cnvcnv 38149 The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.) (Proof shortened by BJ, 16-Oct-2026.) Prefer use of bj-cnvcnv2 38148. (Proof modification is discouraged.)
◡◡𝐴 = (𝐴 ∩ (V × V))
 
16-Oct-2026bj-cnvcnv2 38148 The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.) Do not depend on cnvcnv 6196 in order to prove it from cnvcnv2 6197. (Revised by BJ, 16-Oct-2026.) Do not depend on cnvcnv 6196. (Proof modification is discouraged.)
◡◡𝐴 = (𝐴 ↾ V)
 
16-Oct-2026bj-dfrel2 38147 Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.) (Proof shortened by BJ, 16-Oct-2026.) Do not use opabid2 5812. (Proof modification is discouraged.)
(Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
 
16-Oct-2026bj-dfrel4v 38146 A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6966 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) Do not depend on df-cnv 5663, reverse equality. (Revised by BJ, 16-Oct-2026.) Note: this theorem could be used in place of opabid2 5812, which could be deleted.
(Rel 𝑅 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = 𝑅)
 
16-Oct-2026relssresv 38145 A relation is included in a class if and only if it is included in its "relation part" (see relpart 38143). (Contributed by BJ, 16-Oct-2026.)
(Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ↾ V)))
 
16-Oct-2026bj-dfrel3 38144 Alternate definition of relation: a class is a relation if and only if it is equal to its "relation part" (see relpart 38143). (Contributed by NM, 14-May-2008.) Do not depend on df-cnv 5663 to move it earlier. (Revised by BJ, 16-Oct-2026.)
(Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
 
16-Oct-2026relpart 38143 Two ways of expressing the "relation part" of a class: its restriction to the universal class (that is, its intersection with the Cartesian square of the universal class) equals the class of ordered pairs it contains (the class of ordered pairs whose components are related by it).

The "relation part" of a class is a relation (see df-rel 5662 and relres 6007) and is the largest relation included in that class (that follows from relssresv 38145). Taking the relation part of a class is a projection onto the collection of relations, that is, it is an idempotent operation (residm 6013) with image the collection of relations (by relres 6007 for one inclusion and the forward implication of dfrel3 6205 for the other). As with any projection, a class is in its image if and only if its is a fixpoint of that projection (which is precisely dfrel3 6205).

Other expressions giving the relation part of a class are cnvcnv 6196, coi1res 38151. (Contributed by BJ, 16-Oct-2026.)

(𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
 
16-Oct-2026bj-resss2 38142 The restriction of a class to a class is included in the Cartesian product of the restricting class with the universal class. (Contributed by BJ, 16-Oct-2026.)
(𝑅 ↾ 𝐴) ⊆ (𝐴 × V)
 
16-Oct-2026bj-dfres2 38141 Alternate definition of the restriction of a class to another, as an ordered pair class abstraction: the restriction of a class to another is the class of ordered pairs whose first component is in the restricting class and whose components are related by the restricted class. (Contributed by Mario Carneiro, 5-Nov-2013.) Do not depend on df-rel 5662. (Revised by BJ, 16-Oct-2026.)
(𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)}
 
16-Oct-2026bj-elres 38140 Membership in a restriction. (Contributed by Scott Fenton, 17-Mar-2011.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Use the 𝑥𝐵𝑦 idiom. (Revised by BJ, 16-Oct-2026.)
(𝐴 ∈ (𝐵 ↾ 𝐶) ↔ ∃𝑥 ∈ 𝐶 ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝐵𝑦))
 
16-Oct-2026bj-elinxp 38138 Membership in an intersection with a Cartesian product. (Contributed by Peter Mazsa, 9-Sep-2022.) Use the 𝑥𝑅𝑦 idiom and give a shorter proof not depending on df-rel 5662. (Revised by BJ, 16-Oct-2026.) Note to remove when moving to main section: a shortening of the exact statement elinxp 6020 is possible along the same line.
(𝐶 ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐶 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
 
16-Oct-2026jltaleph 36026 The value of 𝐽 is less than an aleph if its first two arguments are less than that aleph. Proposition 15.11 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 16-Oct-2026.)
((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷))
 
16-Oct-2026jltalephlem 36025 Lemma for jltaleph 36026. (Contributed by BTernaryTau, 16-Oct-2026.)
((𝐶 ∈ 9o ∧ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩)) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ≈ (𝐽0‘⟨𝐴, 𝐵⟩))
 
16-Oct-2026relres 6007 A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by BJ, 16-Oct-2026.)
Rel (𝐴 ↾ 𝐵)
 
16-Oct-2026elvv 5730 Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.) (Proof shortened by BJ, 16-Oct-2026.)
(𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
 
15-Oct-2026pibt2 38532 Theorem T000002 of pi-base, a countably compact topology is also weakly countably compact. See pibp19 38529 and pibp21 38530 for the definitions of the relevant properties. This proof uses the axiom of choice. (Contributed by ML, 30-Mar-2021.) Remove dependency on ax-reg 9613. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
𝑋 = ∪ 𝐽    &   𝐶 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥((∪ 𝑥 = ∪ 𝑦 ∧ 𝑦 ≼ ω) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ 𝑥 = ∪ 𝑧)}    &   𝑊 = {𝑥 ∈ Top ∣ ∀𝑦 ∈ (𝒫 ∪ 𝑥 ∖ Fin)∃𝑧 ∈ ∪ 𝑥𝑧 ∈ ((limPt‘𝑥)‘𝑦)}    ⇒   (𝐽 ∈ 𝐶 → 𝐽 ∈ 𝑊)
 
15-Oct-2026nlpfvineqsn 38524 Given a subset 𝐴 of 𝑋 with no limit points, there exists a function from each point 𝑝 of 𝐴 to an open set intersecting 𝐴 only at 𝑝. This proof uses the axiom of choice. (Contributed by ML, 23-Mar-2021.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
𝑋 = ∪ 𝐽    ⇒   (𝐴 ∈ 𝑉 → ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ ((limPt‘𝐽)‘𝐴) = ∅) → ∃𝑓(𝑓:𝐴⟶𝐽 ∧ ∀𝑝 ∈ 𝐴 ((𝑓‘𝑝) ∩ 𝐴) = {𝑝})))
 
15-Oct-2026cardprcALT 36634 Alternate proof of cardprc 10088 from cardprcr 36633. (Contributed by Adrian Ducourtial, 15-Oct-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V
 
15-Oct-2026cardprcr 36633 Relative version of cardprc 10088. (Contributed by Adrian Ducourtial, 15-Oct-2026.)
𝐴 = {𝑥 ∈ 𝑉 ∣ (card‘𝑥) = 𝑥}    &   (𝐴 ∈ 𝑉 → (har‘∪ 𝐴) ∈ 𝑉)    ⇒    ¬ 𝐴 ∈ 𝑉
 
15-Oct-2026onprcALT 36632 Alternate proof of onprc 7816 from onprcr 36631. (Contributed by Adrian Ducourtial, 15-Oct-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ On ∈ V
 
15-Oct-2026onprcr 36631 Relative version of onprc 7816. (Contributed by Adrian Ducourtial, 15-Oct-2026.)
𝐴 = {𝑥 ∈ 𝑉 ∣ Ord 𝑥}    &   Tr 𝐴    ⇒    ¬ 𝐴 ∈ 𝑉
 
15-Oct-2026ptpconn 36165 The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn)
 
15-Oct-2026omssubadd 35002 A constructed outer measure is countably sub-additive. Lemma 1.5.4 of [Bogachev] p. 17. (Contributed by Thierry Arnoux, 21-Sep-2019.) (Revised by AV, 4-Oct-2020.) Remove dependency on ax-reg 9613. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
𝑀 = (toOMeas‘𝑅)    &   (𝜑 → 𝑄 ∈ 𝑉)    &   (𝜑 → 𝑅:𝑄⟶(0[,]+∞))    &   ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐴 ⊆ ∪ 𝑄)    &   (𝜑 → 𝑋 ≼ ω)    ⇒   (𝜑 → (𝑀‘∪ 𝑦 ∈ 𝑋 𝐴) ≤ Σ*𝑦 ∈ 𝑋(𝑀‘𝐴))
 
15-Oct-2026cmpcref 34552 Equivalent definition of compact space in terms of open cover refinements. Compact spaces are topologies with finite open cover refinements. (Contributed by Thierry Arnoux, 7-Jan-2020.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
Comp = CovHasRefFin
 
15-Oct-2026reff 34541 For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a definition of refinement. Note that this definition uses the axiom of choice through ac6sg 10593. (Contributed by Thierry Arnoux, 12-Jan-2020.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
(𝐴 ∈ 𝑉 → (𝐴Ref𝐵 ↔ (∪ 𝐵 ⊆ ∪ 𝐴 ∧ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑣 ∈ 𝐴 𝑣 ⊆ (𝑓‘𝑣)))))
 
15-Oct-2026fedgmul 34333 The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.) Remove dependency on ax-reg 9613. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
𝐴 = ((subringAlg ‘𝐸)‘𝑉)    &   𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   𝐶 = ((subringAlg ‘𝐹)‘𝑉)    &   𝐹 = (𝐸 ↾s 𝑈)    &   𝐾 = (𝐸 ↾s 𝑉)    &   (𝜑 → 𝐸 ∈ DivRing)    &   (𝜑 → 𝐹 ∈ DivRing)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   (𝜑 → 𝑉 ∈ (SubRing‘𝐹))    ⇒   (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶)))
 
15-Oct-2026elrspunidl 34048 Elementhood in the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑆 ⊆ (LIdeal‘𝑅))    ⇒   (𝜑 → (𝑋 ∈ (𝑁‘∪ 𝑆) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝑆)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘 ∈ 𝑆 (𝑎‘𝑘) ∈ 𝑘)))
 
15-Oct-2026acunirnmpt2f 33327 Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)    &   Ⅎ𝑗𝐴    &   Ⅎ𝑗𝐶    &   Ⅎ𝑗𝐷    &   𝐶 = ∪ 𝑗 ∈ 𝐴 𝐵    &   (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)    &   ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
 
15-Oct-2026acunirnmpt2 33326 Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)    &   (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)    ⇒   (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
 
15-Oct-2026ac6mapd 33289 Axiom of choice equivalent, deduction form. (Contributed by Thierry Arnoux, 13-Oct-2025.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
(𝑦 = (𝑓‘𝑥) → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝜓)    ⇒   (𝜑 → ∃𝑓 ∈ (𝐵 ↑m 𝐴)∀𝑥 ∈ 𝐴 𝜒)
 
15-Oct-2026unidifsnne 33204 The other element of a pair is not the known element. (Contributed by Thierry Arnoux, 26-Aug-2017.) Remove dependency on ax-reg 9613. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
 
15-Oct-2026unidifsnel 33203 The other element of a pair is an element of the pair. (Contributed by Thierry Arnoux, 26-Aug-2017.) Remove dependency on ax-reg 9613. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {𝑋}) ∈ 𝑃)
 
15-Oct-2026foresf1o 33172 From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶onto𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥⟶1-1onto𝐵)
 
15-Oct-2026acsmapd 18757 In an algebraic closure system, if 𝑇 is contained in the closure of 𝑆, there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that the closure of ∪ ran 𝑓 contains 𝑇. This is proven by applying acsficl2d 18755 to each element of 𝑇. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-reg 9613 and ax-inf2 9669. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
(𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   𝑁 = (mrCls‘𝐴)    &   (𝜑 → 𝑆 ⊆ 𝑋)    &   (𝜑 → 𝑇 ⊆ (𝑁‘𝑆))    ⇒   (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)))
 
15-Oct-2026ac6 10585 Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set 𝐵, where 𝜑 depends on 𝑥 (the natural number) and 𝑦 (to specify a member of 𝐵). A stronger version of this theorem, ac6s 10589, allows 𝐵 to be a proper class. Unlike ac6sg 10593, which allows 𝐵 to be a proper class and so uses the Collection principle bnd2 9983, this needs neither ax-reg 9613 nor ax-inf2 9669. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) Replace hypotheses with antecedents. (Revised by Timothy J. O'Malley, 15-Oct-2026.)
(𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
 
14-Oct-2026suppvalbr 8208 The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Sky Winter, 14-Oct-2026.)
((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))})
 
13-Oct-2026mirimanoffr 36627 Relative version of mirimanoff 36628 (which corresponds to the case 𝑉 = V).

Note that the definition of 𝑀 presupposes that ω is available as a set (omex 9671), which is equivalent to assuming ax-inf2 9669. (Contributed by Adrian Ducourtial, 13-Oct-2026.)

𝑀 = {𝑥 ∈ 𝑉 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}    ⇒   𝑀 ∉ 𝑉
 
12-Oct-2026k3n0k2lt 36024 The value of 𝐾2 is strictly bounded by its argument when the value of 𝐾3 is nonzero. Part of Proposition 15.10(2) of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 12-Oct-2026.)
((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐾2‘𝐴) ∈ 𝐴)
 
12-Oct-2026k3n0k1lt 36023 The value of 𝐾1 is strictly bounded by its argument when the value of 𝐾3 is nonzero. Part of Proposition 15.10(2) of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 12-Oct-2026.)
((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐾1‘𝐴) ∈ 𝐴)
 
12-Oct-2026k2le 36022 The value of 𝐾2 is bounded by its argument. Part of Proposition 15.10(1) of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 12-Oct-2026.)
(𝐴 ∈ On → (𝐾2‘𝐴) ⊆ 𝐴)
 
12-Oct-2026k1le 36021 The value of 𝐾1 is bounded by its argument. Part of Proposition 15.10(1) of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 12-Oct-2026.)
(𝐴 ∈ On → (𝐾1‘𝐴) ⊆ 𝐴)
 
12-Oct-2026k3n0j0lt 36020 A strict upper bound on the value of 𝐽0 for nonzero values of 𝐾3. (Contributed by BTernaryTau, 12-Oct-2026.)
((𝐴 ∈ On ∧ ∅ ∈ (𝐾3‘𝐴)) → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ∈ 𝐴)
 
12-Oct-2026j0le 36019 An upper bound on the value of 𝐽0. (Contributed by BTernaryTau, 12-Oct-2026.)
(𝐴 ∈ On → (𝐽0‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴)⟩) ⊆ 𝐴)
 
11-Oct-2026mirimanoff 36628 Mirimanoff's paradox: The class of grounded sets is not a set. Here, a set 𝑥 is grounded if there is no infinite membership-descending sequence of sets beginning with 𝑥. See Paragraph 1 of [Quine] p. 37 . (Contributed by Adrian Ducourtial, 11-Oct-2026.)
𝑀 = {𝑥 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}    ⇒   𝑀 ∉ V
 
11-Oct-2026jk1k2k3 36018 Identity law for the 𝐽 function. Proposition 15.8 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 11-Oct-2026.)
(𝐴 ∈ On → 𝐴 = (𝐽‘⟨(𝐾1‘𝐴), (𝐾2‘𝐴), (𝐾3‘𝐴)⟩))
 
11-Oct-2026k39o 36017 Values of the 𝐾3 function are ordinals less than 9. (Contributed by BTernaryTau, 11-Oct-2026.)
(𝐾3‘𝐴) ∈ 9o
 
11-Oct-2026k2on 36016 Values of the 𝐾2 function are ordinals. (Contributed by BTernaryTau, 11-Oct-2026.)
(𝐾2‘𝐴) ∈ On
 
11-Oct-2026k1on 36015 Values of the 𝐾1 function are ordinals. (Contributed by BTernaryTau, 11-Oct-2026.)
(𝐾1‘𝐴) ∈ On
 
11-Oct-2026noel 4286 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) Remove dependency on ax-10 2180, ax-11 2197, and ax-12 2216. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.) (Proof shortened by BJ, 11-Oct-2026.)
¬ 𝐴 ∈ ∅
 
11-Oct-2026sbcfal 3816 Substituting a class in the false truth constant results in the false truth constant. The analogous result with the true truth constant does not hold in general since substituting a proper class results in falsehood (sbcex 3752), but holds for sets by sbcg 3814. (Contributed by BJ, 11-Oct-2026.)
([𝐴 / 𝑥]⊥ ↔ ⊥)
 
10-Oct-2026prelrrx2bd 50031 Equivalent condition for 𝑍 being a point in a real Euclidean space of dimension 2, determined by its coordinates. (Contributed by Thierry Arnoux, 10-Oct-2026.)
𝐼 = (0..^2)    &   𝑃 = (ℝ ↑m 𝐼)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    ⇒   (𝜑 → (𝑍 = ⟨“𝐴𝐵”⟩ ↔ (𝑍 ∈ 𝑃 ∧ ((𝑍‘0) = 𝐴 ∧ (𝑍‘1) = 𝐵))))
 
10-Oct-2026dfk32 36014 Alternate definition of 𝐾3 using maps-to notation. (Contributed by BTernaryTau, 10-Oct-2026.)
𝐾3 = (𝑥 ∈ On ↦ (2nd ‘(◡𝐽‘𝑥)))
 
10-Oct-2026dfk22 36013 Alternate definition of 𝐾2 using maps-to notation. (Contributed by BTernaryTau, 10-Oct-2026.)
𝐾2 = (𝑥 ∈ On ↦ (2nd ‘(1st ‘(◡𝐽‘𝑥))))
 
10-Oct-2026s2mapd 15070 A doubleton word is a function on the doubleton (0..^2) (which is equal to {0, 1} by fzo0to2pr 13912). See also s3rexrd 15131. (Contributed by Thierry Arnoux, 10-Oct-2026.)
(𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    ⇒   (𝜑 → ⟨“𝑋𝑌”⟩ ∈ (𝑆 ↑m (0..^2)))
 
10-Oct-2026s1mapd 14783 A singleton word is a function on the singleton {0}. See also s2mapd 15070. (Contributed by Thierry Arnoux, 10-Oct-2026.)
(𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑆)    ⇒   (𝜑 → ⟨“𝑋”⟩ ∈ (𝑆 ↑m {0}))
 
9-Oct-2026dfk12 36012 Alternate definition of 𝐾1 using maps-to notation. (Contributed by BTernaryTau, 9-Oct-2026.)
𝐾1 = (𝑥 ∈ On ↦ (1st ‘(1st ‘(◡𝐽‘𝑥))))
 
9-Oct-2026ot1st2nd 35795 Reconstruction of a member of a nested Cartesian product in terms of its ordered triple components. (Contributed by BTernaryTau, 9-Oct-2026.)
(𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) → 𝐴 = ⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴)), (2nd ‘𝐴)⟩)
 
9-Oct-2026mumul 27538 The Möbius function is a multiplicative function. This is one of the primary interests of the Möbius function as an arithmetic function. (Contributed by Mario Carneiro, 3-Oct-2014.) (Proof shortened by Timothy J. O'Malley, 9-Oct-2026.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (μ‘(𝐴 · 𝐵)) = ((μ‘𝐴) · (μ‘𝐵)))
 
9-Oct-2026prmreclem2 17124 Lemma for prmrec 17129. There are at most 2↑𝐾 squarefree numbers which divide no primes larger than 𝐾. (We could strengthen this to 2↑♯(ℙ ∩ (1...𝐾)) but there's no reason to.) We establish the inequality by showing that the prime counts of the number up to 𝐾 completely determine it because all higher prime counts are zero, and they are all at most 1 because no square divides the number, so there are at most 2↑𝐾 possibilities. (Contributed by Mario Carneiro, 5-Aug-2014.) (Proof shortened by Timothy J. O'Malley, 9-Oct-2026.)
𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛}    &   𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))    ⇒   (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾))
 
9-Oct-2026arisum2 16059 Arithmetic series sum of the first 𝑁 nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 2-Aug-2021.) (Proof shortened by Timothy J. O'Malley, 9-Oct-2026.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑁 − 1))𝑘 = (((𝑁↑2) − 𝑁) / 2))
 
8-Oct-2026omexcn 43894 The axioms of the complex numbers imply the Axiom of Infinity (in the form of omex 9671): the map of om2uzf1oi 14123 is a one-to-one onto mapping from ω onto a set of integers, so ω is a set. Compare dcomex 10552 and grothomex 10941. Note that our proof depends on neither ax-inf 9666 nor ax-inf2 9669. (Contributed by Timothy J. O'Malley, 8-Oct-2026.)
ω ∈ V
 
8-Oct-2026cnveq0b 6204 A class is empty iff it is a relation whose converse is empty. (Contributed by SN, 8-Oct-2026.)
(𝐴 = ∅ ↔ (Rel 𝐴 ∧ ◡𝐴 = ∅))
 
7-Oct-2026ordunmax 35792 The union of two ordinals is equal to their maximum. (Contributed by BTernaryTau, 7-Oct-2026.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∪ 𝐵) = if(𝐴 ⊆ 𝐵, 𝐵, 𝐴))
 
6-Oct-2026jf1o 36011 𝐽 is a bijection from ((On × On) × 9o) to On. Proposition 15.3 of [TakeutiZaring] p. 155. (Contributed by BTernaryTau, 6-Oct-2026.)
𝐽:((On × On) × 9o)⟶1-1ontoOn
 
6-Oct-2026jf1 36010 𝐽 is a one-to-one function. Part of Proposition 15.3 of [TakeutiZaring] p. 155. (Contributed by BTernaryTau, 6-Oct-2026.)
𝐽:((On × On) × 9o)⟶1-1On
 
5-Oct-2026dfon2lem1 36723 Lemma for dfon2 36732. (Contributed by Scott Fenton, 28-Feb-2011.) (Proof shortened by SN, 5-Oct-2026.)
Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}
 
5-Oct-2026omeueq 35796 The division algorithm for ordinal multiplication (omeu 8621) produces a unique quotient and remainder. (Contributed by BTernaryTau, 5-Oct-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐴 ≠ ∅)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → 𝐷 ∈ On)    &   (𝜑 → 𝐸 ∈ 𝐴)    &   (𝜑 → ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸))    ⇒   (𝜑 → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))
 
4-Oct-2026bj-pr0moore 38228 A pair containing the empty set is a Moore collection. (Contributed by BJ, 4-Oct-2026.)
{∅, 𝐴} ∈ Moore
 
4-Oct-2026dfj2 36009 Alternate definition of 𝐽 as a one-argument function. (Contributed by BTernaryTau, 4-Oct-2026.)
𝐽 = (𝑥 ∈ ((On × On) × 9o) ↦ ((9o ·o (𝐽0‘(1st ‘𝑥))) +o (2nd ‘𝑥)))
 
2-Oct-2026vsmrep 36090 The universe is a standard model of the strong Axiom Scheme of Replacement. (Contributed by BTernaryTau, 2-Oct-2026.)
V ⊨ Ax-Rep*
 
2-Oct-2026vsmext 36089 The universe is a standard model of the Axiom of Extensionality. (Contributed by BTernaryTau, 2-Oct-2026.)
V ⊨ Ax-Ext
 
2-Oct-2026df-smext 36083 Define the standard model of Extensionality predicate, which is true for a class that is a model of the Axiom of Extensionality under the membership predicate. Our notation is similar to that of df-prv 36278 and df-gzext 36385 (𝑀⊧AxExt), but 𝑀 ⊨ Ax-Ext uses a different double turnstile character and a different format for the axiom name. (Contributed by BTernaryTau, 2-Oct-2026.)
(𝑀 ⊨ Ax-Ext ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
 
30-Sep-2026cocanss1 46116 Cancellation law for composition. Suggested by BJ. (Contributed by Eric Schmidt, 30-Sep-2026.)
(𝜑 → Fun ◡𝐹)    &   (𝜑 → Rel 𝐻)    &   (𝜑 → ran 𝐻 ⊆ dom 𝐹)    ⇒   (𝜑 → ((𝐹 ∘ 𝐻) ⊆ (𝐹 ∘ 𝐾) ↔ 𝐻 ⊆ 𝐾))
 
30-Sep-2026cocanss2 46114 Cancellation law for composition. Suggested by BJ. (Contributed by Eric Schmidt, 30-Sep-2026.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Rel 𝐻)    &   (𝜑 → dom 𝐻 ⊆ ran 𝐹)    ⇒   (𝜑 → ((𝐻 ∘ 𝐹) ⊆ (𝐾 ∘ 𝐹) ↔ 𝐻 ⊆ 𝐾))
 
30-Sep-2026smrepsrep 36088 Derive the strong version of the Axiom Scheme of Replacement used in wfaxrep 46179 (see comments there) from 𝑀 ⊨ Ax-Rep*. (Contributed by BTernaryTau, 30-Sep-2026.)
(𝑀 ⊨ Ax-Rep* → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
 
30-Sep-2026srepdf 36087 Derive the strong version of the Axiom Scheme of Replacement used in wfaxrep 46179 (see comments there) from a particular instance of that version of Replacement. Note that this theorem depends on ax-rep 5235, so it does not follow that we could replace ax-rep 5235 with an axiom that lacks a wff metavariable. (Contributed by BTernaryTau, 30-Sep-2026.)
Ⅎ𝑓𝜑    &   Ⅎ𝑤𝜑    &   Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   Ⅎ𝑧𝜑    &   (𝜑 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))    ⇒   (𝜑 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜓 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜓))))
 
30-Sep-2026srepdflem2 36086 Lemma for srepdf 36087. (Contributed by BTernaryTau, 30-Sep-2026.)
Ⅎ𝑓𝜑    &   Ⅎ𝑤𝜑    &   Ⅎ𝑦𝜑    &   Ⅎ𝑧𝜑    &   (𝜑 → 𝑥 ∈ 𝑀)    &   (𝜑 → (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))    &   (𝜑 → ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜓 → 𝑧 = 𝑦))    &   𝐹 = (𝐺 ↾ 𝑥)    &   𝐺 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀 ∧ ∀𝑦𝜓)}    ⇒   (𝜑 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜓)))
 
30-Sep-2026srepdflem1 36085 Lemma for srepdf 36087. (Contributed by BTernaryTau, 30-Sep-2026.)
Ⅎ𝑤𝜑    &   (𝜑 → ∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜓 → 𝑧 = 𝑦))    &   𝐹 = (𝐺 ↾ 𝑥)    &   𝐺 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀 ∧ ∀𝑦𝜓)}    ⇒   ((𝜑 ∧ 𝑧 ∈ 𝑀) → (∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜓) ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝐹 ∧ 𝑤 ∈ dom 𝐹 ∧ (𝐹‘𝑤) = 𝑧))))
 
30-Sep-2026df-smrep 36084 Define the strong Axiom Scheme of Replacement predicate, which is true for a class that is a standard model of the strong version of the Axiom Scheme of Replacement used in wfaxrep 46179 (see comments there). Our notation includes an asterisk because we cannot guarantee that a standard model of the Axiom Scheme of Replacement will also be a standard model of this strong version. However, proving that a class is a standard model of the strong Axiom Scheme of Replacement is sufficient to prove that it is also a standard model of the ZF Axiom Scheme of Replacement.

The exact expression used in wfaxrep 46179 is unsuitable for use in a definition because it contains a wff metavariable that we cannot quantify over. Instead, this definition uses the specific instance where 𝜑 is replaced with (Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧). This allows the set 𝑓 to effectively take on the role of the function-like wff, and since 𝑓 is a setvar, we are free to add a universal quantifier over it. The full scheme can then be recovered from this definition using smrepsrep 36088.

This approach works because we can use our ax-rep 5235 to guarantee the existence of a suitable 𝑓 in V for any 𝑥 and function-like 𝜑. Note that 𝑀 is not guaranteed to contain a suitable 𝑓 (since it might not model the necessary axioms), so we must use an unrestricted universal quantifier for 𝑓.

(Contributed by BTernaryTau, 30-Sep-2026.)

(𝑀 ⊨ Ax-Rep* ↔ ∀𝑓∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧) → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦(Fun 𝑓 ∧ 𝑤 ∈ dom 𝑓 ∧ (𝑓‘𝑤) = 𝑧)))))
 
29-Sep-2026hfstructcan 46229 Two HFStructs are equal if and only if their converted forms are equal. (Contributed by Eric Schmidt, 29-Sep-2026.)
((𝐹 ∈ HFStruct ∧ 𝐺 ∈ HFStruct) → ((𝐹 ∘ (♯ ↾ ω)) = (𝐺 ∘ (♯ ↾ ω)) ↔ 𝐹 = 𝐺))
 
29-Sep-2026hfstructhf 46228 A converted HFStruct is a hereditarily finite set. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 ∈ HFStruct → (𝐹 ∘ (♯ ↾ ω)) ∈ HF )
 
29-Sep-2026rnhfstructhf 46227 The range of an HFStruct is a hereditarily finite set. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 ∈ HFStruct → ran 𝐹 ∈ HF )
 
29-Sep-2026ishfstruct 46226 Conditions that imply that 𝐹 is an HFStruct. (Contributed by Eric Schmidt, 29-Sep-2026.)
((𝐹 Struct 𝑋 ∧ Rel 𝐹 ∧ ran 𝐹 ⊆ HF ) → 𝐹 ∈ HFStruct)
 
29-Sep-2026rnhfstructsshf 46225 The range of an HFStruct consists of hereditarily finite sets. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 ∈ HFStruct → ran 𝐹 ⊆ HF )
 
29-Sep-2026hfstructfun 46224 An HFStruct is a function. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 ∈ HFStruct → Fun 𝐹)
 
29-Sep-2026hfstructstruct 46223 An HFStruct is an extensible structure. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 ∈ HFStruct → ∃𝑥 𝐹 Struct 𝑥)
 
29-Sep-2026df-hfstruct 46222 Define the class of extensible structures whose components are all hereditarily finite. We will call these "HFStructs". Ideally, an HFStruct would itself be a hereditarily finite set, but this is not possible because the domain of a structure is, in the current formulation, a subset of ℕ. To get around this, we adjust the domain to be a subset of ω, obtaining the "converted form" (𝐹 ∘ (♯ ↾ ω)) of an HFStruct 𝐹. This is indeed hereditarily finite by hfstructhf 46228. Also, unlike general extensible structures, we do not allow ∅ to be a member of an HFStruct. This restriction gives us the property that two HFStructs are equal if and only if their converted forms are equal (hfstructcan 46229). The motivation for introducing this concept is to provide a general way of choosing a canonical representative of a class of finite structures. See df-finfld 46236 for an example of this idea. (Contributed by Eric Schmidt, 29-Sep-2026.)
HFStruct = (dom Struct ∩ 𝒫 (V × HF ))
 
29-Sep-2026rnstructfi 46121 The range of an extensible structure is finite. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 Struct 𝑋 → ran 𝐹 ∈ Fin)
 
29-Sep-2026dmstructfi 46120 The domain of an extensible structure is finite. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 Struct 𝑋 → dom 𝐹 ∈ Fin)
 
29-Sep-2026dmstructnn 46119 The domain of an extensible structure is a subset of ℕ. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝐹 Struct 𝑋 → dom 𝐹 ⊆ ℕ)
 
29-Sep-2026cocan1g 46117 Cancellation law for composition. A version of cocan1 7322 with weaker hypotheses. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝜑 → Fun ◡𝐹)    &   (𝜑 → Rel 𝐻)    &   (𝜑 → ran 𝐻 ⊆ dom 𝐹)    &   (𝜑 → Rel 𝐾)    &   (𝜑 → ran 𝐾 ⊆ dom 𝐹)    ⇒   (𝜑 → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾))
 
29-Sep-2026cocan2g 46115 Cancellation law for composition. A version of cocan2 7323 with weaker hypotheses. (Contributed by Eric Schmidt, 29-Sep-2026.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Rel 𝐻)    &   (𝜑 → dom 𝐻 ⊆ ran 𝐹)    &   (𝜑 → Rel 𝐾)    &   (𝜑 → dom 𝐾 ⊆ ran 𝐹)    ⇒   (𝜑 → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾))
 
29-Sep-2026rmo6 35780 Alternate version of rmo2 3838 where 𝑦 has a restricted quantifier. The antecedent ∃𝑧𝑧 ∈ 𝐴 on the right-hand side handles the case where 𝐴 is empty. (Contributed by BTernaryTau, 29-Sep-2026.)
Ⅎ𝑦𝜑    ⇒   (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑧 𝑧 ∈ 𝐴 → ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)))
 
29-Sep-2026reu6f 35779 Version of reu6 3687 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BTernaryTau, 29-Sep-2026.)
Ⅎ𝑦𝜑    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝑦))
 
29-Sep-2026jech9.3 9849 Every set belongs to some stage of the cumulative hierarchy of sets, expressed using an indexed union. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.) (Proof shortened by BJ, 29-Sep-2026.)
∪ 𝑥 ∈ On (𝑅1‘𝑥) = V
 
29-Sep-2026rankwflemb 9827 Two ways of expressing that a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) (Proof shortened by BJ, 29-Sep-2026.)
(𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
 
29-Sep-2026imadifssrn 6214 Condition for the range of a class to be the range of one of its restrictions. (Contributed by AV, 4-Oct-2025.) Remove antecedent. (Revised by Eric Schmidt, 10-Jul-2026.) (Proof shortened by BJ, 29-Sep-2026.)
((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))
 
29-Sep-2026cnvimarndm 6092 The preimage of the range of a class is equal to the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof shortened by BJ, 29-Sep-2026.)
(◡𝐴 “ ran 𝐴) = dom 𝐴
 
29-Sep-2026cnvimassrndm 6091 The preimage of a superset of the range of a class is equal to the domain of the class. Generalization of cnvimarndm 6092 to supersets of the range. (Contributed by AV, 18-Sep-2024.) (Proof shortened by BJ, 29-Sep-2026.)
(ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
 
28-Sep-2026fwfffunres 35788 If 𝜓 is function-like over 𝐴, then 𝐹 matches its behavior over 𝑥. (Contributed by BTernaryTau, 28-Sep-2026.)
Ⅎ𝑦𝜓    &   (𝜑 → ∀𝑤 ∈ 𝐴 ∃𝑦∀𝑧 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))    &   𝐹 = ({⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)} ↾ 𝑥)    ⇒   (𝜑 → ((𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑥 ∧ 𝜓)) ↔ (𝑤 ∈ dom 𝐹 ∧ (𝐹‘𝑤) = 𝑧)))
 
28-Sep-2026fwfffun 35787 If 𝜓 is function-like over 𝐴, then 𝐹 matches its behavior. (Contributed by BTernaryTau, 28-Sep-2026.)
Ⅎ𝑦𝜓    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦∀𝑧 ∈ 𝐴 (𝜓 → 𝑧 = 𝑦))    &   𝐹 = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓)}    ⇒   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) = 𝑧)))
 
28-Sep-2026mpt3fvot2d 7714 Value of a three-argument function in maps-to notation at an ordered triple. (Contributed by BTernaryTau, 28-Sep-2026.)
(𝜑 → 𝑅 ∈ 𝐴)    &   (𝜑 → 𝑆 ∈ 𝐵)    &   (𝜑 → 𝑇 ∈ 𝐶)    &   (𝜑 → 𝑀 ∈ 𝑉)    &   ((𝜑 ∧ 𝑅 = 𝑥) → 𝐾 = 𝐷)    &   ((𝜑 ∧ 𝑆 = 𝑦) → 𝐿 = 𝐾)    &   ((𝜑 ∧ 𝑇 = 𝑧) → 𝑀 = 𝐿)    &   𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)    ⇒   (𝜑 → (𝐹‘⟨𝑅, 𝑆, 𝑇⟩) = 𝑀)
 
28-Sep-2026mpt3fvotd 7713 Value of a three-argument function in maps-to notation at an ordered triple. (Contributed by BTernaryTau, 28-Sep-2026.)
(𝜑 → 𝑅 ∈ 𝐴)    &   (𝜑 → 𝑆 ∈ 𝐵)    &   (𝜑 → 𝑇 ∈ 𝐶)    &   (𝜑 → 𝑌 ∈ 𝑉)    &   ((𝜑 ∧ ⟨𝑅, 𝑆, 𝑇⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑌 = 𝐷)    &   𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)    ⇒   (𝜑 → (𝐹‘⟨𝑅, 𝑆, 𝑇⟩) = 𝑌)
 
28-Sep-2026mpt3fvd 7712 Value of a three-argument function in maps-to notation. (Contributed by BTernaryTau, 28-Sep-2026.)
(𝜑 → 𝑋 ∈ ((𝐴 × 𝐵) × 𝐶))    &   (𝜑 → 𝑌 ∈ 𝑉)    &   ((𝜑 ∧ 𝑋 = ⟨𝑥, 𝑦, 𝑧⟩) → 𝑌 = 𝐷)    &   𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)    ⇒   (𝜑 → (𝐹‘𝑋) = 𝑌)
 
27-Sep-2026elhf 9934 Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.) Reduce axiom usage and shorten proof. (Revised by BJ, 27-Sep-2026.)
(𝐴 ∈ HF ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
 
27-Sep-2026rnr1 9801 The range of the cumulative hierarchy of sets function. This is the class of all stages of the cumulative hierarchy of sets, not the class of all sets in some stage of the cumulative hierarchy (that is ∪ ran 𝑅1).

This statement does not require the axiom of replacement by avoiding using r1fnon 9800. (Contributed by BJ, 27-Sep-2026.)

ran 𝑅1 = (𝑅1 “ On)
 
27-Sep-2026r1dmlim 9799 The domain of the cumulative hierarchy of sets function is a limit ordinal. This weak form of r1fnon 9800 avoids ax-rep 5235. (Contributed by Mario Carneiro, 16-Nov-2014.) Extract this statement from its conjunction with r1fun 9798. (Revised by BJ, 27-Sep-2026.)
Lim dom 𝑅1
 
27-Sep-2026r1fun 9798 The cumulative hierarchy of sets function is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) Extract this statement from its conjunction with r1dmlim 9799. (Revised by BJ, 27-Sep-2026.)
Fun 𝑅1
 
27-Sep-2026imassrn 6210 Any image by a class is included in the range of the class. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.) (Proof shortened by BJ, 27-Sep-2026.)
(𝐴 “ 𝐵) ⊆ ran 𝐴
 
27-Sep-2026imadmrn 6079 The image of the domain of a class is equal to its range. (Contributed by NM, 14-Aug-1994.) (Proof shortened by BJ, 27-Sep-2026.)
(𝐴 “ dom 𝐴) = ran 𝐴
 
27-Sep-2026dfrn7 6078 The range is equal to the image of any class including the domain. (Contributed by BJ, 27-Sep-2026.)
(dom 𝐴 ⊆ 𝐵 → ran 𝐴 = (𝐴 “ 𝐵))
 
26-Sep-2026omsshf 46217 The class of finite ordinals is included in the class of hereditarily finite sets. (Contributed by Eric Schmidt, 26-Sep-2026.)
ω ⊆ HF
 
26-Sep-2026omhf 46216 Finite ordinals are hereditarily finite sets. (Contributed by Eric Schmidt, 26-Sep-2026.)
(𝐴 ∈ ω → 𝐴 ∈ HF )
 
26-Sep-2026hfrel 46215 A relation is a hereditarily finite set iff its domain and range are. (Contributed by Eric Schmidt, 26-Sep-2026.)
(Rel 𝑅 → (𝑅 ∈ HF ↔ (dom 𝑅 ∈ HF ∧ ran 𝑅 ∈ HF )))
 
26-Sep-2026hfrn 46214 The range of a hereditarily finite set is hereditarily finite. (Contributed by Eric Schmidt, 26-Sep-2026.)
(𝐴 ∈ HF → ran 𝐴 ∈ HF )
 
26-Sep-2026hfdm 46213 The domain of a hereditarily finite set is hereditarily finite. (Contributed by Eric Schmidt, 26-Sep-2026.)
(𝐴 ∈ HF → dom 𝐴 ∈ HF )
 
26-Sep-2026hfxp 46212 The Cartesian product of two hereditarily finite sets is a hereditarily finite set. (Contributed by Eric Schmidt, 26-Sep-2026.)
((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 × 𝐵) ∈ HF )
 
26-Sep-2026hfpr 46211 A pair whose members are hereditarily finite sets is a hereditarily finite set. (Contributed by Eric Schmidt, 26-Sep-2026.)
((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → {𝐴, 𝐵} ∈ HF )
 
25-Sep-2026gblacfnac2d 36030 If 𝐺 is a global choice function, then the Axiom of Choice holds. This is gblacfnacd 36029 expressed using CHOICE. (Contributed by BTernaryTau, 25-Sep-2026.)
(𝜑 → 𝐺 Fn V)    &   (𝜑 → ∀𝑧(𝑧 ≠ ∅ → (𝐺‘𝑧) ∈ 𝑧))    ⇒   (𝜑 → CHOICE)
 
25-Sep-2026j0ltaleph 36008 The value of 𝐽0 is less than an aleph if its two arguments are less than that aleph. Proposition 15.1(2) of [TakeutiZaring] p. 154. (Contributed by BTernaryTau, 25-Sep-2026.)
((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶))
 
25-Sep-2026j0ltalephlem 36007 Lemma for j0ltaleph 36008. (Contributed by BTernaryTau, 25-Sep-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))
 
24-Sep-2026prjspnnorm 43854 In a free module, two nonzero vectors are equivalent iff they have the same normalized representative. (Contributed by SN, 24-Sep-2026.)
∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   𝐹 = (𝑣 ∈ 𝐵 ↦ ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣))    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   𝑆 = (Base‘𝐾)    &   𝐼 = (invr‘𝐾)    &    · = ( ·𝑠 ‘𝑊)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
 
24-Sep-2026prjspnequivnorm 43853 In a free module, a nonzero vector is equivalent to its normalized vector. (Contributed by SN, 24-Sep-2026.)
∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   𝐹 = (𝑣 ∈ 𝐵 ↦ ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣))    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   𝑆 = (Base‘𝐾)    &   𝐼 = (invr‘𝐾)    &    · = ( ·𝑠 ‘𝑊)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 ∼ (𝐹‘𝑋))
 
24-Sep-2026prjspnnormval 43852 Value of 𝐹, a function that 'normalizes' a vector 𝑉 by scaling the first nonzero coordinate to 1. (Contributed by SN, 24-Sep-2026.)
𝐹 = (𝑣 ∈ 𝐵 ↦ ((𝐼‘(𝑣‘(𝐽‘𝑣))) · 𝑣))    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝐹‘𝑉) = ((𝐼‘(𝑉‘(𝐽‘𝑉))) · 𝑉))
 
24-Sep-2026frlmnzcoordsca 43851 In a free module, scaling a vector does not change the index of its first nonzero coordinate. (Contributed by SN, 24-Sep-2026.)
𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &    · = ( ·𝑠 ‘𝑊)    &    0 = (0g‘𝐾)    &   𝑆 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐶 ≠ 0 )    ⇒   (𝜑 → (𝐽‘(𝐶 · 𝑉)) = (𝐽‘𝑉))
 
24-Sep-2026frlmnzcoordcl2 43849 The first nonzero coordinate is a scalar. (Contributed by SN, 24-Sep-2026.)
𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝐾 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   𝑆 = (Base‘𝐾)    ⇒   (𝜑 → (𝑉‘(𝐽‘𝑉)) ∈ 𝑆)
 
24-Sep-2026lmodvsassd 43784 In a module, scalar multiplication is a semigroup action. (Contributed by SN, 24-Sep-2026.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r‘𝐹)    &   (𝜑 → 𝑊 ∈ LMod)    &   (𝜑 → 𝑄 ∈ 𝐾)    &   (𝜑 → 𝑅 ∈ 𝐾)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
 
24-Sep-2026lmodvs0d 43783 A scalar times the zero vector is the zero vector. (Contributed by SN, 24-Sep-2026.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠 ‘𝑊)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝑊 ∈ LMod)    &   (𝜑 → 𝑋 ∈ 𝐾)    ⇒   (𝜑 → (𝑋 · 0 ) = 0 )
 
24-Sep-2026lmod0vsd 43782 Zero times a vector is the zero vector. (Contributed by SN, 24-Sep-2026.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    &   𝑂 = (0g‘𝐹)    &    0 = (0g‘𝑊)    &   (𝜑 → 𝑊 ∈ LMod)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (𝑂 · 𝑋) = 0 )
 
24-Sep-2026frlmvscl 43759 Closure of scalar multiplication in a free module. (Contributed by SN, 24-Sep-2026.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠 ‘𝐹)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐴 ∈ 𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐴 · 𝑋) ∈ 𝐵)
 
24-Sep-2026reldmfrlm 43757 The domain of the free module function is a relation. (Contributed by SN, 24-Sep-2026.)
Rel dom freeLMod
 
24-Sep-2026hftsk 10891 The set of hereditarily finite sets is a Tarski class. (The Tarski-Grothendieck Axiom is not needed for this theorem.) (Contributed by Mario Carneiro, 28-May-2013.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)
HF ∈ Tarski
 
24-Sep-2026hfomALT 10888 Alternate proof of hfom 10348, shorter as a consequence of inar1 10887, but requiring AC. (Contributed by Mario Carneiro, 27-May-2013.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
HF ≈ ω
 
24-Sep-2026tskhf 10880 A nonempty Tarski class contains the whole finite cumulative hierarchy. (This proof does not use ax-inf 9666.) (Contributed by NM, 22-Feb-2011.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)
((𝑇 ∈ Tarski ∧ 𝑇 ≠ ∅) → HF ⊆ 𝑇)
 
24-Sep-2026hfom 10348 The set of hereditarily finite sets is countable. See ackbij2 10347 for an explicit bijection that works without Infinity. See also hfomALT 10888. (Contributed by Stefan O'Rear, 18-Nov-2014.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)
HF ≈ ω
 
24-Sep-2026ackbij2 10347 The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)
𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))    &   𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦))))    &   𝐻 = ∪ (rec(𝐺, ∅) “ ω)    ⇒   𝐻: HF ⟶1-1ontoω
 
24-Sep-2026dfhf2 9933 Alternate definition of the class of hereditarily finite sets as the value of the cumulative hierarchy of sets function at ω. This characterization is simpler but requires the axiom of infinity to hold. (Contributed by BTernaryTau, 25-Jan-2026.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)
HF = (𝑅1‘ω)
 
23-Sep-2026frlmnzcoordn0 43850 The function 𝐽 gives a nonzero coordinate. (Contributed by SN, 23-Sep-2026.)
𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝐾 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝑉‘(𝐽‘𝑉)) ≠ (0g‘𝐾))
 
23-Sep-2026frlmnzcoordcl 43848 Closure of the first nonzero coordinate function 𝐽. (Contributed by SN, 23-Sep-2026.)
𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝐾 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝐽‘𝑉) ∈ (0...𝑁))
 
23-Sep-2026frlmnzcoordinf 43847 The result (𝐽‘𝑉) is the smallest index corresponding to a nonzero coordinate. (Contributed by SN, 23-Sep-2026.)
𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   ((𝜑 ∧ 𝐼 ∈ {𝑖 ∈ (0...𝑁) ∣ (𝑉‘𝑖) ≠ (0g‘𝐾)}) → (𝐽‘𝑉) ≤ 𝐼)
 
23-Sep-2026frlmnzcoordval 43846 Value of 𝐽, a function that returns the index of the first nonzero coordinate of 𝑉. (Contributed by SN, 23-Sep-2026.)
𝐽 = (𝑏 ∈ 𝐵 ↦ inf({𝑖 ∈ (0...𝑁) ∣ (𝑏‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝐽‘𝑉) = inf({𝑖 ∈ (0...𝑁) ∣ (𝑉‘𝑖) ≠ (0g‘𝐾)}, ℝ, < ))
 
23-Sep-2026frlmnzcoordex 43845 Every nonzero vector of a free module has a nonzero coordinate. (Contributed by SN, 23-Sep-2026.)
𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝐾 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → {𝑖 ∈ (0...𝑁) ∣ (𝑉‘𝑖) ≠ (0g‘𝐾)} ≠ ∅)
 
23-Sep-2026elprjspnss 43843 A point of a projective space is a subset of the corresponding free module (an equivalence class of nonzero vectors of that module). (Contributed by SN, 23-Sep-2026.)
𝑃 = (𝑁 ℙrojn 𝐾)    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝐴 ∈ 𝑃)    ⇒   (𝜑 → 𝐴 ⊆ 𝐵)
 
23-Sep-2026frlmbasfn 43758 Elements of the free module are functions. (They map an index to the coordinate at that index, see frlmbasf 22096). (Contributed by SN, 23-Sep-2026.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 Fn 𝐼)
 
22-Sep-2026maxssj0 36006 The value of 𝐽0 is at least the maximum of its two arguments. Proposition 15.1(1) of [TakeutiZaring] p. 154. (Contributed by BTernaryTau, 22-Sep-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ⊆ (𝐽0‘⟨𝐴, 𝐵⟩))
 
22-Sep-2026r0maxss 36003 If two ordered pairs are related by 𝑅0, then the maximum of the first pair is less than or equal to the maximum of the second pair. (The union of two ordinals is proven to be their maximum in ordunmax 35792.) (Contributed by BTernaryTau, 22-Sep-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → 𝐷 ∈ On)    &   (𝜑 → ⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩)    ⇒   (𝜑 → (𝐴 ∪ 𝐵) ⊆ (𝐶 ∪ 𝐷))
 
22-Sep-2026dmr0ss 36002 The domain of 𝑅0 is a subset of On × On. (Contributed by BTernaryTau, 22-Sep-2026.)
dom 𝑅0 ⊆ (On × On)
 
20-Sep-2026brlexod 35999 Condition for two ordered pairs to be related by LexOrd. (Contributed by BTernaryTau, 20-Sep-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → 𝐷 ∈ On)    ⇒   (𝜑 → (⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩ ↔ (𝐴 ∈ 𝐶 ∨ (𝐴 = 𝐶 ∧ 𝐵 ∈ 𝐷))))
 
19-Sep-2026j00smo 36005 𝐽0 is strictly monotonic over its second argument when its first argument is zero. (Contributed by BTernaryTau, 19-Sep-2026.)
𝐹 = (𝑥 ∈ On ↦ (𝐽0‘⟨∅, 𝑥⟩))    ⇒   Smo 𝐹
 
19-Sep-2026j0iso 36004 𝐽0 is the 𝑅0 order isomorphism of On × On onto On. Definition 7.59 of [TakeutiZaring] p. 55. This isomorphism is unique by weisoeq 7388 and r0weseon 36000. (Contributed by BTernaryTau, 19-Sep-2026.)
𝐽0 Isom 𝑅0, E ((On × On), On)
 
19-Sep-2026brr0d 36001 Condition for two ordered pairs to be related by 𝑅0. (Contributed by BTernaryTau, 19-Sep-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → 𝐷 ∈ On)    ⇒   (𝜑 → (⟨𝐴, 𝐵⟩𝑅0⟨𝐶, 𝐷⟩ ↔ ((𝐴 ∪ 𝐵) ∈ (𝐶 ∪ 𝐷) ∨ ((𝐴 ∪ 𝐵) = (𝐶 ∪ 𝐷) ∧ ⟨𝐴, 𝐵⟩LexOrd⟨𝐶, 𝐷⟩))))
 
19-Sep-2026r0weseon 36000 𝑅0 is a set-like well-ordering of On × On. Part of Proposition 7.58 of [TakeutiZaring] p. 54. This theorem is r0weon 10118 expressed using 𝑅0. (Contributed by BTernaryTau, 19-Sep-2026.)
(𝑅0 We (On × On) ∧ 𝑅0 Se (On × On))
 
19-Sep-2026lexoweon 35998 Lexicographical order is a well-ordering of On × On. Part of Proposition 7.56(1) of [TakeutiZaring] p. 54. Unlike 𝑅0, this order is not set-like, as the preimage of ⟨1o, ∅⟩ is the proper class ({∅} × On). This theorem is leweon 10117 expressed using LexOrd. (Contributed by BTernaryTau, 19-Sep-2026.)
LexOrd We (On × On)
 
18-Sep-2026wevsev 36041 If 𝑅 is a well-ordering of the universe, then 𝑆 is a set-like well-ordering of the universe. This is the ZFC version of (3 → 4) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 18-Sep-2026.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}    ⇒   (𝑅 We V → (𝑆 We V ∧ 𝑆 Se V))
 
18-Sep-2026rankse 35823 Construct a relation that is set-like on any class. (Contributed by BTernaryTau, 18-Sep-2026.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝜑))}    ⇒   𝑆 Se 𝐴
 
18-Sep-2026werankwe2 35822 Construct a well-order given a relation that well-orders elements of the same rank. See werankwe 35821 for a version where 𝑅 can depend on 𝑤. (Contributed by BTernaryTau, 18-Sep-2026.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}    ⇒   (∀𝑤 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑤)} → 𝑆 We 𝐴)
 
17-Sep-2026fuconuniqvals 50989 At the same natural-transformation/identity input, the canonical and alternative morphism maps give respectively ∅ and 1o at the unique source object ∅. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐶 = (SetCat‘1o)    &   𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    &   𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))    &   𝑆 = ⟨( I ↾ 2o), 𝑀⟩    &   𝑅 = (𝐸 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))    &   𝐾 = ((1st ‘ ⚬ )‘𝑆)    &   𝐷 = (𝐸Δfunc𝐶)    &   𝐹 = ((1st ‘𝐷)‘∅)    &   𝐺 = ((1st ‘𝐷)‘1o)    &   𝑁 = (1o × {∅})    ⇒   (((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅ ∧ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o)
 
17-Sep-2026fuconuniq 50987 The arrow map 𝐴 fixes the identity arrow 2o and exchanges the two nonidentity parallel arrows ∅ and 1o. It gives a concrete alternative to functor composition with the same object function and a different morphism function. All hypotheses name explicit constructed sets; no category or functor existence is assumed. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐶 = (SetCat‘1o)    &   𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    &   𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))    &   𝑆 = ⟨( I ↾ 2o), 𝑀⟩    &   𝑅 = (𝐸 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))    &   𝐾 = ((1st ‘ ⚬ )‘𝑆)    &   𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))    ⇒   ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)) ∧ (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))
 
17-Sep-2026parmapfunc 50980 A map fixing the identity value and preserving the parallel-arrow class induces an endofunctor fixing every object. Category membership and the hom construction suffice: every composable pair contains an identity. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐶 ∈ Cat    &   𝐻 = (𝑟 ∈ 𝐵, 𝑠 ∈ 𝐵 ↦ if(𝑟 = 𝑠, {𝐽}, if((𝑟 = 𝑋 ∧ 𝑠 = 𝑌), 𝑆, ∅)))    &   𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))    &   (𝐴‘𝐽) = 𝐽    &   ∀𝑡 ∈ 𝑆 (𝐴‘𝑡) ∈ 𝑆    ⇒   ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿
 
17-Sep-2026parmapfunc2 50979 Variant of parmapfunc 50980. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐶 ∈ Cat    &   𝐻 = (𝑟 ∈ 𝐵, 𝑠 ∈ 𝐵 ↦ if(𝑟 = 𝑠, {𝐽}, if((𝑟 = 𝑋 ∧ 𝑠 = 𝑌), 𝑆, ∅)))    &   𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))    &   (𝐴‘𝐽) = 𝐽    &   ∀𝑡 ∈ 𝑆 (𝐴‘𝑡) ∈ 𝑆    &   𝐼 = (Id‘𝐶)    &    · = (comp‘𝐶)    &   𝑆 ∈ V    &   𝐽 ∈ V    &   𝑆 ⊆ 𝒫 𝐽    &   𝑋 ≠ 𝑌    &   𝐾 = (𝑆 ∪ {𝐽})    &   𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔))    &    · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)    &   𝐼 = (𝑡 ∈ 𝐵 ↦ 𝐽)    ⇒   ( I ↾ 𝐵)(𝐶 Func 𝐶)𝐿
 
17-Sep-2026parmapco 50978 An identity-fixing map preserving parallel arrows preserves every typed composition. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝑆 ⊆ 𝒫 𝐽    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    &   (𝜑 → (𝐴‘𝐽) = 𝐽)    &   ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)    &   ((𝜑 ∧ 𝐺 ∈ 𝑆) → (𝐴‘𝐺) ∈ 𝑆)    ⇒   (𝜑 → (𝐴‘(𝐺 ∩ 𝐹)) = ((𝐴‘𝐺) ∩ (𝐴‘𝐹)))
 
17-Sep-2026parmapco2 50977 Variant of parmapco 50978. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → (𝐴‘𝐽) = 𝐽)    &   ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)    &   𝑆 ∈ V    &   𝐽 ∈ V    &   𝑆 ⊆ 𝒫 𝐽    &   𝑋 ≠ 𝑌    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    &   ((𝜑 ∧ 𝐺 ∈ 𝑆) → (𝐴‘𝐺) ∈ 𝑆)    ⇒   (𝜑 → (𝐴‘(𝐺 ∩ 𝐹)) = ((𝐴‘𝐺) ∩ (𝐴‘𝐹)))
 
17-Sep-2026parmapcl 50976 A map fixing the identity value and preserving the parallel-arrow class preserves each hom-set. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → (𝐴‘𝐽) = 𝐽)    &   ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)    ⇒   (𝜑 → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))
 
17-Sep-2026parmapcl2 50975 Variant of parmapcl 50976. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → (𝐴‘𝐽) = 𝐽)    &   ((𝜑 ∧ 𝐹 ∈ 𝑆) → (𝐴‘𝐹) ∈ 𝑆)    &   𝑆 ∈ V    &   𝐽 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝐴‘𝐹) ∈ (𝑈𝐻𝑉))
 
17-Sep-2026parcoval2 50971 Composition in a category with the parallel-arrow hom construction. Every composable pair contains an identity, so the category identity laws determine the value. (Contributed by Zhi Wang, 17-Sep-2026.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐶 ∈ Cat)    &    · = (comp‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    ⇒   (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = if(𝑈 = 𝑉, 𝐺, 𝐹))
 
17-Sep-2026paridval 50970 The identity at any object of a category with the parallel-arrow hom construction is 𝐽. (Contributed by Zhi Wang, 17-Sep-2026.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑈 ∈ 𝐵)    ⇒   (𝜑 → ((Id‘𝐶)‘𝑈) = 𝐽)
 
17-Sep-2026parcatid 50969 Construct a category with a family of parallel arrows from 𝑋 to 𝑌 and only identity endomorphisms. All arrow values are subsets of 𝐽; composition is intersection. Nonidentity arrows can occur only when 𝑋 and 𝑌 are distinct members of 𝐵. If 𝑋 = 𝑌, or either distinguished endpoint is outside 𝐵, the category is discrete. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐵 ∈ V    &   𝐽 ∈ V    &   𝑆 ⊆ 𝒫 𝐽    &   𝐾 = (𝑆 ∪ {𝐽})    &   𝐻 = (𝑚 ∈ 𝐵, 𝑛 ∈ 𝐵 ↦ if(𝑚 = 𝑛, {𝐽}, if((𝑚 = 𝑋 ∧ 𝑛 = 𝑌), 𝑆, ∅)))    &   𝑀 = (𝑟 ∈ 𝐾, 𝑠 ∈ 𝐾 ↦ (𝑟 ∩ 𝑠))    &    · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)    &   𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    ⇒   (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐽))
 
17-Sep-2026parcatid2 50968 Variant of parcatid 50969. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 ∈ V    &   𝐽 ∈ V    &   𝑆 ⊆ 𝒫 𝐽    &   𝐾 = (𝑆 ∪ {𝐽})    &   𝐻 = (𝑚 ∈ 𝐵, 𝑛 ∈ 𝐵 ↦ if(𝑚 = 𝑛, {𝐽}, if((𝑚 = 𝑋 ∧ 𝑛 = 𝑌), 𝑆, ∅)))    &   𝑀 = (𝑟 ∈ 𝐾, 𝑠 ∈ 𝐾 ↦ (𝑟 ∩ 𝑠))    &    · = (𝑝 ∈ (𝐵 × 𝐵), 𝑞 ∈ 𝐵 ↦ 𝑀)    &   𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩}    &   𝑆 ∈ V    &   𝑋 ≠ 𝑌    ⇒   (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐽))
 
17-Sep-2026parhomco 50966 When every member of the parallel-arrow class 𝑆 is a subset of 𝐽, intersection of composable arrows belongs to the hom-set from the first object to the last. The subset hypothesis supplies the required intersection identity laws; it does not follow from the hom construction alone. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    &   𝑆 ⊆ 𝒫 𝐽    ⇒   (𝜑 → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))
 
17-Sep-2026parhomco2 50965 Variant of parhomco 50966. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    &   𝑆 ⊆ 𝒫 𝐽    &   𝑆 ∈ V    &   𝐽 ∈ V    &   𝑋 ≠ 𝑌    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    ⇒   (𝜑 → (𝐺 ∩ 𝐹) ∈ (𝑈𝐻𝑊))
 
17-Sep-2026parhomeqor 50964 In a composable pair, at least one arrow is an endomorphism. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    ⇒   (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))
 
17-Sep-2026parhomeqor2 50963 Variant of parhomeqor 50964. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))    &   𝑆 ∈ V    &   𝐽 ∈ V    &   𝑋 ≠ 𝑌    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    ⇒   (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))
 
17-Sep-2026parhomss 50962 When every member of the parallel-arrow class 𝑆 is a subset of 𝐽, every arrow in a hom-set is a subset of its identity value 𝐽. The subset hypothesis is essential; it does not follow from the hom-set definition alone. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝑆 ⊆ 𝒫 𝐽    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    ⇒   (𝜑 → 𝐹 ⊆ 𝐽)
 
17-Sep-2026parhomss2 50961 Variant of parhomss 50962. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝑆 ⊆ 𝒫 𝐽    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   𝑆 ∈ V    &   𝐽 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → 𝐹 ⊆ 𝐽)
 
17-Sep-2026parhomne 50960 A morphism between different objects has the distinguished source and target. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → ¬ 𝑈 = 𝑉)    ⇒   (𝜑 → (𝑈 = 𝑋 ∧ 𝑉 = 𝑌))
 
17-Sep-2026parhomne2 50959 Variant of parhomne 50960. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → ¬ 𝑈 = 𝑉)    &   𝑆 ∈ V    &   𝐽 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝑈 = 𝑋 ∧ 𝑉 = 𝑌))
 
17-Sep-2026parhomvaln0 50958 Evaluate an inhabited parallel-arrow hom-set. Membership certifies the selected branch even when sethood hypotheses are unavailable. Compare parhomval 50948 for evaluation throughout the object domain. (Contributed by Zhi Wang, 17-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    ⇒   (𝐹 ∈ (𝑈𝐻𝑉) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
 
17-Sep-2026parhomcl2 50950 Variant of parhomcl 50953. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝑆 ∈ V    &   𝐽 ∈ V    &   𝐾 = (𝑆 ∪ {𝐽})    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    ⇒   (𝜑 → 𝐹 ∈ 𝐾)
 
17-Sep-2026fucoaltne 50700 A changed natural transformation witnesses that post-composed functor composition has a different morphism part. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))    &   (𝜑 → ((𝐹(2nd ‘𝐾)𝐺)‘𝐴) ≠ 𝐴)    ⇒   (𝜑 → (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))
 
17-Sep-2026fucoaltval 50698 At a pair consisting of a natural transformation and the identity of the identity functor, post-composed functor composition acts by the post- composing functor on that natural transformation. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))    &   𝐼 = (idfunc‘𝐶)    &   𝐽 = (Id‘(𝐶 FuncCat 𝐶))    ⇒   (𝜑 → (𝐴(⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))
 
17-Sep-2026fucoisfunc2 50695 Full componentwise characterization of an arbitrary morphism part with the object function of functor composition. See also fucoisfunc 50692.

Its three exact function domains, component types and naturality, and pointwise identity and composition laws are necessary and sufficient. Empty object and hom domains are permitted. For a concrete failure of uniqueness, see fuconuniq 50987 and fuconuniqex2 50992. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)

𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))    &   𝑂 = ( ∘func ↾ 𝑊)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Hom ‘𝐶)    &   𝑆 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐸)    &   𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐻 = (Hom ‘𝑇)    &   𝐼 = (Id‘𝑇)    &    · = (comp‘𝑇)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝐾 = (Id‘𝐸)    ⇒   (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))))
 
17-Sep-2026fucoridid 50691 Horizontal composition with the identity natural transformation of the identity functor leaves a natural transformation unchanged. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
(𝜑 → (2nd ‘(⟨𝐶, 𝐶⟩ ∘F 𝐸)) = 𝑃)    &   𝑄 = (𝐶 FuncCat 𝐶)    &   𝐽 = (Id‘𝑄)    &   𝐼 = (idfunc‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐸)𝐻))    ⇒   (𝜑 → (𝐴(⟨𝐺, 𝐼⟩𝑃⟨𝐻, 𝐼⟩)(𝐽‘𝐼)) = 𝐴)
 
17-Sep-2026cofu1stfix 50417 Post-composition by an object-fixing functor preserves the entire object function, including its exact domain. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.)
𝐴 = (Base‘𝐸)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐾 ∈ (𝐸 Func 𝐸))    &   (𝜑 → (1st ‘𝐾) = ( I ↾ 𝐴))    ⇒   (𝜑 → (1st ‘(𝐾 ∘func 𝐹)) = (1st ‘𝐹))
 
17-Sep-2026elovmpoeq 50185 If a value of an operation given by maps-to notation has a member, it equals the substituted body. (Contributed by Zhi Wang, 17-Sep-2026.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝐶 = 𝐷)    ⇒   (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋𝐹𝑌) = 𝐷)
 
17-Sep-2026f1oronf1or 36056 If 𝐹 maps 𝐴 one-to-one into the ordinals or 𝐻 maps the ordinals one-to-one into 𝐴, then either 𝐴 can be mapped one-to-one into the ordinals (by 𝐹), or the ordinals can be mapped one-to-one into 𝐴 (by 𝐻). This is the ZFC version of (10 → 9) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝐴∀𝑋(∃𝐹𝐹:𝐴⟶1-1𝑋 ∨ ∃𝐻𝐻:𝑋⟶1-1𝐴), but since we cannot quantify over classes, we instead consider only the case 𝑋 = On which is sufficient for this proof. The proof is trivial but is included for completeness. (Contributed by BTernaryTau, 17-Sep-2026.)
((𝐹:𝐴⟶1-1On ∨ 𝐻:On⟶1-1𝐴) → (𝐹:𝐴⟶1-1On ∨ 𝐻:On⟶1-1𝐴))
 
17-Sep-2026onprcf1of1ord 36055 If 𝐹 is a bijection from the ordinals to the universe, 𝐻 is a bijection from the ordinals to 𝐴 if 𝐴 is a proper class, and 𝐼 is a bijection from the ordinals to 𝐵 if 𝐵 is a proper class, then 𝐽 either maps 𝐴 one-to-one into 𝐵 or maps 𝐵 one-to-one into 𝐴. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first three hypotheses would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1onto𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is the universe, the case where 𝑋 is an arbitrary proper class 𝐴, and the case where 𝑋 is an arbitrary proper class 𝐵, which together are sufficient for this proof. (Contributed by BTernaryTau, 17-Sep-2026.)
(𝜑 → 𝐹:On⟶1-1ontoV)    &   (𝜑 → (¬ 𝐴 ∈ V → 𝐻:On⟶1-1onto𝐴))    &   (𝜑 → (¬ 𝐵 ∈ V → 𝐼:On⟶1-1onto𝐵))    &   𝐽 = if(𝐴 ∈ V, if(𝐵 ∈ V, 𝐾, (𝐼 ∘ (◡𝐹 ↾ 𝐴))), (𝐻 ∘ (◡𝐹 ↾ 𝐵)))    &   𝐾 = (𝐹‘∩ {𝑥 ∈ On ∣ ((𝐹‘𝑥):𝐴⟶1-1𝐵 ∨ (𝐹‘𝑥):𝐵⟶1-1𝐴)})    ⇒   (𝜑 → (𝐽:𝐴⟶1-1𝐵 ∨ 𝐽:𝐵⟶1-1𝐴))
 
17-Sep-2026onprcf1of1orbd 36054 If 𝐹 is a bijection from the ordinals to the universe and both 𝐴 and 𝐵 are sets, then either 𝐴 can be mapped one-to-one into 𝐵 or 𝐵 can be mapped one-to-one into 𝐴. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac in the case where neither class is a proper class. See onprcf1of1orad 36053 for the case where one of the classes is a proper class, and see onprcf1of1ord 36055 for a theorem that covers both cases. Note that in NBG set theory the first hypothesis would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1onto𝑋)), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 17-Sep-2026.)
(𝜑 → 𝐹:On⟶1-1ontoV)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (∃𝑓 𝑓:𝐴⟶1-1𝐵 ∨ ∃ℎ ℎ:𝐵⟶1-1𝐴))
 
17-Sep-2026onprcf1of1orad 36053 If 𝐹 is a bijection from the ordinals to the universe and 𝐻 is a bijection from the ordinals to the proper class 𝐵, then 𝐼 maps 𝐴 one-to-one into 𝐵. This is the ZFC version of (5 → 10) in https://tinyurl.com/hamkins-gblac in the case where one of the classes is a proper class. See onprcf1of1orbd 36054 for the case where neither class is a proper class, and see onprcf1of1ord 36055 for a theorem that covers both cases. Note that in NBG set theory the first two hypotheses would be replaced by something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1onto𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is the universe and the case where 𝑋 is an arbitrary proper class 𝐵 which together are sufficient for this proof. (Contributed by BTernaryTau, 17-Sep-2026.)
(𝜑 → 𝐹:On⟶1-1ontoV)    &   (𝜑 → (¬ 𝐵 ∈ V → 𝐻:On⟶1-1onto𝐵))    &   (𝜑 → ¬ 𝐵 ∈ V)    &   𝐼 = (𝐻 ∘ (◡𝐹 ↾ 𝐴))    ⇒   (𝜑 → 𝐼:𝐴⟶1-1𝐵)
 
17-Sep-2026dfac16 35846 Axiom of Choice equivalent: for every pair of sets, one is dominated by the other (see also entri3 10670). (Contributed by BTernaryTau, 17-Sep-2026.)
(CHOICE ↔ ∀𝑥∀𝑦(𝑥 ≼ 𝑦 ∨ 𝑦 ≼ 𝑥))
 
17-Sep-2026hfpw 9953 The power class of a hereditarily finite set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) Avoid ax-reg 9613, ax-inf2 9669. (Revised by BTernaryTau, 17-Sep-2026.)
(𝐴 ∈ HF → 𝒫 𝐴 ∈ HF )
 
17-Sep-2026hfuni 9951 The union of a hereditarily finite set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.) Avoid ax-reg 9613, ax-inf2 9669. (Revised by BTernaryTau, 17-Sep-2026.)
(𝐴 ∈ HF → ∪ 𝐴 ∈ HF )
 
17-Sep-2026hfsn 9947 The singleton of a hereditarily finite set is a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015.) Avoid ax-reg 9613, ax-inf2 9669. (Revised by BTernaryTau, 17-Sep-2026.)
(𝐴 ∈ HF → {𝐴} ∈ HF )
 
17-Sep-2026hfun 9945 The union of two hereditarily finite sets is a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015.) Avoid ax-reg 9613, ax-inf2 9669. (Revised by BTernaryTau, 17-Sep-2026.)
((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 ∪ 𝐵) ∈ HF )
 
17-Sep-2026hfsshf 9942 Any subset of a hereditarily finite set is itself a hereditarily finite set. (Contributed by BTernaryTau, 17-Sep-2026.)
((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ HF ) → 𝐴 ∈ HF )
 
17-Sep-2026hfelhf 9941 Any member of a hereditarily finite set is itself a hereditarily finite set. (Contributed by Scott Fenton, 16-Jul-2015.) Avoid ax-reg 9613, ax-inf2 9669. (Revised by BTernaryTau, 17-Sep-2026.)
((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ HF ) → 𝐴 ∈ HF )
 
17-Sep-2026elhf3 9940 A set is hereditarily finite if and only if it is finite and all its members are hereditarily finite. (Contributed by Eric Schmidt, 8-Sep-2026.) Avoid ax-reg 9613, ax-inf2 9669. (Revised by BTernaryTau, 17-Sep-2026.)
(𝐴 ∈ HF ↔ (𝐴 ∈ Fin ∧ 𝐴 ⊆ HF ))
 
17-Sep-2026elhf4 9939 A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026.) Use HF. (Revised by BTernaryTau, 17-Sep-2026.)
(𝐴 ∈ HF ↔ (𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ HF ))
 
16-Sep-2026parhomid 50957 The identity value belongs to each endomorphism hom-set. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 16-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝐽 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    ⇒   (𝜑 → 𝐽 ∈ (𝑈𝐻𝑈))
 
16-Sep-2026parhomideq 50955 An endomorphism in the parallel-arrow construction has the identity value. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 16-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝑈 = 𝑉)    ⇒   (𝜑 → 𝐹 = 𝐽)
 
16-Sep-2026parhomcl 50953 All hom-sets of the parallel-arrow construction lie in its arrow carrier. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 16-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝐾 = (𝑆 ∪ {𝐽})    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    ⇒   (𝜑 → 𝐹 ∈ 𝐾)
 
16-Sep-2026parhomssun 50952 Every value of the parallel-arrow hom operation is a subclass of the arrow carrier. (Contributed by Zhi Wang, 16-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝐾 = (𝑆 ∪ {𝐽})    ⇒   (𝑈𝐻𝑉) ⊆ 𝐾
 
16-Sep-2026parhomidval 50951 On the diagonal, the parallel-arrow hom operation has value {𝐽}. (Contributed by Zhi Wang, 16-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    ⇒   (𝑈 ∈ 𝐵 → (𝑈𝐻𝑈) = {𝐽})
 
16-Sep-2026onf1oracwevd 36052 If 𝐹 maps the proper class 𝑊 one-to-one into the ordinals or 𝐻 maps the ordinals one-to-one into the proper class 𝑊, and if every set can either be mapped one-to-one into the ordinals or have the ordinals mapped one-to-one into it, then 𝑅 well-orders the universe. This is the ZFC version of (9 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first two hypotheses would be replaced by something like (𝜑 → ∀𝑋(∃𝐹𝐹:𝑋⟶1-1On ∨ ∃𝐻𝐻:On⟶1-1𝑋)). However, we cannot quantify over classes, so we split this into the case where 𝑋 is a set and the case where 𝑋 is the proper class 𝑊 which together are sufficient for this proof. (Contributed by BTernaryTau, 16-Sep-2026.)
(𝜑 → (𝐹:𝑊⟶1-1On ∨ 𝐻:On⟶1-1𝑊))    &   (𝜑 → ∀𝑣(∃𝑓 𝑓:𝑣⟶1-1On ∨ ∃ℎ ℎ:On⟶1-1𝑣))    &   𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}    &   𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝐴𝑧))}    &   𝐴 = (𝐼‘∩ {𝑤 ∈ On ∣ (𝐼‘𝑤) We (𝑅1‘suc (rank‘𝑦))})    &   𝐼 = if(𝐹:𝑊⟶1-1On, ◡𝐹, 𝐻)    ⇒   (𝜑 → 𝑅 We V)
 
16-Sep-2026onf1onf1or 36051 If 𝐹 maps 𝐴 one-to-one into the ordinals, then either 𝐴 can be mapped one-to-one into the ordinals (by 𝐹), or the ordinals can be mapped one-to-one into 𝐴. This is the ZFC version of (6 → 9) in https://tinyurl.com/hamkins-gblac. The proof is trivial but is included for completeness. (Contributed by BTernaryTau, 16-Sep-2026.)
(𝐹:𝐴⟶1-1On → (𝐹:𝐴⟶1-1On ∨ 𝐻:On⟶1-1𝐴))
 
16-Sep-2026prconf1acwevd 36048 If 𝐹 maps the proper class 𝑊 one-to-one into the ordinals and the Axiom of Choice holds, then 𝑅 well-orders the universe. (Contributed by BTernaryTau, 16-Sep-2026.)
(𝜑 → (¬ 𝑊 ∈ V → 𝐹:𝑊⟶1-1On))    &   (𝜑 → CHOICE)    &   𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}    &   𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}    &   𝑆 = (◡𝐹‘∩ {𝑤 ∈ On ∣ (◡𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})    ⇒   (𝜑 → 𝑅 We V)
 
16-Sep-2026onprcf1acwevd 36047 If 𝐹 maps the ordinals one-to-one into the proper class 𝑊 and the Axiom of Choice holds, then 𝑅 well-orders the universe. This is the ZFC version of (7 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first hypothesis would be something like (𝜑 → ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:On⟶1-1𝑋)), but since we cannot quantify over classes, we instead consider only the case 𝑋 = 𝑊 which is sufficient for this proof. (Contributed by BTernaryTau, 16-Sep-2026.)
(𝜑 → (¬ 𝑊 ∈ V → 𝐹:On⟶1-1𝑊))    &   (𝜑 → CHOICE)    &   𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}    &   𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}    &   𝑆 = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})    ⇒   (𝜑 → 𝑅 We V)
 
16-Sep-2026onprcf1acwevdlem2 36046 Lemma for onprcf1acwevd 36047 and prconf1acwevd 36048. (Contributed by BTernaryTau, 16-Sep-2026.)
𝑅 = {⟨𝑦, 𝑧⟩ ∣ ((rank‘𝑦) ∈ (rank‘𝑧) ∨ ((rank‘𝑦) = (rank‘𝑧) ∧ 𝑦𝑆𝑧))}    &   𝑆 = (𝐹‘∩ {𝑤 ∈ On ∣ (𝐹‘𝑤) We (𝑅1‘suc (rank‘𝑦))})    &   ((𝜑 ∧ 𝑢 ∈ On) → ∃𝑤 ∈ On (𝐹‘𝑤) We (𝑅1‘𝑢))    ⇒   (𝜑 → 𝑅 We V)
 
16-Sep-2026dfac15 35845 Axiom of Choice equivalent: every set can be mapped one-to-one into the ordinals. (Contributed by BTernaryTau, 16-Sep-2026.)
(CHOICE ↔ ∀𝑥∃𝑓 𝑓:𝑥⟶1-1On)
 
16-Sep-2026werankwe 35821 Construct a well-order given a relation 𝑅(𝑥) that well-orders elements of the same rank. (Contributed by BTernaryTau, 16-Sep-2026.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((rank‘𝑥) ∈ (rank‘𝑦) ∨ ((rank‘𝑥) = (rank‘𝑦) ∧ 𝑥𝑅𝑦))}    &   (𝑤 = (rank‘𝑥) → 𝑇 = 𝑅)    &   (𝑣 = 𝑥 → 𝑈 = 𝑅)    ⇒   (∀𝑥 ∈ 𝐴 𝑅 We {𝑧 ∈ 𝐴 ∣ (rank‘𝑧) = (rank‘𝑥)} → 𝑆 We 𝐴)
 
16-Sep-2026onf1we 35803 If 𝐹 maps 𝐴 one-to-one into the ordinals, then 𝑅 well-orders 𝐴. (Contributed by BTernaryTau, 16-Sep-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}    ⇒   (𝐹:𝐴⟶1-1On → 𝑅 We 𝐴)
 
16-Sep-2026dvdssqnn 16770 Two positive integers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by AV, 16-Sep-2026.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ (𝑀↑2) ∥ (𝑁↑2)))
 
16-Sep-2026ovmptss 8129 If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.) Avoid ax-un 7775 and shorten proof. (Revised by Zhi Wang, 16-Sep-2026.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ⊆ 𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
 
15-Sep-2026fltoprmgt3 28026 Fermat's last theorem holds for any exponent greater than 2 if it holds for all odd prime exponents greater than 3. (TODO-AV: after a proof is available for 𝑁 = 3, see flt3, the hypothesis fltoprmgt3.3 can be removed.) (Contributed by AV, 15-Sep-2026.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ∀𝑝 ∈ ℙ (3 < 𝑝 → ((𝑎↑𝑝) + (𝑏↑𝑝)) ≠ (𝑐↑𝑝)))    &   (𝜑 → ((𝑎↑3) + (𝑏↑3)) ≠ (𝑐↑3))    ⇒   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) ≠ (𝐶↑𝑁))
 
15-Sep-2026fltoprm 28025 Fermat's last theorem holds for any exponent greater than 2 if it holds for odd prime exponents. (Contributed by AV, 15-Sep-2026.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ∀𝑝 ∈ ℙ (2 < 𝑝 → ((𝑎↑𝑝) + (𝑏↑𝑝)) ≠ (𝑐↑𝑝)))    ⇒   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) ≠ (𝐶↑𝑁))
 
15-Sep-2026fltoprmlem2 28024 Lemma 2 for fltoprm 28025: If an integer greater than or equal to 3 is a power of 2, it is a multiple of 4. (Contributed by AV, 15-Sep-2026.)
((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ ℕ0 ∧ 𝑁 = (2↑𝐾)) → 4 ∥ 𝑁)
 
15-Sep-2026fltoprmlem1 28023 Lemma 1 for fltoprm 28025: Every positive integer is either a power of 2 or has an odd prime factor. (Contributed by AV, 15-Sep-2026.)
(𝑁 ∈ ℕ → (∃𝑝 ∈ ℙ (2 < 𝑝 ∧ 𝑝 ∥ 𝑁) ∨ ∃𝑛 ∈ ℕ0 𝑁 = (2↑𝑛)))
 
15-Sep-2026flt4ALT 28022 Fermat's last theorem for the exponent four, derived from Fermat's right triangle theorem (which is not proved yet, see fermrtt: after a proof is available, the hypothesis flt4ALT.r can be removed - TODO-AV). (Contributed by AV, 15-Sep-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ∀𝑎 ∈ ℕ ∀𝑏 ∈ ℕ ∀𝑐 ∈ ℕ ((𝑎↑4) − (𝑏↑4)) ≠ (𝑐↑2))    ⇒   (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑4))
 
15-Sep-2026flt4 28021 Fermat's last theorem for the exponent four. (Contributed by AV, 15-Sep-2026.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    ⇒   (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑4))
 
14-Sep-2026fuconuniqex2 50992 There are categories for which the object function of functor composition does not determine its morphism function. Choose both source categories to be the trivial category and apply fuconuniqex 50991. The full compatibility conditions are given in fucoisfunc 50692; fucoisfunc2 50695 expands the natural-transformation component conditions. (Contributed by Zhi Wang, 14-Sep-2026.)
∃𝑐 ∈ Cat ∃𝑑 ∈ Cat ∃𝑒 ∈ Cat ∃𝑧((1st ‘(⟨𝑐, 𝑑⟩ ∘F 𝑒))(((𝑑 FuncCat 𝑒) ×c (𝑐 FuncCat 𝑑)) Func (𝑐 FuncCat 𝑒))𝑧 ∧ 𝑧 ≠ (2nd ‘(⟨𝑐, 𝑑⟩ ∘F 𝑒)))
 
14-Sep-2026fuconuniqex 50991 The object function of functor composition does not in general determine its morphism function, even when both source categories are the trivial category. The witness category is the explicit two-object, two-parallel-arrow category of par2cat 50973. (Contributed by Zhi Wang, 14-Sep-2026.)
∃𝑒 ∈ Cat ∃𝑧((1st ‘(⟨(SetCat‘1o), (SetCat‘1o)⟩ ∘F 𝑒))((((SetCat‘1o) FuncCat 𝑒) ×c ((SetCat‘1o) FuncCat (SetCat‘1o))) Func ((SetCat‘1o) FuncCat 𝑒))𝑧 ∧ 𝑧 ≠ (2nd ‘(⟨(SetCat‘1o), (SetCat‘1o)⟩ ∘F 𝑒)))
 
14-Sep-2026fuconuniq2 50990 The concrete parallel-arrow category supplies a morphism function different from canonical functor composition, with its object function fixed exactly. See par2cat 50973 and par2hom 50984 for the category and its four hom-sets. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐶 = (SetCat‘1o)    &   𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))    ⇒   ∃𝑧((1st ‘𝐿)(𝑇 Func 𝑄)𝑧 ∧ 𝑧 ≠ (2nd ‘𝐿))
 
14-Sep-2026fuconuniqvalslem 50988 Lemma for fuconuniqvals 50989. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)
𝐶 = (SetCat‘1o)    &   𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    &   𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))    &   𝑆 = ⟨( I ↾ 2o), 𝑀⟩    &   𝑅 = (𝐸 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))    &   𝐾 = ((1st ‘ ⚬ )‘𝑆)    &   𝐷 = (𝐸Δfunc𝐶)    &   𝐹 = ((1st ‘𝐷)‘∅)    &   𝐺 = ((1st ‘𝐷)‘1o)    &   𝑁 = (1o × {∅})    &   𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))    ⇒   (((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘𝐿)⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = ∅ ∧ ((𝑁(⟨𝐹, (idfunc‘𝐶)⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, (idfunc‘𝐶)⟩)((Id‘(𝐶 FuncCat 𝐶))‘(idfunc‘𝐶)))‘∅) = 1o)
 
14-Sep-2026fuconuniqlem 50986 Lemma for fuconuniq 50987. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)
𝐶 = (SetCat‘1o)    &   𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    &   𝑀 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))    &   𝑆 = ⟨( I ↾ 2o), 𝑀⟩    &   𝑅 = (𝐸 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐸⟩ ∘F 𝐸))    &   𝐾 = ((1st ‘ ⚬ )‘𝑆)    &   𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))    &   𝐷 = (𝐸Δfunc𝐶)    &   𝐹 = ((1st ‘𝐷)‘∅)    &   𝐺 = ((1st ‘𝐷)‘1o)    &   𝑁 = (1o × {∅})    ⇒   ((1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)) ∧ (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))
 
14-Sep-2026par2diag 50985 Two constant functors into the concrete parallel-arrow category and a natural transformation whose component is the empty-set arrow. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐶 = (SetCat‘1o)    &   𝐸 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝐿 = (𝐸Δfunc𝐶)    &   𝐹 = ((1st ‘𝐿)‘∅)    &   𝐺 = ((1st ‘𝐿)‘1o)    &   𝐴 = (1o × {∅})    ⇒   ((𝐹 ∈ (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) ∧ (𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺) ∧ (𝐴‘∅) = ∅) ∧ (((1st ‘𝐹)‘∅) = ∅ ∧ ((1st ‘𝐺)‘∅) = 1o))
 
14-Sep-2026par2hom 50984 The four hom-sets of the explicit two-object parallel-arrow category. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐻 = (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))    ⇒   ((∅𝐻∅) = {2o} ∧ (1o𝐻1o) = {2o} ∧ ((∅𝐻1o) = 2o ∧ (1o𝐻∅) = ∅))
 
14-Sep-2026parswapfunc 50983 The explicit exchange of the two parallel nonidentity arrows is an endofunctor with identity object function. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐶 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    &   𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    &   𝐿 = (𝑢 ∈ 2o, 𝑣 ∈ 2o ↦ (𝑎 ∈ (𝑢(𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))𝑣) ↦ (𝐴‘𝑎)))    ⇒   ⟨( I ↾ 2o), 𝐿⟩ ∈ (𝐶 Func 𝐶)
 
14-Sep-2026parswapprop 50982 The concrete transposition fixes the identity, preserves the parallel arrows, and sends the empty-set arrow to the ordinal-one arrow. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    ⇒   ((𝐴‘2o) = 2o ∧ ∀𝑡 ∈ 2o (𝐴‘𝑡) ∈ 2o ∧ (𝐴‘∅) = 1o)
 
14-Sep-2026parswapval 50981 Evaluate the transposition of the two parallel-arrow values. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐴 = (𝑏 ∈ 3o ↦ if(𝑏 = 2o, 2o, if(𝑏 = ∅, 1o, ∅)))    ⇒   (𝑈 ∈ 3o → (𝐴‘𝑈) = if(𝑈 = 2o, 2o, if(𝑈 = ∅, 1o, ∅)))
 
14-Sep-2026par2str 50974 The object set, hom operation, and composition of the concrete two-object parallel-arrow category. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐶 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    ⇒   (2o = (Base‘𝐶) ∧ (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅))) = (Hom ‘𝐶) ∧ (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠))) = (comp‘𝐶))
 
14-Sep-2026par2cat 50973 The explicitly given two-object, two-parallel-arrow category exists. This closed theorem discharges the constructor premises in par2catid 50972. (Contributed by Zhi Wang, 14-Sep-2026.)
{⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩} ∈ Cat
 
14-Sep-2026par2catid 50972 A concrete category with two objects and two parallel nonidentity arrows. The two identity arrows both have raw value 2o; their sources and targets distinguish them. The parallel arrows have raw values the empty set and 1o. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐶 = {⟨(Base‘ndx), 2o⟩, ⟨(Hom ‘ndx), (𝑚 ∈ 2o, 𝑛 ∈ 2o ↦ if(𝑚 = 𝑛, {2o}, if((𝑚 = ∅ ∧ 𝑛 = 1o), 2o, ∅)))⟩, ⟨(comp‘ndx), (𝑝 ∈ (2o × 2o), 𝑞 ∈ 2o ↦ (𝑟 ∈ 3o, 𝑠 ∈ 3o ↦ (𝑟 ∩ 𝑠)))⟩}    ⇒   (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ 2o ↦ 2o))
 
14-Sep-2026parcoval 50967 The composition in the specified parallel-arrow categories is intersection. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑀 = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐾 ↦ (𝑓 ∩ 𝑔))    &    · = (𝑝 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ 𝑀)    &   𝑀 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ 𝐾)    &   (𝜑 → 𝐺 ∈ 𝐾)    ⇒   (𝜑 → (𝐺(⟨𝑈, 𝑉⟩ · 𝑊)𝐹) = (𝐺 ∩ 𝐹))
 
14-Sep-2026parhomid2 50956 Variant of parhomid 50957. (Contributed by Zhi Wang, 14-Sep-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝐽 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    &   𝑆 ∈ V    ⇒   (𝜑 → 𝐽 ∈ (𝑈𝐻𝑈))
 
14-Sep-2026parhomideq2 50954 Variant of parhomideq 50955. (Contributed by Zhi Wang, 14-Sep-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    &   (𝜑 → 𝑈 = 𝑉)    &   𝑆 ∈ V    &   𝐽 ∈ V    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → 𝐹 = 𝐽)
 
14-Sep-2026parhomel 50949 Membership in a hom-set of a parallel-arrow category is equivalent to being its identity value on the diagonal, or a member of 𝑆 at the distinguished source and target. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝑆 ∈ V    &   𝐽 ∈ V    ⇒   ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝐹 ∈ (𝑈𝐻𝑉) ↔ ((𝑈 = 𝑉 ∧ 𝐹 = 𝐽) ∨ (¬ 𝑈 = 𝑉 ∧ ((𝑈 = 𝑋 ∧ 𝑉 = 𝑌) ∧ 𝐹 ∈ 𝑆)))))
 
14-Sep-2026parhomval 50948 At objects in the base, the hom operation has value {𝐽} on the diagonal, 𝑆 from the distinguished source to target, and the empty set otherwise. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))    &   𝑆 ∈ V    ⇒   ((𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → (𝑈𝐻𝑉) = if(𝑈 = 𝑉, {𝐽}, if((𝑈 = 𝑋 ∧ 𝑉 = 𝑌), 𝑆, ∅)))
 
14-Sep-2026termcpostfix 50877 Post-composition by an object-fixing endofunctor induces an object-fixing endofunctor on a functor category with terminal source. (Contributed by Zhi Wang, 14-Sep-2026.)
(𝜑 → 𝐶 ∈ TermCat)    &   𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (𝐷 FuncCat 𝐷)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐷))    &   𝐾 = ((1st ‘ ⚬ )‘𝐹)    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐷))    &   (𝜑 → (1st ‘𝐹) = ( I ↾ 𝐴))    ⇒   (𝜑 → (𝐾 ∈ (𝑄 Func 𝑄) ∧ (1st ‘𝐾) = ( I ↾ (𝐶 Func 𝐷))))
 
14-Sep-2026termccofix 50876 Post-composition by an object-fixing endofunctor leaves a functor with terminal source literally unchanged. (Contributed by Zhi Wang, 14-Sep-2026.)
(𝜑 → 𝐶 ∈ TermCat)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝑆 ∈ (𝐷 Func 𝐷))    &   (𝜑 → (1st ‘𝑆) = ( I ↾ 𝐴))    ⇒   (𝜑 → (𝑆 ∘func 𝐹) = 𝐹)
 
14-Sep-2026termcfunc1eq 50875 Functors with terminal source are equal when they agree on its object. (Contributed by Zhi Wang, 14-Sep-2026.)
(𝜑 → 𝐶 ∈ TermCat)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → ((1st ‘𝐹)‘𝑌) = ((1st ‘𝐺)‘𝑌))    ⇒   (𝜑 → 𝐹 = 𝐺)
 
14-Sep-2026postcof2 50705 A component of the action of a post-composition functor is obtained by applying the morphism part of 𝐹 to the component morphism. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (𝐷 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐾 = ((1st ‘ ⚬ )‘𝐹)    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   (𝜑 → (((𝐺(2nd ‘𝐾)𝐻)‘𝐴)‘𝑋) = ((((1st ‘𝐺)‘𝑋)(2nd ‘𝐹)((1st ‘𝐻)‘𝑋))‘(𝐴‘𝑋)))
 
14-Sep-2026postcof2nd 50704 The morphism part of the explicit post-composition functor sends a pair of functors to the induced map on natural transformations. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (𝐷 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐾 = ((1st ‘ ⚬ )‘𝐹)    ⇒   (𝜑 → (2nd ‘𝐾) = (𝑔 ∈ (𝐶 Func 𝐷), ℎ ∈ (𝐶 Func 𝐷) ↦ (𝑎 ∈ (𝑔(𝐶 Nat 𝐷)ℎ) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((((1st ‘𝑔)‘𝑥)(2nd ‘𝐹)((1st ‘ℎ)‘𝑥))‘(𝑎‘𝑥))))))
 
14-Sep-2026postcof1st 50703 The object part of the explicit post-composition functor sends each functor to its post-composition with 𝐹. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑅 = (𝐷 FuncCat 𝐸)    &    ⚬ = (⟨𝑅, 𝑄⟩ curryF (⟨𝐶, 𝐷⟩ ∘F 𝐸))    &   (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))    &   (𝜑 → 𝐶 ∈ Cat)    &   𝐾 = ((1st ‘ ⚬ )‘𝐹)    ⇒   (𝜑 → (1st ‘𝐾) = (𝑔 ∈ (𝐶 Func 𝐷) ↦ (𝐹 ∘func 𝑔)))
 
14-Sep-2026fucoaltnelem 50699 Lemma for fucoaltne 50700. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)
𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))    &   (𝜑 → ((𝐹(2nd ‘𝐾)𝐺)‘𝐴) ≠ 𝐴)    &   𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))    &   𝐼 = (idfunc‘𝐶)    &   𝐽 = (Id‘(𝐶 FuncCat 𝐶))    ⇒   (𝜑 → (2nd ‘(𝐾 ∘func 𝐿)) ≠ (2nd ‘𝐿))
 
14-Sep-2026fucoaltvallem 50697 Lemma for fucoaltval 50698. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)
𝑄 = (𝐶 FuncCat 𝐸)    &   𝐿 = (⟨𝐶, 𝐶⟩ ∘F 𝐸)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐸))    &   (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐸)𝐺))    &   𝐼 = (idfunc‘𝐶)    &   𝐽 = (Id‘(𝐶 FuncCat 𝐶))    &   𝑇 = ((𝐶 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐶))    ⇒   (𝜑 → (𝐴(⟨𝐹, 𝐼⟩(2nd ‘(𝐾 ∘func 𝐿))⟨𝐺, 𝐼⟩)(𝐽‘𝐼)) = ((𝐹(2nd ‘𝐾)𝐺)‘𝐴))
 
14-Sep-2026fucoaltfunc 50696 Post-compose functor composition with an object-fixing endofunctor of its target. The original object function remains literally fixed for arbitrary source and intermediate categories. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐸)    &   𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   𝐿 = (⟨𝐶, 𝐷⟩ ∘F 𝐸)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   (𝜑 → 𝐾 ∈ (𝑄 Func 𝑄))    &   (𝜑 → (1st ‘𝐾) = ( I ↾ (𝐶 Func 𝐸)))    ⇒   (𝜑 → (1st ‘𝐿)(𝑇 Func 𝑄)(2nd ‘(𝐾 ∘func 𝐿)))
 
14-Sep-2026fucoisfunc2lem2 50694 Lemma for fucoisfunc2 50695. (Contributed by Zhi Wang, 14-Sep-2026.) (New usage is discouraged.)
𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))    &   𝑂 = ( ∘func ↾ 𝑊)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Hom ‘𝐶)    &   𝑆 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐸)    &   𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐻 = (Hom ‘𝑇)    &   𝐼 = (Id‘𝑇)    &    · = (comp‘𝑇)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    &   𝐾 = (Id‘𝐸)    &   𝐽 = (Id‘𝑄)    &    ⚬ = (comp‘𝑄)    ⇒   (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))) ∧ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡))))))
 
14-Sep-2026fucoisfunc2lem1 50693 Lemma for fucoisfunc2lem2 50694. An equivalence that expands the transformation outputs of a composition-shaped hom-family into exact component functions, component typing, and naturality. The component values are arbitrary; no canonical value is imposed. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))    &   𝑂 = ( ∘func ↾ 𝑊)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Hom ‘𝐶)    &   𝑆 = (Hom ‘𝐸)    &    ∙ = (comp‘𝐸)    ⇒   ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ↔ (𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))((𝑝(𝑚𝑃𝑛)𝑞) = (𝑥 ∈ 𝐵 ↦ ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥) ∈ (((1st ‘(𝑂‘𝑚))‘𝑥)𝑆((1st ‘(𝑂‘𝑛))‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝑅𝑦)(((𝑝(𝑚𝑃𝑛)𝑞)‘𝑦)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑚))‘𝑦)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑥(2nd ‘(𝑂‘𝑚))𝑦)‘ℎ)) = (((𝑥(2nd ‘(𝑂‘𝑛))𝑦)‘ℎ)(⟨((1st ‘(𝑂‘𝑚))‘𝑥), ((1st ‘(𝑂‘𝑛))‘𝑥)⟩ ∙ ((1st ‘(𝑂‘𝑛))‘𝑦))((𝑝(𝑚𝑃𝑛)𝑞)‘𝑥))))))
 
14-Sep-2026fucoisfunc 50692 A characterization of functors with the object part of functor composition, in terms of an exact morphism function and compatible component functors. See fucoisfunc2 50695 for the componentwise form. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑊 = ((𝐷 Func 𝐸) × (𝐶 Func 𝐷))    &   𝑇 = ((𝐷 FuncCat 𝐸) ×c (𝐶 FuncCat 𝐷))    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝑂 = ( ∘func ↾ 𝑊)    &   𝐻 = (Hom ‘𝑇)    &   𝐼 = (Id‘𝑇)    &   𝐽 = (Id‘𝑄)    &    · = (comp‘𝑇)    &    ⚬ = (comp‘𝑄)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ Cat)    ⇒   (𝜑 → (𝑂(𝑇 Func 𝑄)𝑃 ↔ ((𝑃 = (𝑢 ∈ 𝑊, 𝑣 ∈ 𝑊 ↦ (𝑢𝑃𝑣)) ∧ ∀𝑚 ∈ 𝑊 ∀𝑛 ∈ 𝑊 ((𝑚𝑃𝑛) = (𝑏 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛)), 𝑎 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛)) ↦ (𝑏(𝑚𝑃𝑛)𝑎)) ∧ ∀𝑝 ∈ ((1st ‘𝑚)(𝐷 Nat 𝐸)(1st ‘𝑛))∀𝑞 ∈ ((2nd ‘𝑚)(𝐶 Nat 𝐷)(2nd ‘𝑛))(𝑝(𝑚𝑃𝑛)𝑞) ∈ ((𝑂‘𝑚)(𝐶 Nat 𝐸)(𝑂‘𝑛)))) ∧ ∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))))))
 
14-Sep-2026fucidcobi 50552 Componentwise functor laws for a correctly typed hom-family into a functor category. The source category 𝑇 and object assignment 𝑂 are arbitrary: the identity and composition laws are equivalent to their component equations. No functor law is assumed in the hypotheses. Functor composition is an application in fucoisfunc2 50695. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑊 = (Base‘𝑇)    &   𝑄 = (𝐶 FuncCat 𝐸)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝑇)    &   𝐼 = (Id‘𝑇)    &   𝐽 = (Id‘𝑄)    &   𝐾 = (Id‘𝐸)    &    · = (comp‘𝑇)    &    ⚬ = (comp‘𝑄)    &    ∙ = (comp‘𝐸)    &   (𝜑 → 𝑇 ∈ Cat)    &   (𝜑 → 𝑃 ∈ X𝑣 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑣))(𝐶 Nat 𝐸)(𝑂‘(2nd ‘𝑣))) ↑m (𝐻‘𝑣)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝑊 (((𝑥𝑃𝑥)‘(𝐼‘𝑥)) = (𝐽‘(𝑂‘𝑥)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚)) = (((𝑦𝑃𝑧)‘𝑛)(⟨(𝑂‘𝑥), (𝑂‘𝑦)⟩ ⚬ (𝑂‘𝑧))((𝑥𝑃𝑦)‘𝑚))) ↔ ∀𝑥 ∈ 𝑊 (∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑥)‘(𝐼‘𝑥))‘𝑡) = (𝐾‘((1st ‘(𝑂‘𝑥))‘𝑡)) ∧ ∀𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)∀𝑡 ∈ 𝐵 (((𝑥𝑃𝑧)‘(𝑛(⟨𝑥, 𝑦⟩ · 𝑧)𝑚))‘𝑡) = ((((𝑦𝑃𝑧)‘𝑛)‘𝑡)(⟨((1st ‘(𝑂‘𝑥))‘𝑡), ((1st ‘(𝑂‘𝑦))‘𝑡)⟩ ∙ ((1st ‘(𝑂‘𝑧))‘𝑡))(((𝑥𝑃𝑦)‘𝑚)‘𝑡)))))
 
14-Sep-2026natcoeq 50551 A natural transformation equals a composite exactly when all its components equal the corresponding composites, including when the object set is empty. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &    ⚬ = (comp‘𝐷)    &    · = (comp‘𝑄)    &   (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   (𝜑 → 𝑅 ∈ (𝐺𝑁𝐻))    &   (𝜑 → 𝐿 ∈ (𝐹𝑁𝐻))    ⇒   (𝜑 → (𝐿 = (𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴) ↔ ∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ ⚬ ((1st ‘𝐻)‘𝑥))(𝐴‘𝑥))))
 
14-Sep-2026natideq 50550 A natural transformation is an identity exactly when all its components are identities. This also covers an empty object set. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐼 = (Id‘𝑄)    &   𝐽 = (Id‘𝐷)    &   (𝜑 → 𝐴 ∈ (𝐹𝑁𝐹))    ⇒   (𝜑 → (𝐴 = (𝐼‘𝐹) ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐽‘((1st ‘𝐹)‘𝑥))))
 
14-Sep-2026nateq 50549 Two natural transformations with the same source and target are equal exactly when all their components agree. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    &   (𝜑 → 𝐿 ∈ (𝐹𝑁𝐺))    ⇒   (𝜑 → (𝐴 = 𝐿 ↔ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) = (𝐿‘𝑥)))
 
14-Sep-2026natfn2 50548 A natural transformation is a function on the objects of the source category. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))    ⇒   (𝜑 → 𝐴 Fn 𝐵)
 
14-Sep-2026isnat3 50547 A natural transformation is an exact component function with typed values satisfying naturality. (Contributed by Zhi Wang, 14-Sep-2026.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)    &   (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))    &   (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))    ⇒   (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 = (𝑥 ∈ 𝐵 ↦ (𝐴‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 (𝐴‘𝑥) ∈ (((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)⟩ · ((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝐺)𝑦)‘ℎ)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
 
14-Sep-2026ixpovf 50402 A member of an indexed product of function spaces has the indicated function type at each pair of indices. Hom-functions of a functor are one application. (Contributed by Zhi Wang, 14-Sep-2026.)
(𝜑 → 𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))𝑁(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))    &   (𝜑 → 𝑈 ∈ 𝑊)    &   (𝜑 → 𝑉 ∈ 𝑊)    ⇒   (𝜑 → (𝑈𝑃𝑉):(𝑈𝐻𝑉)⟶((𝑂‘𝑈)𝑁(𝑂‘𝑉)))
 
14-Sep-2026elixpmpt 50209 Membership in an indexed Cartesian product over a set is equivalent to being its exact function graph with components of the indicated types. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐴 ∈ V    ⇒   (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
 
14-Sep-2026fvmpomptd 50188 At an element of its inner domain, an iterated maps-to function has the value of its body. The operation 𝐻 is arbitrary; evaluation on a morphism in a hom-set is one application. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))    ⇒   (𝜑 → ((𝑈𝐿𝑉)‘𝐹) = (𝐴‘𝐹))
 
14-Sep-2026ovmpomptd 50187 The outer value of an iterated maps-to function is the corresponding inner maps-to function. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    ⇒   (𝜑 → (𝑈𝐿𝑉) = (𝑎 ∈ (𝑈𝐻𝑉) ↦ (𝐴‘𝑎)))
 
14-Sep-2026fnmpompt 50186 Characterization of the outer domain of a chained maps-to expression. (Contributed by Zhi Wang, 14-Sep-2026.)
𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))    ⇒   𝐿 Fn (𝐵 × 𝐵)
 
14-Sep-2026elif0 50127 Membership in a conditional class with an empty set in the false-branch is equivalent to the condition and membership in the first class. (Contributed by Zhi Wang, 14-Sep-2026.)
(𝐴 ∈ if(𝜑, 𝐵, ∅) ↔ (𝜑 ∧ 𝐴 ∈ 𝐵))
 
12-Sep-2026isdrng2 21027 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 12-Sep-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
 
12-Sep-2026sbcfung 6583 Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) Shorten proof and remove dependency on ax-sep 5252 and ax-pr 5395. (Revised by Eric Schmidt, 12-Sep-2026.)
(𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))
 
10-Sep-2026onprcf1acwevdlem1 36045 Lemma for onprcf1acwevd 36047 and prconf1acwevd 36048. (Contributed by BTernaryTau, 10-Sep-2026.)
𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}    ⇒   ((𝐴 ⊆ 𝑊 ∧ 𝐵 ∈ On ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦 We (𝑅1‘𝐵)) → 𝐴 ∈ V)
 
8-Sep-2026elhf3OLD 9950 Obsolete version of elhf3 9940 as of 17-Sep-2026. (Contributed by Eric Schmidt, 8-Sep-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ HF ↔ (𝐴 ∈ Fin ∧ 𝐴 ⊆ HF ))
 
8-Sep-2026hffi 9936 Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 8-Sep-2026.)
(𝐴 ∈ HF → 𝐴 ∈ Fin)
 
8-Sep-2026funmpt3 7711 A function in maps-to notation with three arguments is a function. (Contributed by BTernaryTau, 8-Sep-2026.)
Fun (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷)
 
8-Sep-2026mpt3eqdv 7710 An equality deduction for maps-to notation. (Contributed by BTernaryTau, 8-Sep-2026.)
(𝜑 → 𝐴 = 𝐸)    &   (𝜑 → 𝐵 = 𝐹)    &   (𝜑 → 𝐶 = 𝐺)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → 𝐷 = 𝐻)    ⇒   (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = (𝑥 ∈ 𝐸, 𝑦 ∈ 𝐹, 𝑧 ∈ 𝐺 ↦ 𝐻))
 
8-Sep-2026mosubott 5488 "At most one" remains true inside ordered triple quantification, analogous to mosubopt 5486. (Contributed by BTernaryTau, 8-Sep-2026.)
(∀𝑥∀𝑦∀𝑧∃*𝑤𝜑 → ∃*𝑤∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))
 
8-Sep-2026cotsexgw 5467 Substitution of class 𝐴 for ordered triple ⟨𝑥, 𝑦, 𝑧⟩, analogous to copsexgw 5464. (Contributed by BTernaryTau, 8-Sep-2026.)
(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
 
8-Sep-2026eqvinot 5461 A variable introduction law for ordered triples, analogous to eqvinop 5460. (Contributed by BTernaryTau, 8-Sep-2026.)
𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V    ⇒   (𝐴 = ⟨𝐵, 𝐶, 𝐷⟩ ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))
 
8-Sep-2026ax12ev2c 2220 A commuted form of ax12ev2 2219. (Contributed by BTernaryTau, 8-Sep-2026.)
(𝑥 = 𝑦 → (∃𝑦(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))
 
7-Sep-2026mpt3mpt 7709 Express a three-argument function as a one-argument function, or vice-versa. (Contributed by BTernaryTau, 7-Sep-2026.)
(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ → 𝐷 = 𝐸)    ⇒   (𝑤 ∈ ((𝐴 × 𝐵) × 𝐶) ↦ 𝐷) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐸)
 
6-Sep-2026tz7.48lem 8478 A way of showing an ordinal function is one-to-one. (Contributed by NM, 9-Feb-1997.) Extract onelfvnef1 8477. (Proof shortened by Matthew House, 6-Sep-2026.)
𝐹 Fn On    ⇒   ((𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ 𝐴))
 
6-Sep-2026onelfvnef1 8477 A sufficient condition for a function on ordinals to be one-to-one. (Contributed by NM, 9-Feb-1997.) Extract from tz7.48lem 8478 and generalize statement. (Revised by Matthew House, 6-Sep-2026.)
((𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ On ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑦 ∈ 𝑥 → (𝐹‘𝑥) ≠ (𝐹‘𝑦))) → 𝐹:𝐴⟶1-1𝐵)
 
4-Sep-20269onn 35872 The ordinal 9 is a natural number. (Contributed by BTernaryTau, 4-Sep-2026.)
9o ∈ ω
 
4-Sep-20268onn 35871 The ordinal 8 is a natural number. (Contributed by BTernaryTau, 4-Sep-2026.)
8o ∈ ω
 
4-Sep-20267onn 35870 The ordinal 7 is a natural number. (Contributed by BTernaryTau, 4-Sep-2026.)
7o ∈ ω
 
4-Sep-20266onn 35869 The ordinal 6 is a natural number. (Contributed by BTernaryTau, 4-Sep-2026.)
6o ∈ ω
 
4-Sep-20265onn 35868 The ordinal 5 is a natural number. (Contributed by BTernaryTau, 4-Sep-2026.)
5o ∈ ω
 
4-Sep-20269on 35867 Ordinal 9 is an ordinal number. (Contributed by BTernaryTau, 4-Sep-2026.)
9o ∈ On
 
4-Sep-20268on 35866 Ordinal 8 is an ordinal number. (Contributed by BTernaryTau, 4-Sep-2026.)
8o ∈ On
 
4-Sep-20267on 35865 Ordinal 7 is an ordinal number. (Contributed by BTernaryTau, 4-Sep-2026.)
7o ∈ On
 
4-Sep-20266on 35864 Ordinal 6 is an ordinal number. (Contributed by BTernaryTau, 4-Sep-2026.)
6o ∈ On
 
4-Sep-20265on 35863 Ordinal 5 is an ordinal number. (Contributed by BTernaryTau, 4-Sep-2026.)
5o ∈ On
 
3-Sep-2026df-l 35997 Define the class of all constructible sets. Definition 15.15 of [TakeutiZaring] p. 158. (Contributed by BTernaryTau, 3-Sep-2026.)
𝐿 = (𝐹𝐿 “ On)
 
3-Sep-2026df-fnl 35996 Define a function whose range contains all and only the constructible sets. Based on Definition 15.13 of [TakeutiZaring] p. 158. (Contributed by BTernaryTau, 3-Sep-2026.)
𝐹𝐿 = recs((𝑥 ∈ V ↦ if((𝐾3‘dom 𝑥) = ∅, ran 𝑥, (ℱ ‘⟨(𝐾3‘dom 𝑥), (𝑥‘(𝐾1‘dom 𝑥)), (𝑥‘(𝐾2‘dom 𝑥))⟩))))
 
2-Sep-2026df-k3 35995 Define a function that takes an ordinal and returns the third argument of the ordered triple ⟨𝑥, 𝑦, 𝑛⟩ such that the ordinal equals (𝐽‘⟨𝑥, 𝑦, 𝑛⟩). Based on the third case of Definition 15.7 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 2-Sep-2026.)
𝐾3 = {⟨𝑧, 𝑛⟩ ∣ (𝑛 ∈ 9o ∧ ∃𝑥 ∈ On ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}
 
2-Sep-2026df-k2 35994 Define a function that takes an ordinal and returns the second argument of the ordered triple ⟨𝑥, 𝑦, 𝑛⟩ such that the ordinal equals (𝐽‘⟨𝑥, 𝑦, 𝑛⟩). Based on the second case of Definition 15.7 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 2-Sep-2026.)
𝐾2 = {⟨𝑧, 𝑦⟩ ∣ (𝑦 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑥 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}
 
2-Sep-2026df-k1 35993 Define a function that takes an ordinal and returns the first argument of the ordered triple ⟨𝑥, 𝑦, 𝑛⟩ such that the ordinal equals (𝐽‘⟨𝑥, 𝑦, 𝑛⟩). Based on the first case of Definition 15.7 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 2-Sep-2026.)
𝐾1 = {⟨𝑧, 𝑥⟩ ∣ (𝑥 ∈ On ∧ ∃𝑛 ∈ 9o ∃𝑦 ∈ On 𝑧 = (𝐽‘⟨𝑥, 𝑦, 𝑛⟩))}
 
2-Sep-2026df-j 35992 Define an order isomorphism from (On × On) × 9o to On. Based on Definition 15.2 of [TakeutiZaring] p. 155. (Contributed by BTernaryTau, 2-Sep-2026.)
𝐽 = (𝑥 ∈ On, 𝑦 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
 
2-Sep-2026df-j0 35991 Define the 𝑅0 order isomorphism of On × On onto On. Equivalent to Definition 7.59 of [TakeutiZaring] p. 55 (see j0iso 36004). (Contributed by BTernaryTau, 2-Sep-2026.)
𝐽0 = ◡OrdIso(𝑅0, (On × On))
 
2-Sep-2026df-r0 35990 Define a particular set-like well-ordering of On × On using the lexicographical ordering LexOrd. Based on Definition 7.57 of [TakeutiZaring] p. 54. See r0weseon 36000 for a proof that this relation is a set-like well-order. (Contributed by BTernaryTau, 2-Sep-2026.)
𝑅0 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) ∈ ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∨ (((1st ‘𝑥) ∪ (2nd ‘𝑥)) = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∧ 𝑥LexOrd𝑦)))}
 
2-Sep-2026df-lexo 35989 Define the lexicographical ordering of On × On. Based on Definition 7.55 of [TakeutiZaring] p. 54. (Contributed by BTernaryTau, 2-Sep-2026.)
LexOrd = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))}
 
2-Sep-2026df-gdlopc 35962 Define the combined Gödel operations. This function takes three arguments and uses the first to determine which of the eight Gödel operations df-gdlop1 35954 through df-gdlop8 35961 to apply to the second and third arguments. Based on Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ = (𝑛 ∈ (9o ∖ {∅}), 𝑥 ∈ V, 𝑦 ∈ V ↦ if(𝑛 = 1o, (ℱ1‘⟨𝑥, 𝑦⟩), if(𝑛 = 2o, (ℱ2‘⟨𝑥, 𝑦⟩), if(𝑛 = 3o, (ℱ3‘⟨𝑥, 𝑦⟩), if(𝑛 = 4o, (ℱ4‘⟨𝑥, 𝑦⟩), if(𝑛 = 5o, (ℱ5‘⟨𝑥, 𝑦⟩), if(𝑛 = 6o, (ℱ6‘⟨𝑥, 𝑦⟩), if(𝑛 = 7o, (ℱ7‘⟨𝑥, 𝑦⟩), (ℱ8‘⟨𝑥, 𝑦⟩)))))))))
 
2-Sep-2026df-gdlop8 35961 Define the eighth Gödel operation. This function takes two arguments and returns the intersection of the first argument with the third converse of the second argument (see df-in 4094 and df-cnv3 35953). Based on the eighth case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ8 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∩ (Cnv3‘𝑦)))
 
2-Sep-2026df-gdlop7 35960 Define the seventh Gödel operation. This function takes two arguments and returns the intersection of the first argument with the second converse of the second argument (see df-in 4094 and df-cnv2 35952). Based on the seventh case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ7 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∩ (Cnv2‘𝑦)))
 
2-Sep-2026df-gdlop6 35959 Define the sixth Gödel operation. This function takes two arguments and returns the intersection of the first argument with the converse of the second argument (see df-in 4094 and df-cnv 5663). Based on the sixth case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ6 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∩ ◡𝑦))
 
2-Sep-2026df-gdlop5 35958 Define the fifth Gödel operation. This function takes two arguments and returns the intersection of the first argument with the domain of the second argument (see df-in 4094 and df-dm 5665). Based on the fifth case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ5 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∩ dom 𝑦))
 
2-Sep-2026df-gdlop4 35957 Define the fourth Gödel operation. This function takes two arguments and returns the first argument restricted to the second argument (see df-res 5667). Based on the fourth case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ4 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ↾ 𝑦))
 
2-Sep-2026df-gdlop3 35956 Define the third Gödel operation. This function takes two arguments and returns the first argument minus the second argument (see df-dif 4050). Based on the third case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ3 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∖ 𝑦))
 
2-Sep-2026df-gdlop2 35955 Define the second Gödel operation. This function takes two arguments and returns the intersection of the first argument with the membership relation (see df-in 4094 and df-eprel 5555). The second argument is ignored. Based on the second case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ2 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∩ E ))
 
2-Sep-2026df-gdlop1 35954 Define the first Gödel operation. This function takes two arguments and returns the unordered pair containing both (see df-pr 4591). Based on the first case of Definition 14.2 of [TakeutiZaring] p. 144. (Contributed by BTernaryTau, 2-Sep-2026.)
ℱ1 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑥, 𝑦})
 
2-Sep-2026df-cnv3 35953 Define a function that returns the third converse of a set. The third converse of a set takes all ordered triples in the set and swaps the second and third arguments. Based on Definition 14.1(2) of [TakeutiZaring] p. 143. (Contributed by BTernaryTau, 2-Sep-2026.)
Cnv3 = (𝑤 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨⟨𝑥, 𝑧⟩, 𝑦⟩ ∈ 𝑤})
 
2-Sep-2026df-cnv2 35952 Define a function that returns the second converse of a set. The second converse of a set takes all ordered triples in the set and rotates them so the last argument becomes the first argument. Based on Definition 14.1(1) of [TakeutiZaring] p. 143. (Contributed by BTernaryTau, 2-Sep-2026.)
Cnv2 = (𝑤 ∈ V ↦ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ⟨⟨𝑧, 𝑥⟩, 𝑦⟩ ∈ 𝑤})
 
2-Sep-2026df-9o 35862 Define the ordinal number 9. (Contributed by BTernaryTau, 2-Sep-2026.)
9o = suc 8o
 
2-Sep-2026df-8o 35861 Define the ordinal number 8. (Contributed by BTernaryTau, 2-Sep-2026.)
8o = suc 7o
 
2-Sep-2026df-7o 35860 Define the ordinal number 7. (Contributed by BTernaryTau, 2-Sep-2026.)
7o = suc 6o
 
2-Sep-2026df-6o 35859 Define the ordinal number 6. (Contributed by BTernaryTau, 2-Sep-2026.)
6o = suc 5o
 
2-Sep-2026df-5o 35858 Define the ordinal number 5. (Contributed by BTernaryTau, 2-Sep-2026.)
5o = suc 4o
 
31-Aug-2026angmgmbas 29428 The base set of the angle addition magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &    ∼ = (cgrA‘𝐺)    &   𝐽 = (AngMgm‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    ⇒   (𝜑 → (𝐴 / ∼ ) = (Base‘𝐽))
 
31-Aug-2026angmgm 29427 The angle addition magma is a magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   𝐽 = (AngMgm‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 2 ≤ (♯‘𝑃))    ⇒   (𝜑 → 𝐽 ∈ Mgm)
 
31-Aug-2026angmgm0g 29426 The identity element of the angle addition magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   𝐽 = (AngMgm‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    ⇒   (𝜑 → [⟨“𝑋𝑌𝑋”⟩] ∼ = (0g‘𝐽))
 
31-Aug-2026angmgmlem 29425 Lemma for angmgm 29427. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   𝐽 = (AngMgm‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    ⇒   (𝜑 → (𝐽 ∈ Mgm ∧ [⟨“𝑋𝑌𝑋”⟩] ∼ = (0g‘𝐽)))
 
31-Aug-2026angmgmval 29424 Explicit the value of the angle addition magma for a given geometry 𝐺. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   𝐽 = (AngMgm‘𝐺)    &    ≤ = (≤∠‘𝐺)    ⇒   (𝐺 ∈ 𝑉 → 𝐽 = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /s ∼ ))
 
31-Aug-2026angmgmaddrid 29423 The right identity element for the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   (𝜑 → 𝐸 ∈ 𝐴)    ⇒   (𝜑 → (𝐸 + ⟨“𝑋𝑌𝑋”⟩) ∼ 𝐸)
 
31-Aug-2026angmgmaddlid 29422 The left identity element for the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   (𝜑 → 𝐸 ∈ 𝐴)    ⇒   (𝜑 → (⟨“𝑋𝑌𝑋”⟩ + 𝐸) ∼ 𝐸)
 
31-Aug-2026angmgmaddcl 29421 Closure of the addition of angles. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   (𝜑 → 𝐸 ∈ 𝐴)    &   (𝜑 → 𝐹 ∈ 𝐴)    ⇒   (𝜑 → (𝐸 + 𝐹) ∈ 𝐴)
 
31-Aug-2026cgrabasimass 29408 The angle congruence relation is hereditary. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &    ∼ = (cgrA‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    ⇒   (𝜑 → ( ∼ “ 𝐴) ⊆ 𝐴)
 
31-Aug-2026cgraer 29407 The angle congruence relation is an equivalence relation. (Contributed by Thierry Arnoux, 31-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &    ∼ = (cgrA‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    ⇒   (𝜑 → ( ∼ ∩ (𝐴 × 𝐴)) Er 𝐴)
 
31-Aug-2026qusmnd 19004 Prove that a quotient structure is a monoid. (Contributed by Thierry Arnoux, 31-Aug-2026.)
(𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &   (𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑋)    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))    &   ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) ∼ (𝑥 + (𝑦 + 𝑧)))    &   (𝜑 → 0 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥)    &   ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 + 0 ) ∼ 𝑥)    ⇒   (𝜑 → (𝑈 ∈ Mnd ∧ [ 0 ] ∼ = (0g‘𝑈)))
 
31-Aug-2026qusmgm 18893 Prove that a quotient structure is a unital magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
(𝜑 → 𝑈 = (𝑅 /s ∼ ))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &   (𝜑 → ∼ Er 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑋)    &   (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))    &   ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   (𝜑 → 0 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥)    &   ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 + 0 ) ∼ 𝑥)    ⇒   (𝜑 → (𝑈 ∈ Mgm ∧ [ 0 ] ∼ = (0g‘𝑈)))
 
31-Aug-2026imasmgm2 18892 The image structure of a unital magma is a unital magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)
(𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   (𝜑 → 𝑉 = (Base‘𝑅))    &    + = (+g‘𝑅)    &   (𝜑 → 𝐹:𝑉⟶onto𝐵)    &   ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑 → 𝑅 ∈ 𝑊)    &   ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)    &   (𝜑 → 0 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥))    &   ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥))    ⇒   (𝜑 → (𝑈 ∈ Mgm ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
 
30-Aug-2026smflimlem6 47991 Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. The proof uses fnrndomnum 10644 rather than fnrndomg 10645, and so does not require ax-ac 10564. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Vincent Gonzalez, 30-Aug-2026.)
(𝜑 → 𝑀 ∈ ℤ)    &   𝑍 = (ℤ≥‘𝑀)    &   (𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))    &   𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }    &   𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))    &   (𝜑 → 𝐴 ∈ ℝ)    &   𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})    ⇒   (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))
 
30-Aug-2026subsaliuncl 47573 A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. The proof uses fnrndomnum 10644 rather than fnrndomg 10645, and so does not require ax-ac 10564. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Vincent Gonzalez, 30-Aug-2026.)
(𝜑 → 𝑆 ∈ SAlg)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   𝑇 = (𝑆 ↾t 𝐷)    &   (𝜑 → 𝐹:ℕ⟶𝑇)    ⇒   (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)
 
30-Aug-2026preimaaa 26676 An element of the preimage of the algebraic numbers by a nonconstant polynomial is an algebraic number. (Contributed by SN, 30-Aug-2026.)
(𝜑 → 𝐹 ∈ (Poly‘ℚ))    &   (𝜑 → (deg‘𝐹) ≠ 0)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → (𝐹‘𝐴) ∈ 𝔸)    ⇒   (𝜑 → 𝐴 ∈ 𝔸)
 
30-Aug-2026plyconz 26661 The composition of a nonzero and a nonconstant polynomial is a nonzero polynomial. (Contributed by SN, 30-Aug-2026.)
(𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → 𝐺 ∈ (Poly‘𝑆))    &   (𝜑 → 𝐹 ≠ 0𝑝)    &   (𝜑 → (deg‘𝐺) ≠ 0)    ⇒   (𝜑 → (𝐹 ∘ 𝐺) ≠ 0𝑝)
 
30-Aug-2026rnplynfin 26660 The range of a nonconstant polynomial is not a finite set. (Contributed by SN, 30-Aug-2026.)
(𝜑 → 𝐹 ∈ (Poly‘𝑆))    &   (𝜑 → (deg‘𝐹) ≠ 0)    ⇒   (𝜑 → ¬ ran 𝐹 ∈ Fin)
 
30-Aug-2026idpfv 26558 Value of the identity polynomial. (Contributed by SN, 30-Aug-2026.)
(𝐴 ∈ ℂ → (Xp‘𝐴) = 𝐴)
 
30-Aug-20261q 13119 The number 1 is rational. (Contributed by SN, 30-Aug-2026.)
1 ∈ ℚ
 
30-Aug-2026elmaprd 8898 Deduction associated with elmapd 8888. Reverse direction of elmapdd 8889. (Contributed by Thierry Arnoux, 13-Oct-2025.) Removed redundant hypotheses. (Revised by SN, 30-Aug-2026.)
(𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴))    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
28-Aug-2026dvcot 51138 Derivative of the cotangent function. (Contributed by Jon Pennant, 28-Aug-2026.)
(ℂ D cot) = (𝑥 ∈ dom cot ↦ -((csc‘𝑥)↑2))
 
28-Aug-2026dvcsc 51137 Derivative of the cosecant function. (Contributed by Jon Pennant, 28-Aug-2026.)
(ℂ D csc) = (𝑥 ∈ dom csc ↦ ( -(csc‘𝑥) · (cot‘𝑥)))
 
28-Aug-2026dvsec 51136 Derivative of the secant function. (Contributed by Jon Pennant, 28-Aug-2026.)
(ℂ D sec) = (𝑥 ∈ dom sec ↦ ((sec‘𝑥) · (tan‘𝑥)))
 
27-Aug-2026veroquaddetzerod 51266 The Veronese matrix of six points satisfying a common nonzero homogeneous quadratic equation has determinant zero. (Contributed by Jiamin Zhao, 27-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    &   (𝜑 → 𝐾:(1...6)⟶ℝ)    &   ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)    &   (𝜑 → 𝐾 ≠ ((1...6) × {0}))    ⇒   (𝜑 → (((1...6) maDet ℝfld)‘𝑉) = 0)
 
27-Aug-2026veroquadnolindfd 51265 A nonzero homogeneous quadratic equation satisfied by all six points gives a linear dependence among the columns of the Veronese matrix. (Contributed by Jiamin Zhao, 27-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    &   (𝜑 → 𝐾:(1...6)⟶ℝ)    &   ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)    &   (𝜑 → 𝐾 ≠ ((1...6) × {0}))    ⇒   (𝜑 → ¬ curry tpos 𝑉 LIndF (ℝfld freeLMod (1...6)))
 
27-Aug-2026nellindf 51250 A nonzero coefficient vector whose weighted combination of 𝐹 sums to the zero vector implies that 𝐹 is not linearly independent. (Contributed by Jiamin Zhao, 27-Aug-2026.)
𝐵 = (Base‘𝑊)    &   𝑅 = (Scalar‘𝑊)    &    · = ( ·𝑠 ‘𝑊)    &    0 = (0g‘𝑊)    &   𝑌 = (0g‘𝑅)    &   𝐿 = (Base‘(𝑅 freeLMod 𝐼))    ⇒   (((𝑊 ∈ LMod ∧ 𝐼 ∈ V ∧ 𝐹:𝐼⟶𝐵) ∧ (𝐾 ∈ 𝐿 ∧ 𝐾 ≠ (𝐼 × {𝑌}) ∧ (𝑊 Σg (𝐾 ∘f · 𝐹)) = 0 )) → ¬ 𝐹 LIndF 𝑊)
 
26-Aug-2026dffr7 36898 Alternate quantifier-free definition of a well-founded relation. (Contributed by Scott Fenton, 26-Aug-2026.)
(𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ Fix ( E ∘ (V ∖ (𝑅 ∘ ◡ E ))))
 
26-Aug-2026dffr5 36697 A quantifier-free definition of a well-founded relation. (Contributed by Scott Fenton, 11-Apr-2011.) (Proof shortened by Scott Fenton, 26-Aug-2026.)
(𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))
 
26-Aug-2026drngprops 21026 Properties of a division ring. (Contributed by NM, 4-Apr-2009.) (Revised by AV, 26-Aug-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   (𝑅 ∈ DivRing → (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
 
26-Aug-2026cnvxp 6159 The converse of a Cartesian product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2197. (Revised by SN, 26-Aug-2026.)
◡(𝐴 × 𝐵) = (𝐵 × 𝐴)
 
25-Aug-2026fimact 10642 The image by a function of a countable set is countable. The proof uses imadomnum 10641 rather than imadomg 10640, and so does not require ax-ac 10564. (Contributed by Thierry Arnoux, 27-Mar-2018.) (Revised by Vincent Gonzalez, 25-Aug-2026.)
((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹 “ 𝐴) ≼ ω)
 
25-Aug-2026imadomnum 10641 A version of imadomg 10640 that does not require the axiom of choice ax-ac 10564. (Contributed by Vincent Gonzalez, 25-Aug-2026.)
(𝐴 ∈ dom card → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴))
 
24-Aug-20264p5e9 43517 4 + 5 = 9. (Contributed by SN, 24-Aug-2026.)
(4 + 5) = 9
 
24-Aug-20263p6e9 43516 3 + 6 = 9. (Contributed by SN, 24-Aug-2026.)
(3 + 6) = 9
 
24-Aug-20263p5e8 43515 3 + 5 = 8. (Contributed by SN, 24-Aug-2026.)
(3 + 5) = 8
 
24-Aug-20263p4e7 43514 3 + 4 = 7. (Contributed by SN, 24-Aug-2026.)
(3 + 4) = 7
 
24-Aug-20262p7e9 43513 2 + 7 = 9. (Contributed by SN, 24-Aug-2026.)
(2 + 7) = 9
 
24-Aug-20262p6e8 43512 2 + 6 = 8. (Contributed by SN, 24-Aug-2026.)
(2 + 6) = 8
 
24-Aug-20262p5e7 43511 2 + 5 = 7. (Contributed by SN, 24-Aug-2026.)
(2 + 5) = 7
 
24-Aug-20262p4e6 43510 2 + 4 = 6. (Contributed by SN, 24-Aug-2026.)
(2 + 4) = 6
 
24-Aug-20262p3e5 43509 2 + 3 = 5. (Contributed by SN, 24-Aug-2026.)
(2 + 3) = 5
 
24-Aug-20261p8e9 43508 1 + 8 = 9. (Contributed by SN, 24-Aug-2026.)
(1 + 8) = 9
 
24-Aug-20261p7e8 43507 1 + 7 = 8. (Contributed by SN, 24-Aug-2026.)
(1 + 7) = 8
 
24-Aug-20261p6e7 43506 1 + 6 = 7. (Contributed by SN, 24-Aug-2026.)
(1 + 6) = 7
 
24-Aug-20261p5e6 43505 1 + 5 = 6. (Contributed by SN, 24-Aug-2026.)
(1 + 5) = 6
 
24-Aug-20261p4e5 43504 1 + 4 = 5. (Contributed by SN, 24-Aug-2026.)
(1 + 4) = 5
 
24-Aug-20264p4e8ALT 43502 A shorter proof of 4p4e8 12524 if 6p2e8 12528 was moved up. The most clean way to do this would be to start with 7p2e9 12530, then go 6p2e8 12528, 6p3e9 12529, etc., which is still inelegant. The idea here is that using 4 = 2 + 2 and 2cn 12445 is shorter than using 4 = 3 + 1, 3cn 12451, and ax-1cn 11285. This also works with 5p4e9 12527. (Contributed by SN, 24-Aug-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
(4 + 4) = 8
 
24-Aug-2026rsp2idlid 34001 The ideal span of a two-sided ideal is the ideal itself. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 24-Aug-2026.)
𝐾 = (RSpan‘𝑅)    &   𝑈 = (2Ideal‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝐾‘𝐼) = 𝐼)
 
24-Aug-2026ffsrn 33390 The range of a finitely supported function is finite. The proof uses fnrndomnum 10644 rather than fnrndomg 10645, and so does not require ax-ac 10564. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Vincent Gonzalez, 24-Aug-2026.)
(𝜑 → 𝑍 ∈ 𝑊)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → (𝐹 supp 𝑍) ∈ Fin)    ⇒   (𝜑 → ran 𝐹 ∈ Fin)
 
24-Aug-2026ker2idl 21603 The kernel of a ring homomorphism is a two-sided ideal. (Contributed by Jeff Madsen, 3-Jan-2011.) (Revised by AV, 24-Aug-2026.)
𝐼 = (2Ideal‘𝑅)    &    0 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ { 0 }) ∈ 𝐼)
 
24-Aug-20262idl1el 21579 A two-sided ideal contains 1 iff it is the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 24-Aug-2026.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ( 1 ∈ 𝐼 ↔ 𝐼 = 𝐵))
 
24-Aug-2026dfring3 20548 The predicate "is a (unital) ring" based on a ring being abelian and with the definition of a monoid expanded. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by AV, 24-Aug-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g‘𝑅)    ⇒   (𝑅 ∈ Ring ↔ ((𝑅 ∈ Abel ∧ 𝐺 ∈ Mgm) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)) ∧ (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑦 ∧ (𝑦 · 𝑥) = 𝑦)))
 
24-Aug-2026idvalriota 18870 The unique value of the group identity element. (Contributed by FL, 12-Dec-2009.) (Revised by AV, 24-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    ⇒    0 = (℩𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
 
24-Aug-2026fnct 10647 If the domain of a function is countable, the function is countable. The proof uses fnrndomnum 10644 rather than fnrndomg 10645, and so does not require ax-ac 10564. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Revised by Vincent Gonzalez, 24-Aug-2026.)
((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω)
 
24-Aug-2026dmct 10629 The domain of a countable set is countable. The proof uses fodomnum 10163 rather than fodomg 10627, and so does not require ax-ac 10564. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Revised by Vincent Gonzalez, 24-Aug-2026.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)
 
23-Aug-2026scottidm 35841 The Scott operation is idempotent. (Contributed by BTernaryTau, 23-Aug-2026.)
Scott Scott 𝐴 = Scott 𝐴
 
23-Aug-20260elscott 35837 If the empty set is an element of the input class, then the Scott's trick set is the singleton of the empty set. (Contributed by BTernaryTau, 23-Aug-2026.)
(∅ ∈ 𝐴 → Scott 𝐴 = {∅})
 
23-Aug-2026elscottn0 35836 If the input class has elements, then the Scott's trick set is not empty. (Contributed by BTernaryTau, 23-Aug-2026.)
(𝐴 ∈ 𝐵 → Scott 𝐵 ≠ ∅)
 
23-Aug-2026angmgmaddcpbl 29420 Addition of angles is compatible with angle congruence. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   (𝜑 → 𝐶 ∈ 𝐴)    &   (𝜑 → 𝐷 ∈ 𝐴)    &   (𝜑 → 𝐸 ∈ 𝐴)    &   (𝜑 → 𝐹 ∈ 𝐴)    &   (𝜑 → 𝐸 ∼ 𝐶)    &   (𝜑 → 𝐹 ∼ 𝐷)    ⇒   (𝜑 → (𝐸 + 𝐹) ∼ (𝐶 + 𝐷))
 
23-Aug-2026angmgmaddov2 29419 Value of the addition operation in the angle addition magma, in case the first angle is zero or flat. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → ⟨“𝑊𝑉𝑆”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩)    &   (𝜑 → (𝑉 − 𝑆) = (𝑌 − 𝑋))    ⇒   (𝜑 → (⟨“𝑋𝑌𝑍”⟩ + ⟨“𝑈𝑉𝑊”⟩) = ⟨“𝑈𝑉𝑆”⟩)
 
23-Aug-2026angmgmaddov1 29418 Value of the addition operation in the angle addition magma, in case the first angle is neither zero nor flat. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &    + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))    &   (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → ⟨“𝑍𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)    &   (𝜑 → (𝑌 − 𝑆) = (𝑉 − 𝑈))    &   (𝜑 → ((𝑌𝐿𝑍) ∩ (𝑆𝐼𝑋)) ≠ ∅)    ⇒   (𝜑 → (⟨“𝑋𝑌𝑍”⟩ + ⟨“𝑈𝑉𝑊”⟩) = ⟨“𝑋𝑌𝑆”⟩)
 
23-Aug-2026angmgmaddov2lem 29417 Lemma for angmgmaddov2 29419. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑊𝑉𝑠”⟩ ∼ ⟨“𝑋𝑌𝑍”⟩ ∧ (𝑉 − 𝑠) = (𝑌 − 𝑋)))
 
23-Aug-2026angmgmaddov1lem 29416 Lemma for angmgmaddov1 29418. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
 
23-Aug-2026angmgmaddeu7 29415 There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where both angles are straight angles. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))    &   (𝜑 → 𝑉 ∈ (𝑈𝐼𝑊))    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
 
23-Aug-2026angmgmaddeu6 29414 There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where the first angle is a zero angle, and the second angle is a straight angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))    &   (𝜑 → 𝑈((hlG‘𝐺)‘𝑉)𝑊)    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
 
23-Aug-2026angmgmaddeu5 29413 There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where one angle is a zero angle, and the other angle is a straight angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → 𝑋((hlG‘𝐺)‘𝑌)𝑍)    &   (𝜑 → 𝑉 ∈ (𝑈𝐼𝑊))    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
 
23-Aug-2026angmgmaddeu4 29412 There exists a unique point 𝑠 satisfying the conditions of angle addition. Case where both angles are zero angles. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → 𝑋((hlG‘𝐺)‘𝑌)𝑍)    &   (𝜑 → 𝑈((hlG‘𝐺)‘𝑉)𝑊)    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈)))
 
23-Aug-2026angmgmaddeu3 29411 Existence of a unique point for building angle addition. Case where the second angle is a flat angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   (𝜑 → 𝑉 ∈ (𝑈𝐼𝑊))    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
 
23-Aug-2026angmgmaddeu2 29410 Existence of a unique point for building angle addition. Case where the second angle is a zero angle. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   (𝜑 → 𝑈((hlG‘𝐺)‘𝑉)𝑊)    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
 
23-Aug-2026angmgmaddeu1 29409 There exists a unique point 𝑠 satisfying the conditions of angle addition. General case. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑈 ≠ 𝑉)    &   (𝜑 → 𝑉 ≠ 𝑊)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    &   (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))    &   (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑊))    ⇒   (𝜑 → ∃!𝑠 ∈ 𝑃 (⟨“𝑍𝑌𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ (𝑌 − 𝑠) = (𝑉 − 𝑈) ∧ ((𝑌𝐿𝑍) ∩ (𝑠𝐼𝑋)) ≠ ∅))
 
23-Aug-2026elcgrabasrd 29406 Helper theorem for the membership in the base set of the angle addition monoid. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   (𝜑 → 𝑃 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ 𝐴)
 
23-Aug-2026elcgrabasi 29405 Helper theorem for the membership in the base set of the angle addition monoid. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 ∈ V    &   𝐴 = {𝑑 ∈ (𝑃 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}    &   (𝜑 → 𝐸 ∈ 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 (𝐸 = ⟨“𝑥𝑦𝑧”⟩ ∧ (𝑥 ≠ 𝑦 ∧ 𝑦 ≠ 𝑧)))
 
23-Aug-2026tgaaddcpbl2 29383 The angular addition is compatible with angle congruence: by adding congruent angles together, we obtain congruent angles. Compared with tgaaddcpbl 29382, this version handles cases where 𝑈, 𝑉 and 𝑊 are aligned. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑌 ≠ 𝑆)    &   (𝜑 → 𝑉 ≠ 𝑇)    &   (𝜑 → ((𝑌𝐿𝑆) ∩ (𝑋𝐼𝑍)) ≠ ∅)    &   (𝜑 → ((𝑉𝐿𝑇) ∩ (𝑈𝐼𝑊)) ≠ ∅)    &   (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)    &   (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
 
23-Aug-2026zerocgra 29361 Zero angles are congruent. Zero angles, that is, angles of degree zero, can be expressed by stating that points 𝐴 and 𝐶 are on the same ray starting at 𝐵, that is, 𝐴(𝐾‘𝐵)𝐶. See also flatcgra 29362. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴(𝐾‘𝐵)𝐶)    &   (𝜑 → 𝐷(𝐾‘𝐸)𝐹)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    ⇒   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
 
23-Aug-2026lnoppinn0 29261 The segment between two points 𝑋 and 𝑌 on opposite sides of a line 𝐷 intersects 𝐷. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑋𝑂𝑌)    ⇒   (𝜑 → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)
 
23-Aug-2026tghlsub 29116 Removing identical parts from the end of a ray segment preserves congruence. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐴(𝐾‘𝐵)𝐶)    &   (𝜑 → 𝐷(𝐾‘𝐸)𝐹)    &   (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))    &   (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))    ⇒   (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))
 
23-Aug-2026tgsegconeu 28979 The point constructed in axtgsegcon 28956 is unique. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → ∃!𝑧 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑧) ∧ (𝑌 − 𝑧) = (𝐴 − 𝐵)))
 
23-Aug-2026dfric2 20787 Alternate definition of the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 23-Aug-2026.)
≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ Ring ∧ 𝑠 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingIso 𝑠))}
 
23-Aug-2026s3rexrd 15131 Converse of s3rex 15130, deduction form. (Contributed by Thierry Arnoux, 23-Aug-2026.)
(𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    &   (𝜑 → 𝑍 ∈ 𝑆)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (𝑆 ↑m (0..^3)))
 
23-Aug-2026s3rex 15130 Membership in a family of words of length 3. (Contributed by Thierry Arnoux, 23-Aug-2026.)
𝑆 ∈ V    ⇒   (𝐴 ∈ (𝑆 ↑m (0..^3)) ↔ ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑆 ∃𝑧 ∈ 𝑆 𝐴 = ⟨“𝑥𝑦𝑧”⟩)
 
23-Aug-2026elrelb 5780 A member of a relation expressed by an ordered pair. (Contributed by AV, 23-Aug-2026.)
(Rel 𝑅 → (𝐴 ∈ 𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦)))
 
21-Aug-2026dfprop 38839 The set of sentences of propositional calculus is equal to the set of sentences that are either a variable encoded as a natural number, a negation of a sentence propositional calculus, or an implication between two sentences of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)
PROP = {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
 
21-Aug-2026dfprop2 38838 Every sentence of propositional calculus is either a variable encoded as a natural number, a negation of a sentence of propositional calculus, or an implication between two sentences of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)
PROP ⊆ {𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
 
21-Aug-2026dfprop1 38837 The set of variables encoded as a natural number, negations of sentences of propositional calculus, and implications between sentences of propositional calculus is a subset of PROP. (Contributed by Thomas van Maaren, 21-Aug-2026.)
{𝑥 ∣ (∃𝑧 ∈ PROP (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ PROP 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ⊆ PROP
 
21-Aug-2026impprop 38836 The implication between two sentences of propositional calculus is a sentence of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)
((𝑥 ∈ PROP ∧ 𝑦 ∈ PROP) → (𝑥prop→𝑦) ∈ PROP)
 
21-Aug-2026negprop 38835 The negation of a sentence of propositional calculus is a sentence of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)
(𝑥 ∈ PROP → (prop¬‘𝑥) ∈ PROP)
 
21-Aug-2026varprop 38834 Variables encoded as natural numbers are sentences of propositional calculus. (Contributed by Thomas van Maaren, 21-Aug-2026.)
(𝑛 ∈ ℕ → (propvar ‘𝑛) ∈ PROP)
 
21-Aug-2026dfproplem 38833 Given a set A, the set of all variables encoded as natural numbers, negations of elements in A, and implications between elements of A forms a set. This lemma is used when using fvmptd 7024 on the defining function of PROP. (Contributed by Thomas van Maaren, 21-Aug-2026.)
𝐴 ∈ V    ⇒   {𝑥 ∣ (∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ∈ V
 
21-Aug-2026df-prop 38832 Define the language of propositional calculus. This definition is noncircular. For a more usable and intuitive, but circular, definition see dfprop 38839. (Contributed by Thomas van Maaren, 21-Aug-2026.)
PROP = setrecs((𝑦 ∈ V ↦ {𝑥 ∣ (∃𝑧 ∈ 𝑦 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝑦 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}))
 
21-Aug-2026df-propimp 38830 The implication between two sentences of propositional calculus is encoded by concatenating the two sentences and appending a two at the end. (Contributed by Thomas van Maaren, 21-Aug-2026.)
prop→ = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 ++ 𝑦) ++ ⟨“2”⟩))
 
21-Aug-2026df-propneg 38829 The negation of a sentence of propositional calculus is encoded by appending a one after the sentence. (Contributed by Thomas van Maaren, 21-Aug-2026.)
prop¬ = (𝑥 ∈ V ↦ (𝑥 ++ ⟨“1”⟩))
 
21-Aug-2026df-propvar 38828 Variables in sentences of propositional calculus are encoded by appending a zero after the number of the variable. (Contributed by Thomas van Maaren, 21-Aug-2026.)
propvar = (𝑛 ∈ ℕ ↦ (⟨“𝑛”⟩ ++ ⟨“0”⟩))
 
21-Aug-2026degenmgm2 19169 A degenerate magma: although the operation is not defined for all pairs of elements of the base set, and its domain is not (a subset of) the base set, and the operation is not a function (see degenmgm2nfun 19168), the structure 𝑀 is still a magma according to our definition. (Contributed by AV, 21-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}    ⇒   𝑀 ∈ Mgm
 
21-Aug-2026degenmgm2nfun 19168 The operation of a second degenerate magma is not a function. (Contributed by AV, 21-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}    ⇒    ¬ Fun (+g‘𝑀)
 
21-Aug-2026degenmgm2opdm 19167 The domain of the operation of a second degenerate magma. (Contributed by AV, 21-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}    ⇒   dom (+g‘𝑀) = ({1o} × {1o, 2o})
 
21-Aug-2026degenmgmnfn 19165 The operation of a degenerate magma is not a function on its base set. (Contributed by AV, 21-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    &   𝐵 = (Base‘𝑀)    ⇒    ¬ (+g‘𝑀) Fn (𝐵 × 𝐵)
 
21-Aug-2026degenmgmbas 19164 The base set of a degenerate magma. (Contributed by AV, 21-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    &   𝐵 = (Base‘𝑀)    ⇒   𝐵 = {∅, 1o}
 
21-Aug-2026xpsnprg 7167 The Cartesian product of a singleton and an unordered pair. (Contributed by AV, 21-Aug-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐵, 𝐶}) = {⟨𝐴, 𝐵⟩, ⟨𝐴, 𝐶⟩})
 
19-Aug-2026veroquadmodzerod 51264 The columns of the Veronese matrix, weighted by the coefficients 𝐾, sum to the zero vector of ℝfld freeLMod (1...6). (Contributed by Jiamin Zhao, 19-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    &   (𝜑 → 𝐾:(1...6)⟶ℝ)    &   ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)    ⇒   (𝜑 → ((ℝfld freeLMod (1...6)) Σg (𝐾 ∘f ( ·𝑠 ‘(ℝfld freeLMod (1...6)))curry tpos 𝑉)) = (0g‘(ℝfld freeLMod (1...6))))
 
19-Aug-2026veroquadgsumlem 51263 Lemma for veroquadmodzerod 51264. Express the common homogeneous quadratic equation in ℝfld Σg form using the Veronese matrix 𝑉. (Contributed by Jiamin Zhao, 19-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    &   (𝜑 → 𝐾:(1...6)⟶ℝ)    &   ((𝜑 ∧ 𝑖 ∈ (1...6)) → (((((𝐾‘1) · (((𝐴‘𝑖)‘1)↑2)) + ((𝐾‘2) · (((𝐴‘𝑖)‘2)↑2))) + ((𝐾‘3) · (((𝐴‘𝑖)‘3)↑2))) + ((((𝐾‘4) · (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2))) + ((𝐾‘5) · (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)))) + ((𝐾‘6) · (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))) = 0)    ⇒   ((𝜑 ∧ 𝑖 ∈ (1...6)) → (ℝfld Σg (𝑗 ∈ (1...6) ↦ ((𝐾‘𝑗) · ((curry 𝑉‘𝑖)‘𝑗)))) = 0)
 
19-Aug-2026veronesematrowexpd 51262 Currying the Veronese matrix gives the indexed family of Veronese images, with each image expressed explicitly by coordinates. (Contributed by Jiamin Zhao, 19-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    ⇒   (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (𝑘 ∈ (1...6) ↦ if(𝑘 = 1, (((𝐴‘𝑖)‘1)↑2), if(𝑘 = 2, (((𝐴‘𝑖)‘2)↑2), if(𝑘 = 3, (((𝐴‘𝑖)‘3)↑2), if(𝑘 = 4, (((𝐴‘𝑖)‘1) · ((𝐴‘𝑖)‘2)), if(𝑘 = 5, (((𝐴‘𝑖)‘2) · ((𝐴‘𝑖)‘3)), (((𝐴‘𝑖)‘3) · ((𝐴‘𝑖)‘1))))))))))
 
19-Aug-2026veronesematrowd 51261 Currying the Veronese matrix gives the indexed family of Veronese images of the points 𝐴‘𝑖. (Contributed by Jiamin Zhao, 19-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    ⇒   (𝜑 → curry 𝑉 = (𝑖 ∈ (1...6) ↦ (veronese‘(𝐴‘𝑖))))
 
19-Aug-2026veronesematbasd 51260 The matrix whose 𝑖-th row is the Veronese image of 𝐴‘𝑖 belongs to the base set of (1...6) Mat ℝfld. (Contributed by Jiamin Zhao, 19-Aug-2026.)
𝑉 = (𝑖 ∈ (1...6), 𝑗 ∈ (1...6) ↦ ((veronese‘(𝐴‘𝑖))‘𝑗))    &   (𝜑 → 𝐴:(1...6)⟶(ℝ ↑m (1...3)))    ⇒   (𝜑 → 𝑉 ∈ (Base‘((1...6) Mat ℝfld)))
 
19-Aug-2026veronesefvcl 51252 Every coordinate of the Veronese map of a real 3-vector is real. (Contributed by Jiamin Zhao, 19-Aug-2026.)
((𝑄 ∈ (ℝ ↑m (1...3)) ∧ 𝐾 ∈ (1...6)) → ((veronese‘𝑄)‘𝐾) ∈ ℝ)
 
19-Aug-2026omeiunle 47732 The outer measure of the indexed union of a countable set is less than or equal to the extended sum of the outer measures. The proof uses abrexct 23803 rather than fnrndomg 10645, and so does not require ax-ac 10564. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Revised by Vincent Gonzalez, 19-Aug-2026.)
Ⅎ𝑛𝜑    &   Ⅎ𝑛𝐸    &   (𝜑 → 𝑂 ∈ OutMeas)    &   𝑋 = ∪ dom 𝑂    &   𝑍 = (ℤ≥‘𝑁)    &   (𝜑 → 𝐸:𝑍⟶𝒫 𝑋)    ⇒   (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
 
19-Aug-2026disjinfi 46412 Only a finite number of disjoint sets can have a nonempty intersection with a finite set 𝐶. The proof uses fodomfi 9332 rather than fodomg 10627, and so does not require ax-ac 10564. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Revised by Vincent Gonzalez, 19-Aug-2026.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   (𝜑 → 𝐶 ∈ Fin)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐶) ≠ ∅} ∈ Fin)
 
19-Aug-2026sigaclci 34834 A sigma-algebra is closed under countable intersections. Deduction version. The proof uses abrexct 23803 rather than abrexdom2jm 33176, and so does not require ax-ac 10564. (Contributed by Thierry Arnoux, 19-Sep-2016.) (Revised by Vincent Gonzalez, 19-Aug-2026.)
(((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)
 
19-Aug-2026madefi 28329 The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.) (Proof shortened by Vincent Gonzalez, 19-Aug-2026.)
(𝐴 ∈ ω → (M‘𝐴) ∈ Fin)
 
18-Aug-2026degenmgm 19166 A degenerate magma: although the operation is not defined for all pairs of elements of the base set ((∅(+g‘𝑀)1o) and (∅(+g‘𝑀)∅) are not defined, and therefore are ∅ by definition, which is contained in the base set), and its domain is not (a subset of) the base set (2o is in the domain of the operation, but not in the base set) , the structure 𝑀 is still a magma according to our definition. (Contributed by AV, 18-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    ⇒   𝑀 ∈ Mgm
 
18-Aug-2026degenmgmopdm 19163 The domain of the operation of a degenerate magma. (Contributed by AV, 18-Aug-2026.)
𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, ∅⟩, 1o⟩}⟩}    ⇒   dom (+g‘𝑀) = ({1o} × {∅, 1o, 2o})
 
18-Aug-2026mndideu 18963 The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.) (Proof shortened by AV, 18-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
 
18-Aug-2026mgmideud 18868 Uniqueness of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 18-Aug-2026.)
(𝜑 → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))    ⇒   (𝜑 → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
 
18-Aug-2026fvtp0 7229 The undefined value of a function with a domain of three elements. (Contributed by AV, 18-Aug-2026.)
𝐷 ∈ V    &   𝐸 ∈ V    &   𝐹 ∈ V    &   𝑋 ∈ V    ⇒   ((𝑋 ≠ 𝐴 ∧ 𝑋 ≠ 𝐵 ∧ 𝑋 ≠ 𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝑋) = ∅)
 
18-Aug-2026xpsntpg 7168 The Cartesian product of a singleton and an unordered triple. (Contributed by AV, 18-Aug-2026.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑇)) → ({𝐴} × {𝐵, 𝐶, 𝐷}) = {⟨𝐴, 𝐵⟩, ⟨𝐴, 𝐶⟩, ⟨𝐴, 𝐷⟩})
 
17-Aug-2026veronesevrowd 51259 The Veronese map at a point, expressed explicitly as a piecewise maps-to function on the six coordinates. (Contributed by Jiamin Zhao, 17-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (veronese‘𝑃) = (𝑘 ∈ (1...6) ↦ if(𝑘 = 1, ((𝑃‘1)↑2), if(𝑘 = 2, ((𝑃‘2)↑2), if(𝑘 = 3, ((𝑃‘3)↑2), if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), ((𝑃‘3) · (𝑃‘1)))))))))
 
17-Aug-2026veronesev6lem 51258 Lemma for veronesevrowd 51259. Value of the sixth coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 17-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((veronese‘𝑃)‘6) = ((𝑃‘3) · (𝑃‘1)))
 
17-Aug-2026veronesev5lem 51257 Lemma for veronesevrowd 51259. Value of the fifth coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 17-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((veronese‘𝑃)‘5) = ((𝑃‘2) · (𝑃‘3)))
 
17-Aug-2026veronesev4lem 51256 Lemma for veronesevrowd 51259. Value of the fourth coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 17-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((veronese‘𝑃)‘4) = ((𝑃‘1) · (𝑃‘2)))
 
17-Aug-2026veronesev3lem 51255 Lemma for veronesevrowd 51259. Value of the third coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 17-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((veronese‘𝑃)‘3) = ((𝑃‘3)↑2))
 
17-Aug-2026difelsiga 34837 A sigma-algebra is closed under class differences. The proof goes through difunielsiga 34835 and unelsiga 34836 rather than countable intersection, and so does not use ax-ac 10564. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Proof shortened by Vincent Gonzalez, 17-Aug-2026.)
((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∖ 𝐵) ∈ 𝑆)
 
17-Aug-2026difunielsiga 34835 A sigma-algebra is closed under complement relative to its base set. This is immediate from the definition, see issiga 34814, but the library states it nowhere in this form. (Contributed by Vincent Gonzalez, 17-Aug-2026.)
((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)
 
17-Aug-2026mndfo 18977 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 17-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    ⇒   ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶onto𝐵)
 
17-Aug-2026mndpfo 18976 The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.) (Proof shortened by AV, 17-Aug-2026.)
𝐵 = (Base‘𝐺)    &    ⨣ = (+𝑓‘𝐺)    ⇒   (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶onto𝐵)
 
17-Aug-2026fnrndomg 10645 The range of a function is dominated by its domain. This theorem requires the axiom of choice ax-ac2 10568; see fnrndomnum 10644 for a version that does not. (Contributed by NM, 1-Sep-2004.) (Proof shortened by Vincent Gonzalez, 17-Aug-2026.)
(𝐴 ∈ 𝐵 → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
 
17-Aug-2026fnrndomnum 10644 A version of fnrndomg 10645 that does not require the axiom of choice ax-ac 10564. (Contributed by Vincent Gonzalez, 17-Aug-2026.)
(𝐴 ∈ dom card → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴))
 
16-Aug-2026veronesev2lem 51254 Lemma for veronesevrowd 51259. Value of the second coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 16-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((veronese‘𝑃)‘2) = ((𝑃‘2)↑2))
 
16-Aug-2026coi1res 38151 Precomposing with the identity is the same as taking the relation part of a class. (Contributed by BJ, 16-Aug-2026.)
(𝐴 ∘ I ) = (𝐴 ↾ V)
 
16-Aug-2026elco 38150 Membership in a composition. (Contributed by BJ, 16-Aug-2026.)
(𝐴 ∈ (𝐶 ∘ 𝐵) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐶𝑦)))
 
16-Aug-2026mgmfod 18888 The operation of a magma with identity is an onto function (assuming it is a function). (Contributed by FL, 2-Nov-2009.) (Revised by AV, 16-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mgm)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    &   (𝜑 → + Fn (𝐵 × 𝐵))    ⇒   (𝜑 → + :(𝐵 × 𝐵)⟶onto𝐵)
 
16-Aug-2026mgmidprnd 18887 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 16-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mgm)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    &    ⨣ = (+𝑓‘𝐺)    ⇒   (𝜑 → ran ⨣ = 𝐵)
 
16-Aug-2026mgmidpfod 18886 The operation of a magma with identity as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 16-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mgm)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    &    ⨣ = (+𝑓‘𝐺)    ⇒   (𝜑 → ⨣ :(𝐵 × 𝐵)⟶onto𝐵)
 
16-Aug-2026mgmn0plusgplusf 18857 The group addition function of a magma is the restriction of its group operation to its base set if the base set does not contain the empty set. (Contributed by AV, 16-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mgm)    &   (𝜑 → ∅ ∉ 𝐵)    &    ⨣ = (+𝑓‘𝐺)    ⇒   (𝜑 → ⨣ = ( + ↾ (𝐵 × 𝐵)))
 
16-Aug-2026mgmn0plusgf 18856 The restriction of the group operation of a magma to its base set is a function if the base set does not contain the empty set. Excluding the empty set from the base set is necessary because of the specific definition of an undefined operation value (see also ndmovcl 7630 and ndmovrcl 7631). (Contributed by AV, 16-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mgm)    &   (𝜑 → ∅ ∉ 𝐵)    &   𝑃 = ( + ↾ (𝐵 × 𝐵))    ⇒   (𝜑 → 𝑃:(𝐵 × 𝐵)⟶𝐵)
 
16-Aug-2026relresfld 6296 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.) (Proof shortened by Eric Schmidt, 16-Aug-2026.)
(Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
 
15-Aug-2026veronesev1lem 51253 Lemma for veronesevrowd 51259. Value of the first coordinate of the Veronese map at a point. (Contributed by Jiamin Zhao, 15-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((veronese‘𝑃)‘1) = ((𝑃‘1)↑2))
 
15-Aug-2026f1resfz0f1d 13954 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.) (Revised by Mingli Yuan, 15-Aug-2026.)
(𝜑 → 𝐾 ∈ ℕ0)    &   (𝜑 → 𝐹:(0...𝐾)⟶𝑉)    &   (𝜑 → Fun ◡(𝐹 ↾ (1...𝐾)))    &   (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)    ⇒   (𝜑 → 𝐹:(0...𝐾)⟶1-1𝑉)
 
15-Aug-2026f1resrcmplf1dlem 7301 Lemma for f1resrcmplf1d 7302. (Contributed by BTernaryTau, 27-Sep-2023.) (Revised by Mingli Yuan, 15-Aug-2026.)
(𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝐷 ⊆ 𝐴)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)    &   (𝜑 → 𝑋 ∈ 𝐶)    &   (𝜑 → 𝑌 ∈ 𝐷)    &   (𝜑 → (𝐹‘𝑋) = (𝐹‘𝑌))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
14-Aug-2026veronesevald 51251 Value of the Veronese map at a point, expressed as a maps-to function on the six coordinates. (Contributed by Jiamin Zhao, 14-Aug-2026.)
(𝜑 → 𝑃 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (veronese‘𝑃) = (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑃‘1)↑2), 0) + if(𝑘 = 2, ((𝑃‘2)↑2), 0)) + if(𝑘 = 3, ((𝑃‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑃‘1) · (𝑃‘2)), 0) + if(𝑘 = 5, ((𝑃‘2) · (𝑃‘3)), 0)) + if(𝑘 = 6, ((𝑃‘3) · (𝑃‘1)), 0)))))
 
14-Aug-2026df-veronese 51249 Define the quadratic Veronese map on real 3-vectors, with coordinates ordered as ( x^2 , y^2 , z^2 , x y , y z , z x ). (Contributed by Jiamin Zhao, 14-Aug-2026.)
veronese = (𝑞 ∈ (ℝ ↑m (1...3)) ↦ (𝑘 ∈ (1...6) ↦ (((if(𝑘 = 1, ((𝑞‘1)↑2), 0) + if(𝑘 = 2, ((𝑞‘2)↑2), 0)) + if(𝑘 = 3, ((𝑞‘3)↑2), 0)) + ((if(𝑘 = 4, ((𝑞‘1) · (𝑞‘2)), 0) + if(𝑘 = 5, ((𝑞‘2) · (𝑞‘3)), 0)) + if(𝑘 = 6, ((𝑞‘3) · (𝑞‘1)), 0)))))
 
14-Aug-2026cveronese 51248 Extend class notation to include the quadratic Veronese map on real 3-vectors. (Contributed by Jiamin Zhao, 14-Aug-2026.)
class veronese
 
14-Aug-2026findcard4 38824 Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any set with less elements. The result is then proven to be true for all finite sets. (Contributed by Thomas van Maaren, 14-Aug-2026.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   (𝑦 ∈ Fin → (∀𝑥((♯‘𝑥) < (♯‘𝑦) → 𝜑) → 𝜒))    ⇒   (𝐴 ∈ Fin → 𝜏)
 
13-Aug-20262elfz13 51224 Membership of 2 in the integer interval ( 1 ... 3 ). (Suggested by tirix.) (Contributed by Jiamin Zhao, 1-Aug-2026.) (Proof shortened by Jiamin Zhao, 13-Aug-2026.)
2 ∈ (1...3)
 
13-Aug-20261elfz13 51223 Membership of 1 in the integer interval ( 1 ... 3 ). (Suggested by avekens.) (Contributed by Jiamin Zhao, 1-Aug-2026.) (Proof shortened by Jiamin Zhao, 13-Aug-2026.)
1 ∈ (1...3)
 
12-Aug-2026crossp3d 51247 The vector triple product expansion (BAC-CAB rule): the cross product of 𝑋 with (𝑌⊠𝑍) equals 𝑌 scaled by the dot product of 𝑋 and 𝑍, minus 𝑍 scaled by the dot product of 𝑋 and 𝑌. The dot products are written out as explicit three-term sums of component products, matching the pointwise style of df-crossp 51231 rather than introducing a separate dot product operator. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝑋 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝑌 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝑍 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝑋⊠(𝑌⊠𝑍)) = (𝑘 ∈ (1...3) ↦ ((((((𝑋‘1) · (𝑍‘1)) + ((𝑋‘2) · (𝑍‘2))) + ((𝑋‘3) · (𝑍‘3))) · (𝑌‘𝑘)) − (((((𝑋‘1) · (𝑌‘1)) + ((𝑋‘2) · (𝑌‘2))) + ((𝑋‘3) · (𝑌‘3))) · (𝑍‘𝑘)))))
 
12-Aug-2026crosspaltd 51246 Antisymmetry of the cross product: swapping the two vectors negates the result. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐴⊠𝐵) = (𝑘 ∈ (1...3) ↦ -((𝐵⊠𝐴)‘𝑘)))
 
12-Aug-2026crosspdotd 51245 Value of the scalar triple product, expanded into the standard six-term Sarrus polynomial. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐵(tripp‘𝐴)𝐶) = (((((𝐴‘1) · (𝐵‘2)) · (𝐶‘3)) − (((𝐴‘1) · (𝐵‘3)) · (𝐶‘2))) + (((((𝐴‘2) · (𝐵‘3)) · (𝐶‘1)) − (((𝐴‘2) · (𝐵‘1)) · (𝐶‘3))) + ((((𝐴‘3) · (𝐵‘1)) · (𝐶‘2)) − (((𝐴‘3) · (𝐵‘2)) · (𝐶‘1))))))
 
12-Aug-2026crosspdotsumlem 51244 Lemma for crosspdotd 51245. Expand the group sum over (1...3) into an explicit three-term sum. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))) = (((𝐴‘1) · ((𝐵⊠𝐶)‘1)) + (((𝐴‘2) · ((𝐵⊠𝐶)‘2)) + ((𝐴‘3) · ((𝐵⊠𝐶)‘3)))))
 
12-Aug-2026crosspdot0lem 51243 Lemma for crosspdotd 51245. Unfold the curried scalar triple product application into an explicit group sum. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐶 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐵(tripp‘𝐴)𝐶) = (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝐴‘𝑘) · ((𝐵⊠𝐶)‘𝑘)))))
 
12-Aug-2026crosspv3d 51242 Value of the third component of the cross product. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((𝐴⊠𝐵)‘3) = (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))))
 
12-Aug-2026crosspv2d 51241 Value of the second component of the cross product. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((𝐴⊠𝐵)‘2) = (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))))
 
12-Aug-2026crosspv1d 51240 Value of the first component of the cross product. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → ((𝐴⊠𝐵)‘1) = (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))))
 
12-Aug-2026crosspcld 51239 Closure of the cross product: the cross product of two 3-dimensional real coordinate vectors is again such a vector. (Contributed by Jiamin Zhao, 12-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐴⊠𝐵) ∈ (ℝ ↑m (1...3)))
 
12-Aug-2026tgaas 33116 Angle-angle-side: two triangles with two congruent angles and a congruent side not included between them are congruent. The side included between the two angles is congruent by tgaas1 33115, which reduces the theorem to SAS (tgsas 29430). (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)    ⇒   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
12-Aug-2026tgaas1 33115 Under the AAS hypotheses, the side included between the two given angles is congruent. This is the whole content of AAS: tgaas 33116 simply packages it with SAS (tgsas 29430). The point copied onto ray 𝐸𝐹 by tgaascgra 33113 is eliminated here. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)    ⇒   (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))
 
12-Aug-2026tgaas1h 33114 tgaas1 33115 with the constructed point 𝐻 still explicit: the case split on 𝐹 = 𝐻. Equal, and the side equality is a substitution; distinct, and tgaasne 33112 closes the branch. BOTH branches are proved -- assuming 𝐹 = 𝐻 would be assuming the theorem. (Contributed by JAG , 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝐻 ∈ 𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)    &   (𝜑 → 𝐻(𝐾‘𝐸)𝐹)    &   (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶))    ⇒   (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))
 
12-Aug-2026tgaascgra 33113 The comparison triangle of the AAS route: with 𝐻 placed on ray 𝐸𝐹 at distance 𝐵𝐶 from 𝐸, the triangle 𝐷𝐸𝐻 is congruent to 𝐴𝐵𝐶 (cgrahl2 29354 then tgsas 29430), so its angle at 𝐻 can be read off (cgrcgra 29358) and matched against the given angle at 𝐹. The result is the angle congruence the case split of tgaas1h 33114 consumes. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝐻 ∈ 𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    &   (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐵𝐶𝐴”⟩(cgrA‘𝐺)⟨“𝐸𝐹𝐷”⟩)    &   (𝜑 → 𝐻(𝐾‘𝐸)𝐹)    &   (𝜑 → (𝐸 − 𝐻) = (𝐵 − 𝐶))    ⇒   (𝜑 → ⟨“𝐷𝐹𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝐻𝐸”⟩)
 
12-Aug-2026tgaasne 33112 Two DISTINCT points 𝐹 and 𝐻 on one half-line from 𝐸, subtending congruent angles at a point 𝐷 off the line, are contradictory. ishlg 29098 unfolds the half-line to "one of them is strictly between 𝐸 and the other" -- both sub-branches are proved, each by tgaasexc 33111 on its own triangle, and they are not permutations of one another: the second needs the non-collinearity transported to line 𝐸𝐻 by cgrancol 29367. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝐻 ∈ 𝑃)    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))    &   (𝜑 → 𝐻(𝐾‘𝐸)𝐹)    &   (𝜑 → 𝐹 ≠ 𝐻)    &   (𝜑 → ⟨“𝐷𝐹𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝐻𝐸”⟩)    ⇒   (𝜑 → 𝜓)
 
12-Aug-2026tgaasexc 33111 The contradiction branch of AAS, stated once for both orders of the two candidate points. 𝑆 lies strictly between 𝐸 and 𝑇, and 𝐷 is off the line, so ⟨“𝐷𝑆𝐸”⟩ is an exterior angle of triangle 𝐷𝑇𝑆 and is STRICTLY greater than the remote interior angle at 𝑇-- while the hypothesis says the two are congruent. tgextan 33109 turns that into anything. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝑇) ∨ 𝐸 = 𝑇))    &   (𝜑 → 𝑆 ∈ (𝐸𝐼𝑇))    &   (𝜑 → 𝑆 ≠ 𝐸)    &   (𝜑 → 𝑆 ≠ 𝑇)    &   (𝜑 → ⟨“𝐷𝑇𝐸”⟩(cgrA‘𝐺)⟨“𝐷𝑆𝐸”⟩)    ⇒   (𝜑 → 𝜓)
 
12-Aug-2026tgaasncol 33110 Slide a non-collinearity along the line it is off: if 𝑋 is off line 𝑌𝑍 and 𝑊 is a point of that line other than 𝑍, then 𝑋 is off line 𝑍𝑊 too. Lines 𝑍𝑌 and 𝑍𝑊 coincide (tglineeltr 29129), which is what the exterior-angle hypothesis of tgaasexc 33111 needs at the moving point. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))    &   (𝜑 → 𝑊 ≠ 𝑍)    ⇒   (𝜑 → ¬ (𝑋 ∈ (𝑍𝐿𝑊) ∨ 𝑍 = 𝑊))
 
12-Aug-2026tgextan 33109 tgexta 33108's strictness can be used -- a remote interior angle congruent to the exterior angle is contradictory, so anything follows. This is the shape tgaasexc 33111 consumes: that branch holds an angle congruence and must close on it, which the non-strict ≤∠-only form of tgexta 33108 could never let it do. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    ≤ = (≤∠‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)    ⇒   (𝜑 → 𝜓)
 
12-Aug-2026tgexta 33108 The exterior-angle theorem, GeoCoq l11_41 -- BOTH remote interior angles of the triangle are STRICTLY less than the exterior angle at 𝐶. There is no ltA in set.mm , so strictness is carried inline as ≤∠ together with the non-congruence; the pair is what tgaas 33116's contradiction branch needs, since a bare ≤∠ conclusion is satisfied by a congruence and so contradicts nothing (tgextan 33109 records that the pair is usable). The second conjunct is not a permutation of the first: axtgsegcon 28956 builds a second extension point 𝑦 beyond 𝐶 on line 𝐴𝐶, tgextalt2 33107 reruns the construction there, and the two exterior angles are identified as vertical by tgextaoa 33105. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    ≤ = (≤∠‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → ((⟨“𝐵𝐴𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐵𝐴𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩) ∧ (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)))
 
12-Aug-2026tgextalt2 33107 tgexta 33108's SECOND conjunct, strict, at a supplied second extension point 𝐸. This is NOT a permutation of the first conjunct: 𝐸 is a new point, 𝐶 between 𝐴 and 𝐸, and tgextale 33101 / tgextancgra 33104 are applied a second time at the reconfigured triangle (the roles of 𝐴 and 𝐵 exchanged, exterior angle ⟨“𝐵𝐶𝐸”⟩). tgextaoa 33105 then identifies ⟨“𝐵𝐶𝐸”⟩ with ⟨“𝐴𝐶𝐷”⟩ as vertical angles, and the two halves are transported across that congruence -- leacgra 33094 for ≤∠, and cgratr 29360 under mtand 828 for the negation, which leacgra 33094 cannot move. 𝐸 is existentially constructed by tgexta 33108, not here. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    ≤ = (≤∠‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    &   (𝜑 → 𝐶 ≠ 𝐸)    ⇒   (𝜑 → (⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩))
 
12-Aug-2026tgextalt1 33106 tgexta 33108's FIRST conjunct, strict -- the remote interior angle at 𝐴 is strictly less than the exterior angle at 𝐶. tgextale 33101 and tgextancgra 33104 deliver both halves already, but stated about ⟨“𝐶𝐴𝐵”⟩; tgexta 33108 wants ⟨“𝐵𝐴𝐶”⟩, the same angle read backwards. So this is cgraswap 29357 plus two transports: leacgra1 33093 for the ≤∠ half, and cgratr 29360 under mtand 828 for the negation -- if ⟨“𝐵𝐴𝐶”⟩ were congruent to the exterior angle then so would ⟨“𝐶𝐴𝐵”⟩ be, which tgextancgra 33104 forbids. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    ≤ = (≤∠‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → (⟨“𝐵𝐴𝐶”⟩ ≤ ⟨“𝐴𝐶𝐷”⟩ ∧ ¬ ⟨“𝐵𝐴𝐶”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩))
 
12-Aug-2026tgextaoa 33105 The two exterior angles at 𝐶 are VERTICAL, hence congruent. Ray 𝐶𝐷 is opposite ray 𝐶𝐵 and ray 𝐶𝐸 is opposite ray 𝐶𝐴, so ⟨“𝐵𝐶𝐸”⟩ and ⟨“𝐴𝐶𝐷”⟩ have both legs flipped. One sacgr 29369 (Schwabhaeuser 11.13, GeoCoq l11_13 ) does it in a single shot, because sacgr 29369 flips the first slot of each side independently: the premise it wants is ⟨“𝐷𝐶𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐶𝐷”⟩, a bare angle reversal by cgraswap 29357, needing no non-collinearity at all. oacgr 29370 (11.14) would prove the same thing from two supplements and is not used. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → 𝐶 ∈ (𝐴𝐼𝐸))    &   (𝜑 → 𝐶 ≠ 𝐵)    &   (𝜑 → 𝐶 ≠ 𝐴)    &   (𝜑 → 𝐷 ≠ 𝐶)    &   (𝜑 → 𝐶 ≠ 𝐸)    ⇒   (𝜑 → ⟨“𝐵𝐶𝐸”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
 
12-Aug-2026tgextancgra 33104 The exterior angle is NOT congruent to the remote interior angle -- the strictness tgextale 33101 cannot give, and what tgaas 33116's contradiction branch needs. The exterior angle is ⟨“𝐴𝐶𝐷”⟩, at the extension of 𝐵𝐶 beyond 𝐶. The tail of GeoCoq's l11_41_aux , run through acopyeu 29372 rather than conga_cop_or_out_ts : a congruent exterior angle would, by angle-construction uniqueness, put the midpoint-reflected point of tgmircgra 33096 on the ray 𝐶𝐷-- the two rays make congruent angles with the shared base ray 𝐶𝐴 and, by tgextahp 33103 and hpgid 29274, lie on the same open half-plane of line 𝐴𝐶-- which tgextamirnhl 33102 forbids. Both of acopyeu 29372's non-collinearity hypotheses are discharged, not assumed: source side by rotating the triangle's own ncol , target side by tgextancd 33098. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → ¬ ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
 
12-Aug-2026tgextahp 33103 The reflected point and 𝐷 lie on the same OPEN half-plane of line 𝐴𝐶-- the fact acopyeu 29372 requires and inA does not give. tgextalem1inag 33100 puts the reflected point inside the exterior angle, which is non-strict: isinag 29387's witness clause is (𝑥 = 𝐶 ∨ 𝑥(𝐾‘𝐶)𝑃), so a point inside an angle may sit on a boundary ray. The 𝑥 = 𝐶 branch collapses to (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷) and is killed by tgextancd 33098; the main branch is tgextahpl 33099, whose off-line input is tgextamirn 33097. The $d s are the mandatory ones isinag 29387 carries for the witness variable. (Contributed by JAG , 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
 
12-Aug-2026tgextamirnhl 33102 The midpoint-reflected point is not on the ray from 𝐶 through 𝐷. If it were, then -- ray 𝐶𝐷 being the extension of 𝐵𝐶-- it would lie on line 𝐶𝐵; mirln2 29179, the converse of the mirln 29178 used in tgextamirn 33097, would put the midpoint of 𝐴 and 𝐶 on that line too; and the midpoint is collinear with 𝐴 and 𝐶, so 𝐴 would land on line 𝐶𝐵, contradicting the triangle's non-collinearity. 𝑀 ≠ 𝐶 comes from ismidb 29313 / mirinv 29168 / mirne 29169, not from midbtwn 29314. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → ¬ 𝐷(𝐾‘𝐶)(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵))
 
12-Aug-2026tgextale 33101 The non-strict exterior-angle bound, a checkpoint only. tgmircgra 33096 gives the reflected copy of the interior angle, tgextalem1inag 33100 puts that copy inside the exterior angle; inaglea 33089 turns "inside" into ≤∠ and leacgra1 33093 transports the bound along the congruence (flipped with cgracom 29359). No new geometry. This bound is deliberately too weak for tgaas 33116, whose contradiction branch already holds an angle congruence and so satisfies ≤∠ for free -- strictness comes later, and only tgexta 33108 is the exterior-angle theorem. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(≤∠‘𝐺)⟨“𝐴𝐶𝐷”⟩)
 
12-Aug-2026tgextalem1inag 33100 The point obtained by reflecting 𝐵 through the midpoint of 𝐴 and 𝐶 lies inside the exterior angle ⟨“𝐴𝐶𝐷”⟩. Proved by Pasch on the midpoint and the extension, turned into an in-angle statement with isinagd 29388 and moved from ray 𝐶𝑀 to ray 𝐶𝐴 by inaghl 29394. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)(inA‘𝐺)⟨“𝐴𝐶𝐷”⟩)
 
12-Aug-2026tgextahpl 33099 The generic open half-plane core. From a point 𝑋 between 𝐴 and 𝐷 that is on the half-line from 𝐶 through 𝑊, with 𝑊 off line 𝐴𝐶, both 𝑋 and 𝐷 and both 𝑋 and 𝑊 land on the same side of line 𝐴𝐶 (two colhp 29278 applications), and hpgcom 29275 / hpgtr 29276 compose them. 𝑋 ≠ 𝐶 is read off .xk by hlne1 29101, so no 𝑥 = 𝐶 case reaches here. The $d s are on proof dummies only -- colhp 29278's class abstraction needs its own bound variables -- so nothing propagates to callers. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))    &   (𝜑 → ¬ 𝑊 ∈ (𝐴𝐿𝐶))    &   (𝜑 → 𝑋 ∈ (𝐴𝐼𝐷))    &   (𝜑 → 𝑋(𝐾‘𝐶)𝑊)    ⇒   (𝜑 → 𝑊((hpG‘𝐺)‘(𝐴𝐿𝐶))𝐷)
 
12-Aug-2026tgextancd 33098 The target-side non-collinearity of the exterior angle. Line 𝐶𝐷 is line 𝐶𝐵-- 𝐶 lies between 𝐵 and 𝐷, and 𝐶 ≠ 𝐷-- so 𝐴 is off it exactly because the triangle is non-degenerate. This is acopyeu 29372's acopy.2 frame hypothesis in tgextancgra 33104, and it is what excludes isinag 29387's 𝑥 = 𝐶 branch in tgextahp 33103. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    &   (𝜑 → 𝐶 ≠ 𝐷)    ⇒   (𝜑 → ¬ (𝐴 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷))
 
12-Aug-2026tgextamirn 33097 The midpoint-reflected point is off line 𝐴𝐶. If it were on that line, mirln 29178-- whose centre-on-the-line hypothesis is the midpoint, itself on 𝐴𝐶-- would put its reflection there too, and mirmir 29164 says that reflection is 𝐵, contradicting the triangle's non-collinearity. This is the strengthening inA cannot give: it is what turns "inside the angle" into an open half-plane fact. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    ⇒   (𝜑 → ¬ (((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵) ∈ (𝐴𝐿𝐶))
 
12-Aug-2026tgmircgra 33096 The angle congruence the exterior-angle theorem opens with: the interior angle at 𝐴 in triangle 𝐶𝐴𝐵 is congruent to the angle at 𝐶 in the midpoint-reflected triangle. This is cgrcgra 29358 over tgmircgrg 33095; the vertex distinctnesses come from the non-collinearity via tgbcne 33076 and tgacne 33075. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    ⇒   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐴𝐶(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)”⟩)
 
12-Aug-2026tgmircgrg 33095 Reflecting 𝐵 through the midpoint of 𝐴 and 𝐶 carries triangle 𝐶𝐴𝐵 onto 𝐴𝐶𝑃, where 𝑃 is the reflected point. The dimension condition the mid* family carries is discharged from the non-collinearity through ncoltgdim2 29058, not taken as a hypothesis. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))    ⇒   (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝐴𝐶(((pInvG‘𝐺)‘(𝐴(midG‘𝐺)𝐶))‘𝐵)”⟩)
 
12-Aug-2026leacgra 33094 GeoCoq l11_30 analogue: ≤∠ is preserved when both compared angles are replaced by congruent angles. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    ≤ = (≤∠‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝐻 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ≤ ⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    ⇒   (𝜑 → ⟨“𝐻𝑈𝐽”⟩ ≤ ⟨“𝑋𝑌𝑍”⟩)
 
12-Aug-2026leacgra1 33093 Source-side ≤∠ transport: replacing the smaller angle by a congruent angle preserves the angle order. The counterpart of leacgra2 33092, proved directly from the witness form of ≤∠ (isleag 29396, isleagd 29397). (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝐻 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐻𝑈𝐽”⟩)    ⇒   (𝜑 → ⟨“𝐻𝑈𝐽”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
12-Aug-2026leacgra2 33092 Target-side ≤∠ transport: replacing the larger angle by a congruent angle preserves the angle order. Proved from leawittrg 33091 and isleag 29396, the witness form of ≤∠. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝑋𝑌𝑍”⟩)    &   (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
12-Aug-2026leawittrg 33091 Target-side ≤∠ witness transport: move a witness inside ⟨“𝐷𝐸𝐹”⟩ across a congruent target angle ⟨“𝑋𝑌𝑍”⟩, carrying the angle ⟨“𝐻𝑈𝐽”⟩ copied at that witness with it. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐻 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐻𝑈𝐽”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑊”⟩)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐻𝑈𝐽”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026cgrainag 33090 Target-side in-angle transport: if 𝑊 lies inside angle ⟨“𝐷𝐸𝐹”⟩ and that angle is congruent to ⟨“𝑋𝑌𝑍”⟩, then some point inside ⟨“𝑋𝑌𝑍”⟩ makes ⟨“𝑋𝑌𝑣”⟩ congruent to the sub-angle ⟨“𝐷𝐸𝑊”⟩. The geometric core of leawittrg 33091. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inaglea 33089 If 𝐶 lies inside angle ⟨“𝐴𝐵𝑌”⟩, then angle ⟨“𝐴𝐵𝐶”⟩ is less than or equal to ⟨“𝐴𝐵𝑌”⟩. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝐶(inA‘𝐺)⟨“𝐴𝐵𝑌”⟩)    ⇒   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(≤∠‘𝐺)⟨“𝐴𝐵𝑌”⟩)
 
12-Aug-2026inagcgratrgn 33088 In-angle transport, non-collinear case: unfold isinag 29387, eliminate its segment-point existential, and dispatch on the disjunct it carries -- 𝑥 = 𝐸 to inagcgrabtwn 33081, 𝑥(𝐾‘𝐸)𝑊 to inagcgratrgncol 33087. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ¬ (𝑊 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagcgratrgncol 33087 In-angle transport once the segment point 𝐽 of isinag 29387 is on the ray 𝐸𝑊: iscgrag 33082 normalizes the target arms and its two existentials are eliminated here, leaving inagcgratrgxfr 33086 to do the transfer. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → 𝐷 ≠ 𝐸)    &   (𝜑 → 𝐽 ∈ (𝐷𝐼𝐹))    &   (𝜑 → 𝐽(𝐾‘𝐸)𝑊)    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagcgratrgxfr 33086 In-angle transport with the target arms already normalized to 𝑆 and 𝑇: cgrgptxfr 33083 transfers the segment point 𝐽 to a point of (𝑆𝐼𝑇), and that one existential is eliminated here. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝐷 ≠ 𝐸)    &   (𝜑 → 𝐽 ∈ (𝐷𝐼𝐹))    &   (𝜑 → 𝐽(𝐾‘𝐸)𝑊)    &   (𝜑 → 𝑆(𝐾‘𝑌)𝑋)    &   (𝜑 → 𝑇(𝐾‘𝑌)𝑍)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrG‘𝐺)⟨“𝑆𝑌𝑇”⟩)    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagcgratrgw 33085 inagcgratrgmain 33084 with the existential introduced at its own witness 𝑄. Split out so that the substitution instance is built once, at the top level, rather than under the existential eliminations that produce 𝑆, 𝑇 and 𝑄. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝐷 ≠ 𝐸)    &   (𝜑 → 𝐽(𝐾‘𝐸)𝑊)    &   (𝜑 → 𝑆(𝐾‘𝑌)𝑋)    &   (𝜑 → 𝑇(𝐾‘𝑌)𝑍)    &   (𝜑 → 𝑄 ∈ (𝑆𝐼𝑇))    &   (𝜑 → ⟨“𝐷𝐸𝐽”⟩(cgrG‘𝐺)⟨“𝑆𝑌𝑄”⟩)    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagcgratrgmain 33084 In-angle transport, the non-degenerate main case, with its three constructions carried as hypotheses: 𝐽 the segment point from isinag 29387, 𝑆 and 𝑇 the normalized target arms from iscgrag 33082, and 𝑄 the transferred point from cgrgptxfr 33083. Concludes the body of cgrainag 33090's existential at the witness 𝑄. **No acopy 29371 and no half-plane fact**: the witness is the transferred point, not a fresh angle copy, which is what removes the non-collinearity obligations. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝐷 ≠ 𝐸)    &   (𝜑 → 𝐽(𝐾‘𝐸)𝑊)    &   (𝜑 → 𝑆(𝐾‘𝑌)𝑋)    &   (𝜑 → 𝑇(𝐾‘𝑌)𝑍)    &   (𝜑 → 𝑄 ∈ (𝑆𝐼𝑇))    &   (𝜑 → ⟨“𝐷𝐸𝐽”⟩(cgrG‘𝐺)⟨“𝑆𝑌𝑄”⟩)    &    − = (dist‘𝐺)    ⇒   (𝜑 → (𝑄(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑄”⟩))
 
12-Aug-2026cgrgptxfr 33083 Transfer a segment point across a triangle congruence: given ⟨“𝐴𝐵𝐶”⟩cgrG⟨“𝑋𝐸𝑍”⟩ and 𝐽 between 𝐴 and 𝐶, the corresponding point on (𝑋𝐼𝑍) exists, with its betweenness, its two distances and the sub-triangle congruence. GeoCoq reaches this through l4_6 and cong2_conga_cong ; here cgrg3col4 29402 (Schwabhauser 11.28) constructs the point with every pairwise distance and tgbtwnxfr 29023 -- which *is* l4_6 -- supplies the image betweenness. The 𝐴 = 𝐶 case takes cgrg3col4 29402's right disjunct, since tgellng 29046 needs 𝐴 ≠ 𝐶 and cgrA does not supply it. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &    − = (dist‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐽 ∈ 𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑋𝐸𝑍”⟩)    &   (𝜑 → 𝐽 ∈ (𝐴𝐼𝐶))    ⇒   (𝜑 → ∃𝑦 ∈ 𝑃 ((𝑦 ∈ (𝑋𝐼𝑍) ∧ ⟨“𝐴𝐵𝐽”⟩(cgrG‘𝐺)⟨“𝑋𝐸𝑦”⟩) ∧ ((𝐴 − 𝐽) = (𝑋 − 𝑦) ∧ (𝐽 − 𝐶) = (𝑦 − 𝑍))))
 
12-Aug-2026iscgrag 33082 Ray normalization: an angle congruence yields two points on the target rays carrying a full triangle congruence, together with the two arm lengths. This is the forward direction of iscgra 29346 (Schwabhauser 11.3) repackaged -- nothing is constructed; the distances are read off the cgrG with cgr3simp1 29013 and cgr3simp2 29014. No non-degeneracy hypothesis is needed or wanted: cgrA carries vertex distinctness intrinsically (cgrane1 29349 through cgrane4 29352), and an extra hypothesis would obstruct the degenerate branches. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    − = (dist‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    ⇒   (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩ ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹) ∧ ((𝐸 − 𝑥) = (𝐵 − 𝐴) ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
 
12-Aug-2026inagcgrabtwn 33081 In-angle transport, degenerate branch (iii): the containing angle is **flat** while the sub-angle is not. This is the 𝑥 = 𝐸 disjunct of isinag 29387, which the non-degenerate spine cannot reach. Unlike branches (i) and (ii) the copied angle must be *constructed*: the congruence hypothesis relates 𝑊 to nothing on the target side, and acopy 29371 needs a point off line 𝑋𝑌 to pick a side, which the collinear 𝑍 cannot be. ncoltgdim2 29058 takes the dimension from this branch's own non-collinearity and tglowdim2ln 29150 produces that point. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    &   (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))    &   (𝜑 → ¬ (𝑊 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagcgrasubbtwn 33080 In-angle transport, degenerate branch (ii): the sub-angle ⟨“𝐷𝐸𝑊”⟩ is **flat**, 𝐸 lying between 𝐷 and 𝑊. The witness is built beyond 𝑌 on the ray opposite 𝑌𝑋 with axtgsegcon 28956, and flatcgra 29362 supplies the copied angle from the two betweennesses. The in-angle conjunct goes through inagbtwn 33078, which carries the flatness up to the containing angle, then cgrabtwn 29364 across the congruence and isinagd 29388 on the target. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    &   (𝜑 → 𝐸 ∈ (𝐷𝐼𝑊))    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagcgrasubhl 33079 In-angle transport, degenerate branch (i): the sub-angle ⟨“𝐷𝐸𝑊”⟩ is **null**, 𝑊 lying on ray 𝐸𝐷. The witness is not constructed -- it is 𝑋, the target's own first arm -- and cgrahlb 33077 supplies the copied angle from the two ray degeneracies. No GeoCoq counterpart: GeoCoq's branch only locates a witness supplied by its universal statement. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)    &   (𝜑 → 𝐷(𝐾‘𝐸)𝑊)    ⇒   (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
 
12-Aug-2026inagbtwn 33078 A flat sub-angle forces the containing angle flat: if 𝑊 is inside angle ⟨“𝐷𝐸𝐹”⟩ and 𝐸 lies between 𝐷 and 𝑊, then 𝐸 lies between 𝐷 and 𝐹. This is GeoCoq bet_in_angle_bet. Proved by unfolding isinag 29387 and following its segment point. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)    &   (𝜑 → 𝐸 ∈ (𝐷𝐼𝑊))    ⇒   (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
 
12-Aug-2026cgrahlb 33077 Two null angles are congruent: if 𝐴 and 𝐶 lie on one half-line from 𝐵, and 𝐷 and 𝐹 on one half-line from 𝐸, then angle ⟨“𝐴𝐵𝐶”⟩ is congruent to ⟨“𝐷𝐸𝐹”⟩. This is GeoCoq l11_21_b ; set.mm has only the converse, cgrahl 29365 (GeoCoq l11_21_a ). Proved by copying both arms onto the target rays with hlcgrex 29112 and closing the third side by segment subtraction (tgcgrsub2 29088), so it needs no non-collinear frame. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &    − = (dist‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐷 ∈ 𝑃)    &   (𝜑 → 𝐸 ∈ 𝑃)    &   (𝜑 → 𝐹 ∈ 𝑃)    &   (𝜑 → 𝐴(𝐾‘𝐵)𝐶)    &   (𝜑 → 𝐷(𝐾‘𝐸)𝐹)    ⇒   (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
 
12-Aug-2026tgbcne 33076 Non-collinearity of triangle 𝐴𝐵𝐶 implies distinct vertices 𝐵 and 𝐶. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    ⇒   (𝜑 → 𝐵 ≠ 𝐶)
 
12-Aug-2026tgacne 33075 Non-collinearity of triangle 𝐴𝐵𝐶 implies distinct vertices 𝐴 and 𝐶. (Contributed by JAG, 12-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))    ⇒   (𝜑 → 𝐴 ≠ 𝐶)
 
12-Aug-2026ress0g 18983 0g is unaffected by restriction. This is a bit more generic than submnd0 18985. (Contributed by Thierry Arnoux, 23-Oct-2017.) (Proof shortened by AV, 12-Aug-2026.)
𝑆 = (𝑅 ↾s 𝐴)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 0 = (0g‘𝑆))
 
12-Aug-2026idressid 18891 The restriction of a structure with an identity element to a subset containing the identity element has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 12-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    &   𝑆 = (𝐺 ↾s 𝐴)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 0 ∈ 𝐴)    ⇒   (𝜑 → (0g‘𝑆) = 0 )
 
11-Aug-2026crosspclem 51238 Lemma for crosspcld 51239. Closure of the three-way coordinate case split used in the cross product's mapping rule. (Contributed by Jiamin Zhao, 11-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → if(𝑘 = 1, (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))), if(𝑘 = 2, (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))), (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))))) ∈ ℝ)
 
11-Aug-2026crosspcle3d 51237 Closure of the third component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1))) ∈ ℝ)
 
11-Aug-2026crosspcle2d 51236 Closure of the second component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))) ∈ ℝ)
 
11-Aug-2026crosspcle1d 51235 Closure of the first component of the cross product's coordinate formula. (Contributed by Jiamin Zhao, 11-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    &   (𝜑 → 𝐵 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))) ∈ ℝ)
 
11-Aug-2026wrdf1d 51220 A one-to-one word maps its domain into its alphabet. (Contributed by Mingli Yuan, 11-Aug-2026.)
(𝜑 → 𝑊 ∈ Word 𝐷)    &   (𝜑 → Fun ◡𝑊)    ⇒   (𝜑 → 𝑊:dom 𝑊⟶1-1𝐷)
 
11-Aug-2026idressidex 18890 The restriction of a structure with an identity element to a subset containing the identity element has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 11-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    &   𝑆 = (𝐺 ↾s 𝐴)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 0 ∈ 𝐴)    ⇒   (𝜑 → ∃𝑒 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
 
11-Aug-2026idressidex0 18889 The restriction of a structure with an identity element to a subset containing the identity element has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 11-Aug-2026.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    0 = (0g‘𝐺)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    &   𝑆 = (𝐺 ↾s 𝐴)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 0 ∈ 𝐴)    &   𝐶 = (Base‘𝑆)    ⇒   (𝜑 → ∃𝑒 ∈ 𝐶 ∀𝑥 ∈ 𝐶 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
 
11-Aug-20260gisid 18877 In a structure with an identity element, the group identity element is an identity element of the structure. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 11-Aug-2026.)
𝐵 = (Base‘𝐺)    &    0 = (0g‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    ⇒   (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
 
10-Aug-2026rr3fv3cld 51229 Third component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐴‘3) ∈ ℝ)
 
10-Aug-2026rr3fv2cld 51228 Second component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐴‘2) ∈ ℝ)
 
10-Aug-2026rr3fv1cld 51227 First component of a 3-dimensional real coordinate vector is real. (Contributed by Jiamin Zhao, 10-Aug-2026.)
(𝜑 → 𝐴 ∈ (ℝ ↑m (1...3)))    ⇒   (𝜑 → (𝐴‘1) ∈ ℝ)
 
10-Aug-2026had0 1634 If the first input is false, then the adder sum is equivalent to the exclusive disjunction of the other two inputs, and conversely. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jul-2020.) Strengthen to a biconditional. (Revised by BJ, 10-Aug-2026.)
(¬ 𝜑 ↔ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒)))
 
10-Aug-2026had1 1633 If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs, and conversely. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) Strengthen to a biconditional. (Revised by BJ, 10-Aug-2026.)
(𝜑 ↔ (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))
 
10-Aug-2026birot 389 Rotation of the arguments of the nested implication (. ↔ (. ↔ .)) (a general phenomenon for a commutative associative binary operation, see e.g., inrot 4132) . (Contributed by BJ, 10-Aug-2026.)
((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ (𝜓 ↔ (𝜒 ↔ 𝜑)))
 
8-Aug-2026dfring2 20547 The predicate "is a unital ring" based on a ring being abelian. Definition of "ring with unit" in [Lang] p. 83. (Contributed by Jeff Hankins, 21-Nov-2006.) (Revised by AV, 8-Aug-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g‘𝑅)    ⇒   (𝑅 ∈ Ring ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))
 
8-Aug-2026relcnvtrg 6286 Subclass law for converse of a composition. A particular case is ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅), which says that a relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) Generalize to a statement about three classes rather than one. (Revised by Peter Mazsa, 17-Oct-2023.) Remove antecedent. (Revised by Eric Schmidt, 8-Aug-2026.)
((𝑅 ∘ 𝑆) ⊆ 𝑇 ↔ (◡𝑆 ∘ ◡𝑅) ⊆ ◡𝑇)
 
5-Aug-2026f1owe 7384 Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.) Strengthen to biconditional. (Revised by Eric Schmidt, 5-Aug-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   (𝐹:𝐴⟶1-1onto𝐵 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
 
4-Aug-2026f1we 7386 Pull back a well ordering by a one-to-one function. (Contributed by Eric Schmidt, 4-Aug-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)}    ⇒   (𝐹:𝐴⟶1-1𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴))
 
3-Aug-2026nadddird 37153 Natural multiplication distributes over natural addition. Deduction form. (Contributed by Scott Fenton, 3-Aug-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    ⇒   (𝜑 → ((𝐴 +no 𝐵) ·no 𝐶) = ((𝐴 ·no 𝐶) +no (𝐵 ·no 𝐶)))
 
3-Aug-2026nadddid 37152 Natural multiplication distributes over natural addition. Deduction form. (Contributed by Scott Fenton, 3-Aug-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    ⇒   (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
 
3-Aug-2026nadddilem4 37150 Lemma for nadddi 37151. Prove the forward implication. (Contributed by Scott Fenton, 3-Aug-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))    &   (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))    &   (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))    &   (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))    ⇒   (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
 
3-Aug-2026nadddilem3 37149 Lemma for nadddi 37151. Prove a subcase of the forward implication. (Contributed by Scott Fenton, 3-Aug-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ (𝐵 +no 𝐶))    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑌 ⊆ (𝑍 +no 𝐶))    &   (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))    &   (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))    ⇒   (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)))
 
2-Aug-2026abweex 35801 The class of well-orders of a set 𝐴 and its subsets is a set. (Contributed by BTernaryTau, 2-Aug-2026.)
(𝐴 ∈ 𝑉 → {𝑟 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ∈ V)
 
2-Aug-2026tgaaddcpbl 29382 The angular addition is compatible with angle congruence: by adding congruent angles together, we obtain congruent angles. Theorem 11.22 of [Schwabhauser] p. 99. The angles ⟨“𝑋𝑌𝑆”⟩ and ⟨“𝑆𝑌𝑍”⟩ are added to result in ⟨“𝑋𝑌𝑍”⟩, and ⟨“𝑈𝑉𝑇”⟩ and ⟨“𝑇𝑉𝑊”⟩ are added to result in ⟨“𝑈𝑉𝑊”⟩. (Contributed by Thierry Arnoux, 2-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}    &   𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑌 ≠ 𝑆)    &   (𝜑 → 𝑉 ≠ 𝑇)    &   (𝜑 → 𝑋𝑂𝑍)    &   (𝜑 → 𝑈𝑄𝑊)    &   (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)    &   (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
 
2-Aug-2026tgaaddcpbllem3 29381 Lemma for tgaaddcpbl 29382. (Contributed by Thierry Arnoux, 2-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}    &   𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑌 ≠ 𝑆)    &   (𝜑 → 𝑉 ≠ 𝑇)    &   (𝜑 → 𝑋𝑂𝑍)    &   (𝜑 → 𝑈𝑄𝑊)    &   (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)    &   (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)    &   (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
 
2-Aug-2026tgaaddcpbllem2 29380 Lemma for tgaaddcpbl 29382. (Contributed by Thierry Arnoux, 2-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}    &   𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑌 ≠ 𝑆)    &   (𝜑 → 𝑉 ≠ 𝑇)    &   (𝜑 → 𝑋𝑂𝑍)    &   (𝜑 → 𝑈𝑄𝑊)    &   (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)    &   (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)    &   (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))    &   (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))    &   (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))    &   (𝜑 → 𝑌 ∈ (𝑆𝐼𝑅))    &   𝑀 = ((pInvG‘𝐺)‘𝑉)    &   𝐾 = (hlG‘𝐺)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
 
2-Aug-2026tgaaddcpbllem1 29379 Lemma for tgaaddcpbl 29382. (Contributed by Thierry Arnoux, 2-Aug-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∼ = (cgrA‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}    &   𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑆 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑃)    &   (𝜑 → 𝑈 ∈ 𝑃)    &   (𝜑 → 𝑉 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑌 ≠ 𝑆)    &   (𝜑 → 𝑉 ≠ 𝑇)    &   (𝜑 → 𝑋𝑂𝑍)    &   (𝜑 → 𝑈𝑄𝑊)    &   (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)    &   (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)    &   (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))    &   (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))    &   (𝜑 → 𝑅(𝐾‘𝑌)𝑆)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
 
1-Aug-20263elfz13 51225 Membership of 3 in the integer interval ( 1 ... 3 ). (Contributed by Jiamin Zhao, 1-Aug-2026.)
3 ∈ (1...3)
 
1-Aug-20262ne3 51222 2 is not equal to 3. (Contributed by Jiamin Zhao, 1-Aug-2026.)
2 ≠ 3
 
1-Aug-20261ne3 51221 1 is not equal to 3. (Contributed by Jiamin Zhao, 1-Aug-2026.)
1 ≠ 3
 
31-Jul-2026crosspval 51234 Value of the cross product of two 3-dimensional real coordinate vectors as a function on (1...3). (Contributed by Jiamin Zhao, 31-Jul-2026.)
((𝐴 ∈ (ℝ ↑m (1...3)) ∧ 𝐵 ∈ (ℝ ↑m (1...3))) → (𝐴⊠𝐵) = (𝑘 ∈ (1...3) ↦ if(𝑘 = 1, (((𝐴‘2) · (𝐵‘3)) − ((𝐴‘3) · (𝐵‘2))), if(𝑘 = 2, (((𝐴‘3) · (𝐵‘1)) − ((𝐴‘1) · (𝐵‘3))), (((𝐴‘1) · (𝐵‘2)) − ((𝐴‘2) · (𝐵‘1)))))))
 
31-Jul-2026df-tripp 51233 Define the scalar triple product of three 3-dimensional real coordinate vectors as the dot product of the first vector 𝑥 with the cross product of the other two (𝑦 and 𝑧). Apply as (𝑦(tripp‘𝑥)𝑧). Vectors are represented as functions on (1...3). (Contributed by Jiamin Zhao, 31-Jul-2026.)
tripp = (𝑥 ∈ (ℝ ↑m (1...3)) ↦ (𝑦 ∈ (ℝ ↑m (1...3)), 𝑧 ∈ (ℝ ↑m (1...3)) ↦ (ℝfld Σg (𝑘 ∈ (1...3) ↦ ((𝑥‘𝑘) · ((𝑦⊠𝑧)‘𝑘))))))
 
31-Jul-2026ctripp 51232 Extend class notation to include the scalar triple product of 3-dimensional real coordinate vectors. (Contributed by Jiamin Zhao, 31-Jul-2026.)
class tripp
 
31-Jul-2026df-crossp 51231 Define the cross product of two 3-dimensional real coordinate vectors. Vectors are represented as functions on (1...3). (Contributed by Jiamin Zhao, 31-Jul-2026.)
⊠ = (𝑢 ∈ (ℝ ↑m (1...3)), 𝑣 ∈ (ℝ ↑m (1...3)) ↦ (𝑘 ∈ (1...3) ↦ if(𝑘 = 1, (((𝑢‘2) · (𝑣‘3)) − ((𝑢‘3) · (𝑣‘2))), if(𝑘 = 2, (((𝑢‘3) · (𝑣‘1)) − ((𝑢‘1) · (𝑣‘3))), (((𝑢‘1) · (𝑣‘2)) − ((𝑢‘2) · (𝑣‘1)))))))
 
31-Jul-2026ccrossp 51230 Extend class notation to include the cross product operation. (Contributed by Jiamin Zhao, 31-Jul-2026.)
class ⊠
 
31-Jul-2026rr3fvcl 51226 The components of a 3-dimensional real coordinate vector are real numbers. (Contributed by Jiamin Zhao, 31-Jul-2026.)
(𝐴 ∈ (ℝ ↑m (1...3)) → ((𝐴‘1) ∈ ℝ ∧ (𝐴‘2) ∈ ℝ ∧ (𝐴‘3) ∈ ℝ))
 
31-Jul-2026numtowerdt 48121 Certain number sets and fields form a tower. In particular, singleton 1, natural numbers, natural numbers with zero, integers, rationals, algebraic reals (notice that current definition allows algebraic numbers to be complex thus the restriction), reals and complex number sets are a tower of proper subsets. (Contributed by Ender Ting, 31-Jul-2026.)
⟨“{1}ℕℕ0ℤℚ(𝔸 ∩ ℝ)ℝℂ”⟩ ∈ ( [⊊] Chain V)
 
31-Jul-2026nadddilem2 37148 Lemma for nadddi 37151. Prove the reverse implication. (Contributed by Scott Fenton, 31-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))    &   (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))    &   (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))    &   (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))    ⇒   (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ⊆ (𝐴 ·no (𝐵 +no 𝐶)))
 
31-Jul-2026nadddilem1 37147 Lemma for nadddi 37151. Prove a subcase of the reverse implication. (Contributed by Scott Fenton, 31-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))    &   (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))    &   (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))    ⇒   ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
 
31-Jul-2026onelssd 37128 An element of an ordinal number is a subset of the number. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ 𝐴)    ⇒   (𝜑 → 𝐵 ⊆ 𝐴)
 
31-Jul-2026ontr2d 37127 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝐵 ∈ 𝐶)    ⇒   (𝜑 → 𝐴 ∈ 𝐶)
 
31-Jul-2026onelond 37126 An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. Lemma 1.3 of [Schloeder] p. 1. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ 𝐴)    ⇒   (𝜑 → 𝐵 ∈ On)
 
31-Jul-2026acwer1prc 35852 The class of all well-orderings of the stages of the cumulative hierarchy is a proper class. (Contributed by BTernaryTau, 31-Jul-2026.)
𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}    ⇒   (CHOICE → ¬ 𝑊 ∈ V)
 
31-Jul-2026acwer1prclem 35851 Lemma for acwer1prc 35852. (Contributed by BTernaryTau, 31-Jul-2026.)
𝑊 = {𝑟 ∣ ∃𝑥 ∈ On (𝑟 ⊆ ((𝑅1‘𝑥) × (𝑅1‘𝑥)) ∧ 𝑟 We (𝑅1‘𝑥))}    ⇒   ((CHOICE ∧ ω ≼ 𝐴 ∧ (card‘(𝑅1‘𝐵)) = 𝐴) → ∃𝑠((card‘𝑠) ∈ (card “ 𝑊) ∧ (card‘𝑠) = 𝐴))
 
31-Jul-2026weexenwe 35848 If a well-ordering of an infinite set exists, then a well-ordering equinumerous to the set exists. (Contributed by BTernaryTau, 31-Jul-2026.)
((∃𝑟 𝑟 We 𝐴 ∧ ω ≼ 𝐴) → ∃𝑠(𝑠 ⊆ (𝐴 × 𝐴) ∧ 𝑠 We 𝐴 ∧ 𝑠 ≈ 𝐴))
 
30-Jul-2026nadd32d 37138 Commutative/associative law that swaps the last two terms in a triple sum. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    ⇒   (𝜑 → ((𝐴 +no 𝐵) +no 𝐶) = ((𝐴 +no 𝐶) +no 𝐵))
 
30-Jul-2026naddassd 37137 Natural addition associates. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    &   (𝜑 → 𝐶 ∈ On)    ⇒   (𝜑 → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))
 
30-Jul-2026naddcomd 37136 Natural addition commutes. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    ⇒   (𝜑 → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
 
30-Jul-2026naddlidd 37135 Identity law for natural addition. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (∅ +no 𝐴) = 𝐴)
 
30-Jul-2026naddridd 37134 Identity law for natural addition. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (𝐴 +no ∅) = 𝐴)
 
30-Jul-2026nmulcomd 37133 Natural multiplication commutes. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    ⇒   (𝜑 → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))
 
30-Jul-2026nmullidd 37132 Identity law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (1o ·no 𝐴) = 𝐴)
 
30-Jul-2026nmulridd 37131 Identity law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (𝐴 ·no 1o) = 𝐴)
 
30-Jul-2026nmull0d 37130 Natural multiplication by zero. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (∅ ·no 𝐴) = ∅)
 
30-Jul-2026nmulr0d 37129 Natural multiplication by zero. Deduction form. (Contributed by Scott Fenton, 30-Jul-2026.)
(𝜑 → 𝐴 ∈ On)    ⇒   (𝜑 → (𝐴 ·no ∅) = ∅)
 
28-Jul-2026tmachfullfin 48166 Folk theorem. For any algorithm deterministically processing a stream of data (essentially, an infinite tape with cell indices 𝐼 and finite alphabet 𝑈), if it terminates on every possible input, then it never looks beyond a finite portion ∪ ran 𝑆 of the input.

Termination is expressed here with a weaker condition: that an execution may only look at a finite number of cells. Obviously, a program which finishes in a finite number of steps can only scan finite set of cells.

This theorem has many corollaries, such as: any encoding scheme able to represent all integers has at least one non-decodable tape (in other terms, encoding of the infinity).

I no longer have the source for this theorem but I believe I first read about it on LessWrong. My gratitude to Grok for suggesting that this theorem will require Axiom of Choice, and to DeepSeek for suggesting the topology-based proof route. (Contributed by Ender Ting, 28-Jul-2026.)

(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ∪ ran 𝑆 ∈ Fin)
 
28-Jul-2026tmachlem-fssscan 48165 Any scan set is finite. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ran 𝑆 ⊆ Fin)
 
28-Jul-2026tmachlem-franscan 48164 There is a finite number of different scan sets. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ran 𝑆 ∈ Fin)
 
28-Jul-2026tmachlem-agreefin 48163 Any agreement set has a finite (singleton) list of possible scans. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑏 ∈ ran 𝐴) → (𝑆 “ 𝑏) ∈ Fin)
 
28-Jul-2026tmachlem-agreesn 48162 Scans for all tapes of a single agreement set are identical. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘𝑎)})
 
28-Jul-2026tmachlem-exagreecover 48161 Particular properties of the finite cover of agreesets. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ∃𝑎(𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎))
 
28-Jul-2026tmachlem-extpcover 48160 Product topology of tapes admits finite cover. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)
 
28-Jul-2026tmachlem-exlargecover 48159 Product topology of tapes has an open cover. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ∪ ran 𝐴 = 𝑇)
 
28-Jul-2026tmachlem-uassst 48158 Union of all agreement sets only includes tapes. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ∪ ran 𝐴 ⊆ 𝑇)
 
28-Jul-2026tmachlem-tpbase 48154 The base set of product topology of tapes is the set of tapes. (Contributed by Ender Ting, 28-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
 
27-Jul-2026tmachlem-tpopen2 48157 Variable-renaming lemma connecting tmachlem-agreeprod 48152 and tmachlem-tpopen 48156. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) ∈ (∏t‘(𝑏 ∈ 𝐼 ↦ 𝒫 𝑈)))
 
27-Jul-2026tmachlem-tpopen 48156 Agreement sets are open in the product topology of tapes. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝑇) → X𝑏 ∈ 𝐼 if(𝑏 ∈ (𝑆‘𝑎), {(𝑎‘𝑏)}, 𝑈) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
 
27-Jul-2026tmachlem-tpitem 48155 Topology lemma. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)‘𝑎) = 𝒫 𝑈)
 
27-Jul-2026tmachlem-tpcomp 48153 Product (discrete) topology of tapes is compact by Tychonoff's theorem. To work for infinite index sets (such as ℤ for 𝐼 which is the main interpretation), it requires Choice. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∈ Comp)
 
27-Jul-2026tmachlem-agreeprod 48152 Agreement set can be written as infinite product of acceptable values for tape's cells. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) = X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))
 
27-Jul-2026tmachlem-agreeself 48151 Any tape belongs to its own agreement set. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))
 
27-Jul-2026tmachlem-finscan 48150 Execution on any tape only scanned a finite number of cells. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ Fin)
 
27-Jul-2026tmachlem-extapes 48149 The class of all tapes is a set. (Contributed by Ender Ting, 27-Jul-2026.)
(𝜑 → 𝑈 ∈ Fin)    &   (𝜑 → 𝐼 ∈ V)    &   (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   (𝜑 → 𝑇 ∈ V)
 
27-Jul-2026nadddi 37151 Natural multiplication distributes over natural addition. (Contributed by Scott Fenton, 27-Jul-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ·no (𝐵 +no 𝐶)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))
 
24-Jul-2026goldratval 48141 Value of the golden ratio. (Contributed by Ender Ting, 24-Jul-2026.)
𝐹 = (2 · (cos‘(π / 5)))    ⇒   𝐹 = ((1 + (√‘5)) / 2)
 
24-Jul-2026goldratmolem4 48140 Lemma 4 for determining the value of golden ratio. (Contributed by Ender Ting, 24-Jul-2026.)
𝐹 = (2 · (cos‘(π / 5)))    ⇒   (((𝐹↑2) − 𝐹) − 1) = 0
 
24-Jul-2026goldpolyfactor 48132 Factorization of a polynomial which has golden ratio among its roots, done by term-by-term-by-term multiplying and summing a few shorter polynomials. (Contributed by Ender Ting, 24-Jul-2026.)
𝐹 ∈ ℂ    ⇒   (((((𝐹↑2) − 𝐹) − 1) · (((𝐹↑2) − 𝐹) − 1)) · (𝐹 + 2)) = ((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2)
 
24-Jul-2026chnrun2 48115 Superadditivity of chain constructor over relation parameter. (Contributed by Ender Ting, 24-Jul-2026.)
((𝑅 Chain 𝐵) ∪ ( < Chain 𝐵)) ⊆ ((𝑅 ∪ < ) Chain 𝐵)
 
24-Jul-2026chnrun 48114 Satisfying either of two chain relations is sufficient to make a chain under their union. (Contributed by Ender Ting, 24-Jul-2026.)
((𝐴 ∈ (𝑅 Chain 𝐵) ∨ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))
 
24-Jul-2026chnrin2 48113 Distribution of chain class constructor over relation intersection. (Contributed by Ender Ting, 24-Jul-2026.)
((𝑅 ∩ < ) Chain 𝐵) = ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵))
 
24-Jul-2026chnrrin 48112 A chain of elements satisfying two relations at once is a chain under either of them. (Contributed by Ender Ting, 24-Jul-2026.)
(𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵) → (𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)))
 
24-Jul-2026chnrin 48111 Satisfying two chain relations makes a chain under their intersection. (Contributed by Ender Ting, 24-Jul-2026.)
((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵))
 
24-Jul-2026chndun2 48110 Superaddivity of chain constructor over alphabet parameter. (Contributed by Ender Ting, 24-Jul-2026.)
((𝑅 Chain 𝐵) ∪ (𝑅 Chain 𝐶)) ⊆ (𝑅 Chain (𝐵 ∪ 𝐶))
 
24-Jul-2026chndun 48109 Chains in either of two alphabets are chains in their union. (Contributed by Ender Ting, 24-Jul-2026.)
((𝐴 ∈ (𝑅 Chain 𝐵) ∨ 𝐴 ∈ (𝑅 Chain 𝐶)) → 𝐴 ∈ (𝑅 Chain (𝐵 ∪ 𝐶)))
 
24-Jul-2026chndin2 48108 Distribution of chain class constructor over alphabet intersection. (Contributed by Ender Ting, 24-Jul-2026.)
(𝑅 Chain (𝐵 ∩ 𝐶)) = ((𝑅 Chain 𝐵) ∩ (𝑅 Chain 𝐶))
 
24-Jul-2026chndrin 48107 A chain whose alphabet is intersection of two classes is also a chain in each of those alphabets. (Contributed by Ender Ting, 24-Jul-2026.)
(𝐴 ∈ (𝑅 Chain (𝐵 ∩ 𝐶)) → (𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ (𝑅 Chain 𝐶)))
 
24-Jul-2026chndin 48106 A chain in two alphabets at once is also a chain in their intersection. (Contributed by Ender Ting, 24-Jul-2026.)
((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ (𝑅 Chain 𝐶)) → 𝐴 ∈ (𝑅 Chain (𝐵 ∩ 𝐶)))
 
24-Jul-2026wrddun2 48105 Superadditivity of word constructor. Class of words over union alphabet includes all words over either alphabet in the union; if 𝐵 and 𝐶 are non-empty and different, then this subclass relation is strict because of the words which have symbols both from 𝐵 and from 𝐶. (Contributed by Ender Ting, 24-Jul-2026.)
(Word 𝐵 ∪ Word 𝐶) ⊆ Word (𝐵 ∪ 𝐶)
 
24-Jul-2026wrddun 48104 Words in either of two alphabets are words in their union. (Contributed by Ender Ting, 24-Jul-2026.)
((𝐴 ∈ Word 𝐵 ∨ 𝐴 ∈ Word 𝐶) → 𝐴 ∈ Word (𝐵 ∪ 𝐶))
 
24-Jul-2026wrddin2 48103 Distribution of word class constructor over class intersection. (Contributed by Ender Ting, 24-Jul-2026.)
Word (𝐵 ∩ 𝐶) = (Word 𝐵 ∩ Word 𝐶)
 
24-Jul-2026wrddrin 48102 A word whose alphabet is intersection of two classes is also a word in each of those alphabets. (Contributed by Ender Ting, 24-Jul-2026.)
(𝐴 ∈ Word (𝐵 ∩ 𝐶) → (𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶))
 
24-Jul-2026wrddin 48101 A word in two alphabets is also a word under their intersection. (Contributed by Ender Ting, 24-Jul-2026.)
((𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶) → 𝐴 ∈ Word (𝐵 ∩ 𝐶))
 
24-Jul-2026ricer 20786 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 24-Jul-2026.)
≃𝑟 Er Ring
 
24-Jul-2026ricref 20778 Ring isomorphism is reflexive. (Contributed by by AV, 24-Jul-2026.)
(𝑅 ∈ Ring → 𝑅 ≃𝑟 𝑅)
 
24-Jul-2026ricrel 20774 The domain of the ring isomorphism relation is a relation. (Contributed by AV, 24-Jul-2026.)
Rel ≃𝑟
 
24-Jul-2026rhmkerinj 20770 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 24-Jul-2026.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &    0 = (0g‘𝑅)    &   𝑍 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵⟶1-1𝐶 ↔ (◡𝐹 “ {𝑍}) = { 0 }))
 
24-Jul-2026rimval 20760 The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 24-Jul-2026.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    ⇒   (𝑅 RingIso 𝑆) = {𝑓 ∈ (𝑅 RingHom 𝑆) ∣ 𝑓:𝐵⟶1-1onto𝐶}
 
24-Jul-2026isrhm0 20736 The predicate "is a ring homomorphism from 𝑅 to 𝑆". (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 24-Jul-2026.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &    1 = (1r‘𝑅)    &   𝑁 = (1r‘𝑆)    &    · = (.r‘𝑅)    &    × = (.r‘𝑆)    &    + = (+g‘𝑅)    &    ⨣ = (+g‘𝑆)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ (𝐹:𝐵⟶𝐶 ∧ (𝐹‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))))))
 
24-Jul-2026rhmval0 20735 The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by AV, 24-Jul-2026.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &    1 = (1r‘𝑅)    &   𝑁 = (1r‘𝑆)    &    · = (.r‘𝑅)    &    × = (.r‘𝑆)    &    + = (+g‘𝑅)    &    ⨣ = (+g‘𝑆)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ((𝑓‘ 1 ) = 𝑁 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) × (𝑓‘𝑦))))})
 
23-Jul-2026goldratmolem3 48139 Lemma 3 for determining the value of golden ratio. (Contributed by Ender Ting, 23-Jul-2026.)
𝐹 = (2 · (cos‘(π / 5)))    ⇒   ((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2) = 0
 
23-Jul-2026isdrng5 21038 A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 23-Jul-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 (𝑦 · 𝑥) = 1 ))
 
23-Jul-2026brric2 20784 The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 23-Jul-2026.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆)))
 
23-Jul-2026ricrcl 20780 Ring isomorphism implies the right side is a ring. (Contributed by AV, 23-Jul-2026.)
(𝑅 ≃𝑟 𝑆 → 𝑆 ∈ Ring)
 
23-Jul-2026riclcl 20779 Ring isomorphism implies the left side is a ring. (Contributed by AV, 23-Jul-2026.)
(𝑅 ≃𝑟 𝑆 → 𝑅 ∈ Ring)
 
23-Jul-2026rhm0 20753 A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.) (Revised by AV, 23-Jul-2026.)
0 = (0g‘𝑅)    &   𝑍 = (0g‘𝑆)    ⇒   (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 0 ) = 𝑍)
 
22-Jul-2026sqrtnpoly 48148 Square root function is not polynomial with complex coefficients either. Otherwise, its composition with a square monomial - the identity - would have to be of even degree. (Contributed by Ender Ting, 22-Jul-2026.)
¬ √ ∈ (Poly‘ℂ)
 
22-Jul-2026sqrtrrnpoly 48147 Real square root is not a polynomial with real coefficients, because its value is imaginary for negative arguments. (Contributed by Ender Ting, 22-Jul-2026.)
¬ √ ∈ (Poly‘ℝ)
 
22-Jul-2026sqrtqaa 48120 Square root of a rational number is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)
(𝐴 ∈ ℚ → (√‘𝐴) ∈ 𝔸)
 
22-Jul-2026sqrtnzqaa 48119 Square root of a nonzero rational is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ 𝔸)
 
22-Jul-2026zrdrng 21056 A zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Revised by AV, 22-Jul-2026.)
0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   ( 0 = 1 → ¬ 𝑅 ∈ DivRing)
 
22-Jul-2026isdrng3 21037 A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 22-Jul-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ (𝐵 ∖ { 0 })(𝑦 · 𝑥) = 1 ))
 
22-Jul-2026isdrng3lem2 21036 Lemma for isdrng3 21037 (for the right to left implication). Formerly part of proof for isdrng3 21037. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 22-Jul-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ (𝐵 ∖ { 0 })(𝑦 · 𝑥) = 1 ) → ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp)
 
22-Jul-2026isdrng3lem1 21035 Lemma for isdrng3 21037 (for the left to right implication). Formerly part of proof for isdrng3 21037. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 22-Jul-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp) → ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ (𝐵 ∖ { 0 })(𝑦 · 𝑥) = 1 )
 
22-Jul-2026isdrng3lem0 21034 Lemma for isdrng3 21037: The base set of a multipication group restricted to a subset of the original base set. (Contributed by AV, 22-Jul-2026.)
𝐵 = (Base‘𝑅)    ⇒   (Base‘((mulGrp‘𝑅) ↾s (𝐵 ∖ 𝑋))) = (𝐵 ∖ 𝑋)
 
22-Jul-2026hta 9989 A ZFC emulation of Hilbert's transfinite axiom. The set 𝐵 has the properties of Hilbert's epsilon, except that it also depends on a well-ordering 𝑅. This theorem arose from discussions with Raph Levien on 5-Mar-2004 about translating the HOL proof language, which uses Hilbert's epsilon. See https://us.metamath.org/downloads/choice.txt (copy of obsolete link http://ghilbert.org/choice.txt) and https://us.metamath.org/downloads/megillaward2005he.pdf.

Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires 𝑅 We 𝐴 as an antecedent. Class 𝐴 collects the sets of the least rank for which 𝜑(𝑥) is true. Class 𝐵, which emulates Hilbert's epsilon, is the minimum element in a well-ordering 𝑅 on 𝐴.

If a well-ordering 𝑅 on 𝐴 can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace 𝑅 with a dummy setvar variable, say 𝑤, and attach 𝑤 We 𝐴 as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, 𝐵 (which will have 𝑤 as a free variable) will no longer be present, and we can eliminate 𝑤 We 𝐴 by applying exlimiv 1965 and weth 10600, using scottex 9960 to establish the existence of 𝐴.

For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 9988. (Contributed by NM, 11-Mar-2004.) (Revised by Mario Carneiro, 25-Jun-2015.) Use the Scott operation. (Revised by BTernaryTau, 22-Jul-2026.)

𝐴 = Scott {𝑥 ∣ 𝜑}    &   𝐵 = (℩𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑧𝑅𝑦)    ⇒   (𝑅 We 𝐴 → (𝜑 → [𝐵 / 𝑥]𝜑))
 
22-Jul-2026scott0bs 9971 Theorem scheme version of scott0b 9964. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 22-Jul-2026.)
(∃𝑥𝜑 ↔ Scott {𝑥 ∣ 𝜑} ≠ ∅)
 
21-Jul-2026alseu-no-surprise 51214 Demonstrate that there is never a "surprise" when using the "all some one" quantifier, that is, it is never possible for the consequent to be both always true and always false. This follows from als-no-surprise 51182 by alseuals 51200. For a contrast, see alimp-surprise 51156. (Contributed by David A. Wheeler, 21-Jul-2026.)
¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))
 
21-Jul-2026alseueu 51213 "The 𝜑 is 𝜓 " implies that exactly one thing is both 𝜑 and 𝜓. This is the half of dfalseu2 51212 that drops the universal conjunct; it does not reverse, so ∃!𝑥(𝜑 ∧ 𝜓) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 21-Jul-2026.)
(∀∃!𝑥(𝜑 → 𝜓) → ∃!𝑥(𝜑 ∧ 𝜓))
 
21-Jul-2026dfalseu2 51212 An "all some one" statement is equivalent to its universal part conjoined with the claim that exactly one 𝑥 satisfies both 𝜑 and 𝜓. In other words, given ∀𝑥(𝜑 → 𝜓), requiring exactly one 𝑥 to satisfy 𝜑, which is what df-alseu 51197 requires, and requiring exactly one 𝑥 to satisfy (𝜑 ∧ 𝜓) come to the same thing. Read 𝜑 as "is a king" and 𝜓 as "is hungry": if every king is hungry, then "there is exactly one king" and "there is exactly one hungry king" say the same thing, so either of them, together with "every king is hungry", gives "the king is hungry".

The universal conjunct is what makes that work, and it cannot be dropped. ∃!𝑥(𝜑 ∧ 𝜓) on its own is strictly weaker than ∀∃!𝑥(𝜑 → 𝜓), since it is satisfied when many things are 𝜑 and just one of those is 𝜓, as in a region with five kings exactly one of whom is hungry; see alseueu 51213 for the one direction that does hold without it. Uniqueness attaches to the antecedent, not to the conjunction. Russell's analysis of a definite description is built the same way: its uniqueness clause constrains the description predicate alone, while the predication is a separate conjunct. See his worked example of "the father of Charles II was executed", [Russell1905] p. 482. (Contributed by David A. Wheeler, 21-Jul-2026.)

(∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥(𝜑 ∧ 𝜓)))
 
21-Jul-2026nfralseu 51211 Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 51180. (Contributed by David A. Wheeler, 21-Jul-2026.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃!𝑦 ∈ 𝐴(𝜑 → 𝜓)
 
21-Jul-2026nfalseu 51210 Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 51179. Unlike nfals 51179 it requires 𝑥 and 𝑦 to be disjoint, because the corresponding builder for ∃! is nfeuw 2621, which requires it; the version without that requirement, nfeu 2622, depends on ax-13 2404 and its use is discouraged. (Contributed by David A. Wheeler, 21-Jul-2026.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃!𝑦(𝜑 → 𝜓)
 
21-Jul-2026ralseubii 51209 Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 51177. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))
 
21-Jul-2026alseubii 51208 Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 51176. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))
 
21-Jul-2026ralseu2d 51207 Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
 
21-Jul-2026ralseu1d 51206 Deduction rule: Given "all some one" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
 
21-Jul-2026alseu2d 51205 Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   (𝜑 → ∃!𝑥𝜓)
 
21-Jul-2026alseu1d 51204 Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 → ∀∃!𝑥(𝜓 → 𝜒))    ⇒   (𝜑 → ∀𝑥(𝜓 → 𝜒))
 
21-Jul-2026ralseud 51203 Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 51206 and ralseu2d 51207 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∀∃!𝑥 ∈ 𝐴(𝜓 → 𝜒))
 
21-Jul-2026alseud 51202 Introduction rule: "all some one" holds if the "for all" part holds and the antecedent has exactly one witness. This is the converse of alseu1d 51204 and alseu2d 51205 taken together. (Contributed by David A. Wheeler, 21-Jul-2026.)
(𝜑 → ∀𝑥(𝜓 → 𝜒))    &   (𝜑 → ∃!𝑥𝜓)    ⇒   (𝜑 → ∀∃!𝑥(𝜓 → 𝜒))
 
21-Jul-2026ralseurals 51201 "All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 51200. (Contributed by David A. Wheeler, 21-Jul-2026.)
(∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))
 
21-Jul-2026alseuals 51200 "All some one" implies "all some": requiring exactly one witness is stronger than requiring at least one. Any consequence of an allsome statement is therefore a consequence of the corresponding "all some one" statement, which is how alseu-no-surprise 51214 is proved. (Contributed by David A. Wheeler, 21-Jul-2026.)
(∀∃!𝑥(𝜑 → 𝜓) → ∀∃𝑥(𝜑 → 𝜓))
 
21-Jul-2026dfralseu2 51199 The bounded "all some one" form is the general form with the class membership folded into the antecedent. This is the "all some one" counterpart of dfrals2 51166. (Contributed by David A. Wheeler, 21-Jul-2026.)
(∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
 
21-Jul-2026df-ralseu 51198 Define "all some one" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and exactly one 𝑥 in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 21-Jul-2026.)
(∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
 
21-Jul-2026df-alseu 51197 Define "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 21-Jul-2026.)
(∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
 
21-Jul-2026wralseu 51196 Extend wff definition to include "all some one" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and exactly one 𝑥 in 𝐴 satisfies 𝜑. (Contributed by David A. Wheeler, 21-Jul-2026.)
wff ∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓)
 
21-Jul-2026walseu 51195 Extend wff definition to include "all some one" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and exactly one 𝑥 satisfies 𝜑. (Contributed by David A. Wheeler, 21-Jul-2026.)
wff ∀∃!𝑥(𝜑 → 𝜓)
 
21-Jul-2026sqrtnnaa 48118 Square root of a natural number is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
(𝐴 ∈ ℕ → (√‘𝐴) ∈ 𝔸)
 
21-Jul-2026naddle 37146 Condition for bounding natural addition above. (Contributed by Scott Fenton, 21-Jul-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))
 
21-Jul-2026ltnadd 37145 Condition for bounding a natural sum below. (Contributed by Scott Fenton, 21-Jul-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 +no 𝐶) ↔ (∃𝑏 ∈ 𝐵 𝐴 ⊆ (𝑏 +no 𝐶) ∨ ∃𝑐 ∈ 𝐶 𝐴 ⊆ (𝐵 +no 𝑐))))
 
21-Jul-2026nmullid 37125 Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
(𝐴 ∈ On → (1o ·no 𝐴) = 𝐴)
 
21-Jul-2026nmulrid 37124 Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
(𝐴 ∈ On → (𝐴 ·no 1o) = 𝐴)
 
21-Jul-2026aaliou3r 26709 The sum presented above is convergent, which means that the "Liouville number" is indeed real. (Contributed by Ender Ting, 21-Jul-2026.)
Σ𝑘 ∈ ℕ (2↑ -(!‘𝑘)) ∈ ℝ
 
21-Jul-20261aa 26680 One is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
1 ∈ 𝔸
 
21-Jul-20260aa 26679 Zero is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
0 ∈ 𝔸
 
20-Jul-2026ralsanmo 51187 An "all some" statement restricted to a class, conjoined with the claim that at most one 𝑥 in 𝐴 satisfies its antecedent, is equivalent to the universal part conjoined with the claim that exactly one 𝑥 in 𝐴 satisfies the antecedent. This is the restricted counterpart of alsanmo 51186. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
((∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
 
20-Jul-2026alsanmo 51186 An "all some" statement conjoined with the claim that at most one 𝑥 satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one 𝑥 satisfies the antecedent. The "all some" quantifier supplies the existence of such an 𝑥 and ∃*𝑥𝜑 supplies the at-most-one part, so together they yield ∃!𝑥𝜑. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
 
20-Jul-2026rexrals 51185 If a member of 𝐴 satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 51191. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
(∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))
 
20-Jul-2026ralrals 51184 If the universal part of a restricted "all some" statement holds, then the statement reduces to the existence of a member of 𝐴 satisfying its antecedent. This is the restricted counterpart of ralals 51190. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
(∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜑))
 
20-Jul-2026tgaltai 29475 Parallelism implies alternate angles congruence: if a line (𝑋𝐿𝑍) crosses two parallel lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊), then the alternate angles are congruent. First direction of Theorem 12.21 of [Schwabhauser] p. 126. This is also Proposition 1.29 of Euclid's Elements . (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   (𝜑 → 𝑌𝑂𝑊)    &   (𝜑 → 𝑋 ≠ 𝑍)    ⇒   (𝜑 → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)
 
20-Jul-2026quadcgrprlng 29474 Nontrivial quadrilaterals with congruent and parallel opposite sides are parallelograms. Theorem 12.20 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   (𝜑 → 𝑌𝑂𝑊)    ⇒   (𝜑 → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
 
20-Jul-2026prlngsymquadopp 29473 In parallelograms, opposing vertices are on opposite sides of the diagonal. Second part of Theorem 12.19 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐼 = (Itv‘𝐺)    ⇒   (𝜑 → 𝑊𝑂𝑌)
 
20-Jul-2026prlngsymquad 29472 All parallelograms are symmetric quadrilaterals. First part of Theorem 12.19 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))    ⇒   (𝜑 → ((𝑋 − 𝑌) = (𝑍 − 𝑊) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))
 
20-Jul-2026prlngsymquadlem 29471 Lemma for prlngsymquad 29472. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))    &   (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋))    &   𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌)    ⇒   (𝜑 → 𝑇 = 𝑊)
 
20-Jul-2026symquadprlng 29470 Symmetrical quadrilaterals are parallelograms. Theorem 12.18 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → 𝑌 ≠ 𝑊)    &   (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))    &   (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))    ⇒   (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))
 
20-Jul-2026prlngeq 29465 Playfair's axiom, written as an equality: if two different lines are parallel to a given line at a given point, they are equal. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ∥ 𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐶)    ⇒   (𝜑 → 𝐵 = 𝐶)
 
20-Jul-2026df-angmgm 29404 Definition of the angle addition magma. See following theorems for better readable properties. Because the textbook's definition of angle congruence does not consider orientation (cf. cgraswap 29357) , our notion of angle does not include reflex angles (angles larger than a straight angle). Therefore, like with ℕ0, at this point, we can only build a magma , which is e.g. not isomorphic with the rotation group SO(2) . (Contributed by Thierry Arnoux, 20-Jul-2026.)
AngMgm = (𝑔 ∈ V ↦ ⦋(Base‘𝑔) / 𝑝⦌⦋{𝑑 ∈ (𝑝 ↑m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} / 𝑎⦌({⟨(Base‘ndx), 𝑎⟩, ⟨(+g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑧 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑧”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑧) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑧 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑧”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑧) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑧(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤∠‘𝑔)⟩} /s (cgrA‘𝑔)))
 
20-Jul-2026symquadmid 29334 In a symmetrical quadrilateral, the midpoints of the diagonals coincide. Corollary of Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑀 = (midG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → 𝑌 ≠ 𝑊)    &   (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))    &   (𝜑 → 𝑌𝑂𝑊)    ⇒   (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))
 
20-Jul-2026hlopp 29280 If two points 𝑋 and 𝑌 lie on opposite sides of a line 𝐴, then given a point 𝑍 on 𝐴, any point 𝑊 on the line (𝑋𝐿𝑍) opposite to 𝑌 lies on the half line (𝑍𝑋) (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑋𝑂𝑌)    &   (𝜑 → 𝑍 ∈ 𝐴)    &   (𝜑 → 𝑊𝑂𝑌)    &   (𝜑 → 𝑊 ∈ (𝑋𝐿𝑍))    ⇒   (𝜑 → 𝑊(𝐾‘𝑍)𝑋)
 
20-Jul-2026symquadprlnglem 29195 Lemma for symquadprlnglem 29195. (Contributed by Thierry Arnoux, 20-Jul-2026.)
𝑃 = (Base‘𝐺)    &    − = (dist‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊))    &   (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋))    &   (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))    &   (𝜑 → 𝑌 ≠ 𝑊)    &   (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍))    &   (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊))    ⇒   (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))
 
20-Jul-2026crng4 20518 Commutative/associative law for commutative rings. See also mul4d 11549. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 20-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑈 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 · 𝑌) · (𝑍 · 𝑈)) = ((𝑋 · 𝑍) · (𝑌 · 𝑈)))
 
19-Jul-2026crngrhmfo 20756 The image of a surjective homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.) (Revised by AV, 19-Jul-2026.)
𝐵 = (Base‘𝑆)    ⇒   ((𝑅 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:dom 𝐹⟶onto𝐵) → 𝑆 ∈ CRing)
 
19-Jul-2026karden 9986 If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 10662). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9984 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥 ∣ 𝑥 ≈ 𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
𝐴 ∈ V    &   𝐶 = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}    &   𝐷 = Scott {𝑥 ∣ 𝑥 ≈ 𝐵}    ⇒   (𝐶 = 𝐷 ↔ 𝐴 ≈ 𝐵)
 
19-Jul-2026kardex 9984 The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
Scott {𝑥 ∣ 𝑥 ≈ 𝐴} ∈ V
 
19-Jul-2026cplem2 9979 Lemma for the Collection Principle cp 9981. (Contributed by NM, 17-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
𝐴 ∈ V    ⇒   ∃𝑦∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝑦) ≠ ∅)
 
19-Jul-2026cplem1 9977 Lemma for the Collection Principle cp 9981. (Contributed by NM, 17-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
𝐶 = ∪ 𝑥 ∈ 𝐴 Scott 𝐵    ⇒   ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐶) ≠ ∅)
 
19-Jul-2026scott0b 9964 Applying Scott's trick yields the empty set iff it was applied to the empty set. (Contributed by NM, 15-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 19-Jul-2026.)
(𝐴 = ∅ ↔ Scott 𝐴 = ∅)
 
19-Jul-2026disjdifg 4428 A class does not intersect a relative complement of a superclass. (Contributed by NM, 24-Mar-1998.) Generalize from disjdif 4429. (Revised by BJ, 19-Jul-2026.)
(𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 ∖ 𝐵)) = ∅)
 
19-Jul-2026sseq0b 4356 The only subclass of the empty class is itself. (Contributed by NM, 7-Mar-2007.) Strengthen sseq0 4357 to a biconditional. (Revised by BJ, 19-Jul-2026.)
(𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 = ∅))
 
18-Jul-2026scottex 9960 Scott's trick produces a set. (Contributed by NM, 13-Oct-2003.) Use the Scott operation. (Revised by BTernaryTau, 18-Jul-2026.)
Scott 𝐴 ∈ V
 
18-Jul-2026sepab 5298 Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 5303. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)
(𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)
 
18-Jul-2026ssex 5286 A subclass of a set is a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 5252 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) (Proof shortened by BJ, 18-Jul-2026.)
𝐵 ∈ V    ⇒   (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
 
18-Jul-2026ssexg 5285 A subclass of a set is a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) (Proof shortened by BJ, 18-Jul-2026.)
((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)
 
18-Jul-2026inssdif0 4325 Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) (Proof shortened by BJ, 18-Jul-2026.)
((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)
 
16-Jul-2026nmulle 37144 A condition for bounding a natural product above. Converse of ltnmul 37143. (Contributed by Scott Fenton, 16-Jul-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·no 𝐵) ⊆ 𝐶 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝐶 +no (𝑎 ·no 𝑏))))
 
15-Jul-20262alsraln0id 51194 Nested general "all some" quantifiers with class membership as their antecedents, for the same class 𝐴: 𝜑 holds for every 𝑥 and every 𝑦 in 𝐴, and 𝐴 is not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))
 
15-Jul-20262alsraln0 51193 Nested general "all some" quantifiers with class membership as their antecedents: 𝜑 holds for every 𝑥 in 𝐴 and every 𝑦 in 𝐵, and both 𝐴 and 𝐵 are not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)))
 
15-Jul-2026n0als 51192 If 𝐴 is not empty, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(𝐴 ≠ ∅ → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
 
15-Jul-2026rexals 51191 If some 𝑥 in 𝐴 satisfies 𝜑, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. See rexrals 51185 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))
 
15-Jul-2026ralals 51190 If 𝜑 holds for every 𝑥 in 𝐴, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that some 𝑥 in 𝐴 satisfies 𝜑. See ralrals 51184 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∀𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑))
 
15-Jul-2026alsraln0 51189 The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and 𝐴 is not empty. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))
 
15-Jul-2026alsralrex 51188 The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and some 𝑥 in 𝐴 satisfies 𝜑. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
(∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))
 
15-Jul-2026ltnmul 37143 Characterize less-than a natural product. (Contributed by Scott Fenton, 15-Jul-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 ·no 𝐶) ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 (𝐴 +no (𝑏 ·no 𝑐)) ⊆ ((𝑏 ·no 𝐶) +no (𝐵 ·no 𝑐))))
 
15-Jul-2026nmulel1 37142 Natural multiplication by a non-zero number preserves less-than. (Contributed by Scott Fenton, 15-Jul-2026.)
(((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ≠ ∅)) → (𝐶 ·no 𝐴) ∈ (𝐶 ·no 𝐵))
 
15-Jul-2026nmulss1 37141 Natural multiplication preserves less-than or equal. (Contributed by Scott Fenton, 15-Jul-2026.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ·no 𝐴) ⊆ (𝐶 ·no 𝐵))
 
15-Jul-2026nmuladdss 37140 Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ⊆ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
 
15-Jul-2026nmuladdel 37139 Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·no 𝐵) +no (𝐴 ·no 𝐷)) ∈ ((𝐴 ·no 𝐵) +no (𝐶 ·no 𝐷)))
 
15-Jul-2026infinfnum 35849 Equivalence between two infiniteness criteria for numerable sets. (Contributed by BTernaryTau, 15-Jul-2026.)
(𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))
 
14-Jul-2026bj-inex1gALT 38015 Proof of inex1g 5283 from sepg 5254 to then allow proving inex1 5281 from it. That does not reduce the combined proof size of inex1 5281 and inex1g 5283. (Contributed by BJ, 14-Jul-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)
 
14-Jul-2026soinfdom 35800 A strict order relation on an infinite set dominates that set. (Contributed by BTernaryTau, 14-Jul-2026.)
((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉 ∧ ω ≼ 𝐴) → 𝐴 ≼ 𝑅)
 
14-Jul-2026jaeqifi 35782 Inference combining four equality antecedents into one equality of conditional operators antecedent. (Contributed by BTernaryTau, 14-Jul-2026.)
(𝐴 = 𝐶 → 𝜑)    &   (𝐴 = 𝐷 → 𝜑)    &   (𝐵 = 𝐶 → 𝜑)    &   (𝐵 = 𝐷 → 𝜑)    ⇒   (if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷) → 𝜑)
 
14-Jul-2026uniex2 7778 The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) (Proof shortened by BJ, 14-Jul-2026.)
∃𝑦 𝑦 = ∪ 𝑥
 
14-Jul-2026sepgi 5255 Inference associated with sepg 5254. The requirement that 𝑦 not occur in 𝜑 is necessary, as notsep 5329 shows. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)
𝐴 ∈ V    ⇒   ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
 
13-Jul-2026idomcanr 49650 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 13-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑍 ≠ 0 ) → ((𝑋 · 𝑍) = (𝑌 · 𝑍) → 𝑋 = 𝑌))
 
13-Jul-2026idomcanl 49649 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 13-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 ≠ 0 ) → ((𝑋 · 𝑌) = (𝑋 · 𝑍) → 𝑌 = 𝑍))
 
13-Jul-2026prlngmid2 29469 If the midpoints of two segments (𝑋𝐼𝑍) and (𝑌𝐼𝑊) coincide, the points 𝑋, 𝑌, 𝑍 and 𝑊 form a parallelogram, i.e. the lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊) are parallel. Theorem 12.17 of [Schwabhauser] p. 125. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   𝑀 = (midG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))
 
13-Jul-2026prlnginn0 29468 A line 𝐶 intersecting another line 𝐴 also intersects any line 𝐵 parallel to 𝐴. Theorem 12.16 of [Schwabhauser] p. 125. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝐶 ∈ ran 𝐿)    &   (𝜑 → (𝐴 ∩ 𝐶) ≠ ∅)    &   (𝜑 → 𝐴 ≠ 𝐶)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝐵 ⊆ 𝐻)    &   (𝜑 → 𝐶 ⊆ 𝐻)    ⇒   (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)
 
13-Jul-2026prlngplngtr 29467 Transitivity of parallelism, for lines in the same plane 𝐻. This is case 1 of Theorem 12.15 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   (𝜑 → 𝐶 ⊆ 𝐻)    &   (𝜑 → 𝐵 ∥ 𝐶)    ⇒   (𝜑 → 𝐴 ∥ 𝐶)
 
13-Jul-2026prlngpln4 29466 Building a parallel line conserves planes, i.e. given a line 𝐴 and a point 𝑋 not on 𝐴, the (unique) parallel 𝐵 to 𝐴 through 𝑋 lies completely within the plane defined by 𝐴 and 𝑋. Theorem 12.14 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐻)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝐵 ⊆ 𝐻)
 
13-Jul-2026prlngmo2 29464 Playfair's axiom, without the restriction that the point 𝑋 is outside of the line 𝐴. Theorem 12.11 of [Schwabhauser] p. 123. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
 
13-Jul-2026dfprlng3 29456 Alternate definition of (strict) parallelism. Theorem 12.7 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌))    ⇒   (𝜑 → (𝐴 ∥ (𝑋𝐿𝑌) ↔ (𝑋((hpG‘𝐺)‘𝐴)𝑌 ∧ (𝐴 ∩ (𝑋𝐿𝑌)) = ∅)))
 
13-Jul-2026dfprlng2 29455 Alternate definition of (strict) parallelism. Theorem 12.7 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋}))    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ (𝑃 ∖ {𝑍}))    &   (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊))    ⇒   (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ↔ (𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∧ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)))
 
13-Jul-2026ragsupplcgra 29375 An angle ⟨“𝑋𝑌𝑍”⟩ is a right angle exactly when it is congruent to its supplementary angle ⟨“𝑋𝑌𝑊”⟩. Theorem 11.18 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ {𝑌}))    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ (𝑃 ∖ {𝑌}))    &   (𝜑 → 𝑊 ∈ (𝑃 ∖ {𝑌}))    &   (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))    ⇒   (𝜑 → (⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑊”⟩))
 
13-Jul-2026cgrarag 29374 Any angle ⟨“𝐴𝐵𝐶”⟩ congruent with a right angle ⟨“𝑋𝑌𝑍”⟩ is a right angle. Theorem 11.17 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)    ⇒   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
 
13-Jul-2026mirlni 29197 The mirror of a point 𝑋 on a line (𝑌𝐿𝑍) is on the mirrored line ((𝑀‘𝑌)𝐿(𝑀‘𝑍)). (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆‘𝐴)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑍 ≠ 𝑌)    &   (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍))    ⇒   (𝜑 → (𝑀‘𝑋) ∈ ((𝑀‘𝑌)𝐿(𝑀‘𝑍)))
 
13-Jul-2026mirleqb 29196 Equality theorem for point mirroring. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆‘𝐴)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    ⇒   (𝜑 → (𝑋 = 𝑌 ↔ (𝑀‘𝑋) = (𝑀‘𝑌)))
 
13-Jul-2026hlgrcl2 29097 Reverse closure for rays. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐴(𝐾‘𝐶)𝐵)    ⇒   (𝜑 → 𝐵 ∈ 𝑃)
 
13-Jul-2026hlgrcl1 29096 Reverse closure for rays. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → 𝐴(𝐾‘𝐶)𝐵)    ⇒   (𝜑 → 𝐴 ∈ 𝑃)
 
13-Jul-2026ishlg2 29095 Alternate version of ishlg 29098, including closure. (Contributed by Thierry Arnoux, 13-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐾 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑃)    ⇒   (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))
 
13-Jul-2026eldifsnbd 4749 Membership in a set with an element removed implies non-equality with that element. (Contributed by Thierry Arnoux, 13-Jul-2026.)
(𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶}))    ⇒   (𝜑 → 𝐴 ≠ 𝐶)
 
13-Jul-2026nrmod 3843 Deduce the negation of a restricted "at most one" quantifier. (Contributed by Thierry Arnoux, 13-Jul-2026.)
(𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑌 → (𝜓 ↔ 𝜃))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝜒)    &   (𝜑 → 𝜃)    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → ¬ ∃*𝑥 ∈ 𝐴 𝜓)
 
13-Jul-20263orim123da 1474 Disjoin antecedents and consequents of three premises. (Contributed by Thierry Arnoux, 13-Jul-2026.)
(𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜂))    &   ((𝜑 ∧ 𝜓) → 𝜒)    &   ((𝜑 ∧ 𝜃) → 𝜏)    &   ((𝜑 ∧ 𝜂) → 𝜁)    ⇒   (𝜑 → (𝜒 ∨ 𝜏 ∨ 𝜁))
 
12-Jul-2026rals-no-surprise 51183 Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of 𝐴 that satisfy the antecedent. This is the restricted counterpart of als-no-surprise 51182, and follows from it by dfrals2 51166. Note that this needs no assumption that 𝐴 is nonempty, because allsome requires a member of 𝐴 satisfying 𝜑, and that member would have to satisfy both 𝜓 and ¬ 𝜓. The ordinary restricted "for all" requires no such member and can be vacuously true, as shown in empty-surprise2 51159; that is the point of allsome. (Contributed by David A. Wheeler, 12-Jul-2026.)
¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))
 
12-Jul-2026cbvals 51181 Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    ⇒   (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))
 
12-Jul-2026nfrals 51180 Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)
 
12-Jul-2026nfals 51179 Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝜑    &   Ⅎ𝑥𝜓    ⇒   Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)
 
12-Jul-2026alsbid 51178 Deduction form of alsbii 51176. (Contributed by David A. Wheeler, 12-Jul-2026.)
Ⅎ𝑥𝜑    &   (𝜑 → (𝜓 ↔ 𝜃))    &   (𝜑 → (𝜒 ↔ 𝜏))    ⇒   (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))
 
12-Jul-2026ralsbii 51177 Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))
 
12-Jul-2026alsbii 51176 Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 ↔ 𝜒)    &   (𝜓 ↔ 𝜃)    ⇒   (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑥(𝜒 → 𝜃))
 
12-Jul-2026ralsex 51175 The consequent of an "all some" restricted to a class is witnessed: some member of 𝐴 satisfying 𝜑 also satisfies 𝜓. Restricted counterpart of alsex 51174. (Contributed by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
 
12-Jul-2026alsex 51174 The consequent of an "all some" is witnessed: if 𝜓 holds of every 𝑥 satisfying 𝜑, and some 𝑥 satisfies 𝜑, then some 𝑥 satisfies 𝜓. This is the positive counterpart of als-no-surprise 51182, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, as alimp-surprise 51156 shows. It is the reason the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)
 
12-Jul-2026ralsn0d 51173 Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → 𝐴 ≠ ∅)
 
12-Jul-2026rals2d 51172 Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
 
12-Jul-2026rals1d 51171 Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))    ⇒   (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
 
12-Jul-2026ralsd 51168 Introduction rule for "all some" restricted to a class. This is the converse of rals1d 51171 and rals2d 51172 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))    &   (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))
 
12-Jul-2026alsd 51167 Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 51169 and als2d 51170 taken together, and is what lets an "all some" statement be proved rather than merely taken apart. (Contributed by David A. Wheeler, 12-Jul-2026.)
(𝜑 → ∀𝑥(𝜓 → 𝜒))    &   (𝜑 → ∃𝑥𝜓)    ⇒   (𝜑 → ∀∃𝑥(𝜓 → 𝜒))
 
12-Jul-2026dfrals2 51166 The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
 
12-Jul-2026df-rals 51165 Define "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true.

An older definition of the "all some" quantifier when scoped to a class, named df-alsc and now removed, instead applied a bare formula 𝜑 to the members of a class, asserting only (∀𝑥 ∈ 𝐴𝜑 ∧ ∃𝑥𝑥 ∈ 𝐴), that is, that the formula held throughout 𝐴 and that 𝐴 had at least one member. I've now decided that that was a mistake. Its older existence conjunct ∃𝑥𝑥 ∈ 𝐴 did not require any member of 𝐴 to satisfy the antecedent, so if the formula was itself an implication, that inner implication could still be vacuously true, which is precisely what the allsome quantifier exists to prevent. For example, the older definition meant that "among Martians, all tall ones are green" could be considered true if there are Martians, but no tall Martians. This version of the definition instead ensures that claims of the form "among Martians, all tall ones are green" can only be true if all tall Martians are green and that there is at least one tall Martian. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)

(∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
 
12-Jul-2026wrals 51163 Extend wff definition to include "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓)
 
12-Jul-2026wals 51162 Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥(𝜑 → 𝜓)
 
12-Jul-2026idomnzd 49648 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 12-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) = 0 )) → (𝑋 = 0 ∨ 𝑌 = 0 ))
 
12-Jul-2026isidom3 49647 The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 12-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 0 ≠ 1 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 ))))
 
12-Jul-2026bj-vn0ALT 38178 Alternate proof of vn0 4294 which does not use eqabbw 2837 (and is shorter than vn0 4294 when eqabbw 2837 is inlined). (Contributed by BJ, 12-Jul-2026.) Using the same dummy variable for 𝑦 and 𝑧 slightly reduces the proof size. (Proof modification is discouraged.) (New usage is discouraged.)
V ≠ ∅
 
12-Jul-2026cmprmidlmcl 21661 The complement of a prime ideal is multiplicatively closed. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ (𝐵 ∖ 𝑃))) → (𝐼 · 𝐽) ∈ (𝐵 ∖ 𝑃))
 
12-Jul-2026prmidlc2 21660 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → 𝐽 ∈ 𝑃)
 
12-Jul-2026rspsn0 21556 A principal ideal (an ideal generated by one element) in a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 12-Jul-2026.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝐾 = (RSpan‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐾‘{𝑋}) = {𝑖 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐵 𝑖 = (𝑥 · 𝑋)})
 
12-Jul-2026nvpss 4366 No class strictly includes the universal class. Dual of npss0 4364. (Contributed by BJ, 12-Jul-2026.)
¬ V ⊊ 𝐴
 
12-Jul-2026vvin 4362 Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 4360. (Contributed by BJ, 12-Jul-2026.)
((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)
 
12-Jul-2026vn0 4294 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2149, df-clel 2839. (Revised by GG, 6-Sep-2024.) (Proof shortened by BJ, 12-Jul-2026.)
V ≠ ∅
 
10-Jul-202625or6to4 43448 Question 67 of 68 from a lecture Prof. Loof Lirpa held last Saturday in Lincoln Park. When asked why the smaller root wasn't reduced to 3/2, Lirpa responded "It really doesn't matter anyhow." (Contributed by Luke Murphy, 10-Jul-2026.)
(𝜑 → 𝐴 = 1)    &   (𝜑 → 𝐵 = -(53 / 2))    &   (𝜑 → 𝐶 = (75 / 2))    &   (𝜑 → 𝑋 ∈ ℂ)    ⇒   (𝜑 → (((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ↔ (𝑋 = 25 ∨ 𝑋 = (6 / 4))))
 
10-Jul-2026quadfac 43447 The solution of a quadratic equation via factoring. (Contributed by Luke Murphy, 10-Jul-2026.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝑀 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℂ)    &   (𝜑 → (𝑀 + 𝑁) = -(𝐵 / 𝐴))    &   (𝜑 → (𝑀 · 𝑁) = (𝐶 / 𝐴))    ⇒   (𝜑 → (((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ↔ (𝑋 = 𝑀 ∨ 𝑋 = 𝑁)))
 
10-Jul-2026nelscottrankgt 35833 If a member of the input class is not a member of the Scott's trick set, then its rank is greater than the rank of a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ∈ (rank‘𝐶))
 
10-Jul-2026scottrankeqel 35832 If a member of the input class has the same rank as a member of the Scott's trick set, then it is also a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)
 
10-Jul-2026elscott2 35828 Membership in a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
(𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ (rank‘𝐴) = ∩ (rank “ 𝐵)))
 
10-Jul-2026dfscott3 35827 Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
Scott 𝐴 = (𝐴 ∩ (𝑅1‘suc ∩ (rank “ 𝐴)))
 
10-Jul-2026imadifssranOLD 6215 Obsolete version of imadifssrn 6214 as of 29-Sep-2026. (Contributed by AV, 4-Oct-2025.) Remove antecedent. (Revised by Eric Schmidt, 10-Jul-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))
 
9-Jul-2026elscottrankeq 35830 Elements in a Scott's trick set have the same rank. (Contributed by BTernaryTau, 9-Jul-2026.)
((𝐴 ∈ Scott 𝐶 ∧ 𝐵 ∈ Scott 𝐶) → (rank‘𝐴) = (rank‘𝐵))
 
8-Jul-2026rankscott 35838 The rank of a nonempty Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
(𝐴 ≠ ∅ → (rank‘Scott 𝐴) = suc ∩ (rank “ 𝐴))
 
8-Jul-2026elscottrank 35829 The rank of an element in a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
(𝐴 ∈ Scott 𝐵 → (rank‘𝐴) = ∩ (rank “ 𝐵))
 
8-Jul-2026dfscott2 35826 Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}
 
7-Jul-2026hashomiso 46210 The ♯ function yields an order isomorphism between ω and ℕ0. (Contributed by Eric Schmidt, 7-Jul-2026.)
(♯ ↾ ω) Isom E , < (ω, ℕ0)
 
7-Jul-2026hashomf1o 46209 The ♯ function yields a bijection from ω to ℕ0. (Contributed by Eric Schmidt, 7-Jul-2026.)
(♯ ↾ ω):ω⟶1-1ontoℕ0
 
7-Jul-2026hashnnltb 46208 The ♯ function on ω preserves the ordering. (Contributed by Eric Schmidt, 7-Jul-2026.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (♯‘𝐴) < (♯‘𝐵)))
 
7-Jul-2026hashnnlt 46207 The ♯ function on ω preserves the ordering. (Contributed by Eric Schmidt, 7-Jul-2026.)
((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (♯‘𝐵) < (♯‘𝐴))
 
7-Jul-2026hashnnm 46206 The ♯ function on ω preserves multiplication. (Contributed by Eric Schmidt, 7-Jul-2026.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (♯‘(𝐴 ·o 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))
 
7-Jul-2026hashnnsuc 46205 The ♯ function on ω turns successor into adding 1. (Contributed by Eric Schmidt, 7-Jul-2026.)
(𝐴 ∈ ω → (♯‘suc 𝐴) = ((♯‘𝐴) + 1))
 
7-Jul-2026hashnna 46204 The ♯ function on ω preserves addition. (Contributed by Eric Schmidt, 7-Jul-2026.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (♯‘(𝐴 +o 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
 
7-Jul-2026kardexen 35931 One set is equinumerous to another iff an element in its kard cardinality is equinumerous to an element in the second set's kard cardinality. See kardeng 35925 for a version with equality of cardinals. (Contributed by BTernaryTau, 7-Jul-2026.)
(𝐴 ≈ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≈ 𝑦)
 
6-Jul-2026kardsdom 35930 One set strictly dominates another iff an element in its kard cardinality strictly dominates an element in the second set's kard cardinality. (Contributed by BTernaryTau, 6-Jul-2026.)
(𝐴 ≺ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≺ 𝑦)
 
5-Jul-2026prlngeu 29463 Given a line 𝐴 and a point 𝑋 not on 𝐴, a unique line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.13 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   (𝜑 → ∃!𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
 
5-Jul-2026prlngmo 29462 Playfair's axiom. Given a line 𝐴 and a point 𝑋 not on 𝐴, at most one line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.11 of [Schwabhauser] p. 123. Note that this is the first instance of a theorem where the geometry is required to be Euclidean, as expressed by 𝐺 ∈ TarskiGE. Theorem A10 of [Schwabhauser] p. 24 is used, in the form of axtgeucl 28964, in the proof of prlngmolem1 29460. See prlngex 29459 for the corresponding existence theorem. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝐺 ∈ TarskiGE)    ⇒   (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
 
5-Jul-2026prlngmolem2 29461 Lemma for prlngmo 29462. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝐺 ∈ TarskiGE)    &   𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑟 ∈ 𝑏 𝑟 ∈ (𝑥(Itv‘𝐺)𝑦))}    &   𝑄 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))}    ⇒   (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
 
5-Jul-2026prlngmolem1 29460 Lemma for prlngmo 29462: Contradiction: Assuming two different parallels 𝐵 and 𝐶 having a common point 𝑋 exist to a line 𝐴, the geometry cannot be Euclidean (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐵) ∧ 𝑏 ∈ (𝑃 ∖ 𝐵)) ∧ ∃𝑦 ∈ 𝐵 𝑦 ∈ (𝑎𝐼𝑏))}    &   𝑄 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑤 ∈ 𝐴 𝑤 ∈ (𝑎𝐼𝑏))}    &   𝐼 = (Itv‘𝐺)    &   (𝜑 → 𝐵 ∈ ran 𝐿)    &   (𝜑 → 𝐶 ∈ ran 𝐿)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ∥ 𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐶)    &   (𝜑 → 𝑇 ∈ 𝐴)    &   (𝜑 → 𝑊 ∈ (𝐶 ∖ 𝐵))    &   (𝜑 → 𝐵 ≠ 𝐶)    &   (𝜑 → 𝑊𝑂𝑇)    ⇒   (𝜑 → ¬ 𝐺 ∈ TarskiGE)
 
5-Jul-2026prlngex 29459 There exists at least one parallel line 𝑏 to a given line 𝐴 through a given point 𝑋. Theorem 12.10 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝑃)    ⇒   (𝜑 → ∃𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))
 
5-Jul-2026perpprlng 29458 If two lines 𝐴 and 𝐵 have a common perpendicular 𝐶 and lie in the same plane 𝐻, then they are parallel. Theorem 12.9 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝐵 ⊆ 𝐻)    &   (𝜑 → 𝐶 ⊆ 𝐻)    &   (𝜑 → 𝐴(⟂G‘𝐺)𝐶)    &   (𝜑 → 𝐵(⟂G‘𝐺)𝐶)    ⇒   (𝜑 → 𝐴 ∥ 𝐵)
 
5-Jul-2026prlngpln3 29457 Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝐵 ⊆ (𝐴𝐸𝑋))
 
5-Jul-2026prlnghpg 29454 If two lines 𝐴 and 𝐵 are parallel, then any two points 𝑋 and 𝑌 of 𝐵 lie on the same half-plane limited by 𝐴. Theorem 12.6 of [Schwabhauser] p. 122. . (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑋((hpG‘𝐺)‘𝐴)𝑌)
 
5-Jul-2026prlngpln 29453 Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ≠ 𝐵)    ⇒   (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))
 
5-Jul-2026prlngin0 29452 Two parallel lines do not intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∥ 𝐵)    &   (𝜑 → 𝐴 ≠ 𝐵)    ⇒   (𝜑 → (𝐴 ∩ 𝐵) = ∅)
 
5-Jul-2026prlngrcl2 29451 Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∥ 𝐵)    ⇒   (𝜑 → 𝐵 ∈ ran 𝐿)
 
5-Jul-2026prlngrcl1 29450 Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝐿 = (LineG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∥ 𝐵)    ⇒   (𝜑 → 𝐴 ∈ ran 𝐿)
 
5-Jul-2026perpeq 29378 Uniqueness of the perpendicular to a line 𝐴 within a plane 𝐻 at a point 𝑋. Theorem 11.20 of [Schwabhauser] p. 99. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝑌 ∈ 𝐻)    &   (𝜑 → 𝑍 ∈ 𝐻)    &   (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴)    &   (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴)    ⇒   (𝜑 → (𝑋𝐿𝑌) = (𝑋𝐿𝑍))
 
5-Jul-2026perpeqlem 29377 Lemma for perpeq 29378. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝑌 ∈ 𝐻)    &   (𝜑 → 𝑍 ∈ 𝐻)    &   (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴)    &   (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴)    &   (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)𝑍)    ⇒   (𝜑 → (𝑋𝐿𝑌) = (𝑋𝐿𝑍))
 
5-Jul-2026ragraghl 29376 Drawing two right angles at a point 𝑋 on the same side of a line (𝑋𝐿𝑌) leads to points 𝑊 and 𝑍 on the same ray from 𝑋. Theorem 11.19 of [Schwabhauser] p. 99. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → ⟨“𝑌𝑋𝑍”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝑌𝑋𝑊”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → 𝑍((hpG‘𝐺)‘(𝑌𝐿𝑋))𝑊)    ⇒   (𝜑 → 𝑍((hlG‘𝐺)‘𝑋)𝑊)
 
5-Jul-2026ragcgra 29373 Right angles are congruent with each other. Theorem 11.16 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ 𝑃)    &   (𝜑 → 𝐴 ∈ 𝑃)    &   (𝜑 → 𝐵 ∈ 𝑃)    &   (𝜑 → 𝐶 ∈ 𝑃)    &   (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝐵 ≠ 𝐶)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝑌 ≠ 𝑍)    ⇒   (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
 
5-Jul-2026nhpmirhp 29306 If a point 𝑍 is on the plane defined by a line 𝐴 and a point 𝑌, but not on the same half-plane as 𝑌, then its mirror point (𝑀‘𝑍) by a point 𝑋 on 𝐴 is on the same half-plane as 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   𝑀 = (𝑆‘𝑋)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝑍 ∈ ((𝐴𝐸𝑌) ∖ 𝐴))    &   (𝜑 → ¬ 𝑌((hpG‘𝐺)‘𝐴)𝑍)    ⇒   (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)(𝑀‘𝑍))
 
5-Jul-2026hpgssplng 29304 Any point 𝑋 on a half plane defined by a line 𝐴 and another point 𝑌 is on the plane defined by 𝐴 and 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋((hpG‘𝐺)‘𝐴)𝑌)    ⇒   (𝜑 → 𝑋 ∈ (𝐴𝐸𝑌))
 
5-Jul-2026mirplncl 29303 The mirror of a point with regard to another point is in the same plane as the two points. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆‘𝑋)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝑋 ∈ 𝐻)    &   (𝜑 → 𝑌 ∈ 𝐻)    ⇒   (𝜑 → (𝑀‘𝑌) ∈ 𝐻)
 
5-Jul-2026plngmiropp 29302 Given a line 𝐴 and a point 𝑋 not on 𝐴, then a point 𝑌 on the plane defined by 𝐴 and 𝑋 is either opposite to 𝑋, or opposite to the mirror point of 𝑋 by any point 𝑍 of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆‘𝑍)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝑌 ∈ ((𝐴𝐸𝑋) ∖ 𝐴))    &   (𝜑 → 𝑍 ∈ 𝐴)    ⇒   (𝜑 → (𝑋𝑂𝑌 ∨ (𝑀‘𝑋)𝑂𝑌))
 
5-Jul-2026oppmir 29262 The mirror point with regard to a point 𝑋 on a line 𝐴 lies on the other side of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝑆 = (pInvG‘𝐺)    &   𝑀 = (𝑆‘𝑋)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴))    &   𝐿 = (LineG‘𝐺)    ⇒   (𝜑 → 𝑌𝑂(𝑀‘𝑌))
 
5-Jul-2026perpin 29230 If two lines 𝐴 and 𝐵 are perpendicular, then they intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)
(𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴(⟂G‘𝐺)𝐵)    ⇒   (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)
 
5-Jul-2026ecase33d 1505 Deduction for elimination by cases. (Contributed by Thierry Arnoux, 5-Jul-2026.)
(𝜑 → ¬ 𝜓)    &   (𝜑 → ¬ 𝜒)    &   (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃))    ⇒   (𝜑 → 𝜃)
 
5-Jul-20263anasss 1381 Associative law for conjunction applied to antecedent (eliminates syllogism). Converse of 3anassrs 1382. (Contributed by Thierry Arnoux, 5-Jul-2026.)
((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)
 
4-Jul-20261enumkard 35940 The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets.

The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10273 for context on disjoint union as a representation of cardinal addition.

This theorem is not limited to numerable sets, but it also does not depend on AC. See 1enumen 35807 for a version that uses equinumerosity , 1enumcard 35808 for a version that uses the card function, and 1enum 36061 for a version that uses an explicit sum of complex number 1s.

(Contributed by BTernaryTau, 4-Jul-2026.)

(𝐴 ∈ V → (kard‘𝐴) = (kard‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)))
 
4-Jul-2026rankkardu 35939 An upper bound on the rank of a kard cardinal. (Contributed by BTernaryTau, 4-Jul-2026.)
(rank‘(kard‘𝐴)) ⊆ suc (rank‘𝐴)
 
4-Jul-2026karddom 35929 One set dominates another iff an element in its kard cardinality dominates an element in the second set's kard cardinality. (Contributed by BTernaryTau, 4-Jul-2026.)
(𝐴 ≼ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≼ 𝑦)
 
4-Jul-2026kardsn 35928 A singleton has cardinality one. (Contributed by BTernaryTau, 4-Jul-2026.)
(𝐴 ∈ 𝑉 → (kard‘{𝐴}) = (kard‘1o))
 
4-Jul-2026kardenir 35926 If two sets are equinumerous, then their kard cardinal numbers are equal. (Contributed by BTernaryTau, 4-Jul-2026.)
(𝐴 ≈ 𝐵 → (kard‘𝐴) = (kard‘𝐵))
 
4-Jul-2026rankscottu 35839 An upper bound on the rank of a Scott's trick set. (Contributed by BTernaryTau, 4-Jul-2026.)
(𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))
 
3-Jul-2026kardfi 35938 The kard cardinal number of a finite set is finite. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ Fin → (kard‘𝐴) ∈ Fin)
 
3-Jul-2026kardnnfi 35937 The kard cardinal number of a finite ordinal is finite. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ ω → (kard‘𝐴) ∈ Fin)
 
3-Jul-2026kardcard 35936 Two sets have equal kard cardinalities iff they have equal card cardinalities. This theorem depends on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵)))
 
3-Jul-2026ackardcard 35935 The Axiom of Choice implies that two sets have equal kard cardinalities iff they have equal card cardinalities. (Contributed by BTernaryTau, 3-Jul-2026.)
(CHOICE → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵))))
 
3-Jul-2026kardcard2 35934 Two numerable sets have equal kard cardinalities iff they have equal card cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵)))
 
3-Jul-2026kardcard2b 35933 If two sets have equal kard cardinalities, then they have equal card cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
((kard‘𝐴) = (kard‘𝐵) → (card‘𝐴) = (card‘𝐵))
 
3-Jul-2026kardcard2a 35932 If two sets have equal nonzero card cardinalities, then they have equal kard cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
(((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (kard‘𝐴) = (kard‘𝐵))
 
3-Jul-2026kard0b 35927 The empty set is the only set with cardinality zero. This is the kard version of cardeq0 10663. (Contributed by BTernaryTau, 3-Jul-2026.)
((kard‘𝐴) = (kard‘∅) ↔ 𝐴 = ∅)
 
3-Jul-2026kardeng 35925 Two sets are equinumerous iff their kard cardinal numbers are equal. Unlike carden 10662, this theorem does not depend on the Axiom of Choice, but it does depend on the Axiom of Regularity and the Axiom of Infinity. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ 𝑉 → ((kard‘𝐴) = (kard‘𝐵) ↔ 𝐴 ≈ 𝐵))
 
3-Jul-2026kardeq0 35924 Applying kard to a class yields the empty set iff the class is a proper class. (Contributed by BTernaryTau, 3-Jul-2026.)
((kard‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ V)
 
3-Jul-2026elkarden 35923 Any member of the kard cardinal number of a set is equinumerous to the set. Contrast with cardne 10073 for card cardinals. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ (kard‘𝐵) → 𝐴 ≈ 𝐵)
 
3-Jul-2026kard0 35922 The kard cardinality of the empty set is the singleton of the empty set. (Contributed by BTernaryTau, 3-Jul-2026.)
(kard‘∅) = {∅}
 
3-Jul-2026kardval2 35921 The value of the kard function. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. See also kardval 35920. (Contributed by BTernaryTau, 3-Jul-2026.)
(kard‘𝐴) = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
 
3-Jul-2026kardval 35920 The value of the kard function. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. See also kardval2 35921. (Contributed by BTernaryTau, 3-Jul-2026.)
(kard‘𝐴) = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}
 
3-Jul-2026kardfn 35919 The kard class is a function on the universe. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
kard Fn V
 
3-Jul-2026acnum 35847 The Axiom of Choice implies that any set is numerable. (Contributed by BTernaryTau, 3-Jul-2026.)
(CHOICE → (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card))
 
3-Jul-2026scottssr1 35840 Relationship between a Scott's trick set and the cumulative hierarchy. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ 𝐵 → Scott 𝐵 ⊆ (𝑅1‘suc (rank‘𝐴)))
 
3-Jul-2026scott0bOLD 35835 Obsolete version of scott0b 9964 as of 18-Jul-2026. (Contributed by BTernaryTau, 3-Jul-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = ∅ ↔ Scott 𝐴 = ∅)
 
3-Jul-2026scottsn 35834 Applying Scott's trick to a singleton leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)
Scott {𝐴} = {𝐴}
 
3-Jul-2026elscottrankss 35831 Relationship between the ranks of an element in a Scott's trick set and an element in the input class. (Contributed by BTernaryTau, 3-Jul-2026.)
((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))
 
3-Jul-2026elscott 35825 Membership in a Scott's trick set. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (rank‘𝐴) ⊆ (rank‘𝑥)))
 
3-Jul-2026scotteqi 35824 Equality theorem for the Scott operation. Inference form of scotteq 9958. (Contributed by BTernaryTau, 3-Jul-2026.)
𝐴 = 𝐵    ⇒   Scott 𝐴 = Scott 𝐵
 
3-Jul-2026onrankid 35811 The rank of an ordinal number is itself. (Contributed by BTernaryTau, 3-Jul-2026.)
(𝐴 ∈ On ↔ (rank‘𝐴) = 𝐴)
 
3-Jul-2026scott0 9963 Applying Scott's trick to the empty set leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)
Scott ∅ = ∅
 
2-Jul-2026df-kard 35918 Define the alternative cardinal number function. Under this definition, the cardinal number of a set is the set of all sets equinumerous to it and having the least possible rank. Definition of [Enderton] p. 222. See kardval 35920 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35925. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 10047. (Contributed by BTernaryTau, 2-Jul-2026.)
kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})
 
2-Jul-2026rncardr1prc 35850 The Axiom of Choice implies that the cardinalities of the layers of the cumulative hierarchy form a proper class. (Contributed by BTernaryTau, 2-Jul-2026.)
(CHOICE → ¬ ran (card ∘ 𝑅1) ∈ V)
 
30-Jun-2026isfieldidl2 21571 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 30-Jun-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    ⇒   (𝑅 ∈ Field ↔ (𝑅 ∈ CRing ∧ 𝐵 ≠ { 0 } ∧ 𝐼 = {{ 0 }, 𝐵}))
 
30-Jun-2026isfieldidl 21570 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 30-Jun-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝑅 ∈ Field ↔ (𝑅 ∈ CRing ∧ 0 ≠ 1 ∧ 𝐼 = {{ 0 }, 𝐵}))
 
30-Jun-2026rspprop 21554 Properties of a class to be the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 30-Jun-2026.)
𝐵 = (Base‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((𝐾‘𝑆) = 𝑋 ↔ (𝑋 ∈ 𝐼 ∧ 𝑆 ⊆ 𝑋 ∧ ∀𝑖 ∈ 𝐼 (𝑆 ⊆ 𝑖 → 𝑋 ⊆ 𝑖))))
 
30-Jun-2026rspvalint 21553 The ideal generated by a subset of a ring as intersection of ideals including the subset. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 30-Jun-2026.)
𝐵 = (Base‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝐾‘𝑆) = ∩ {𝑡 ∈ 𝐼 ∣ 𝑆 ⊆ 𝑡})
 
28-Jun-2026inlidl 34041 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 28-Jun-2026.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (LIdeal‘𝑅))
 
28-Jun-2026unichnlidl 21546 The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by AV, 28-Jun-2026.)
((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (LIdeal‘𝑅))
 
28-Jun-2026lidlunin0 21545 The union of a nonempty subset of ideals in a ring is nonempty. (Contributed by AV, 28-Jun-2026.)
((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∪ 𝐶 ≠ ∅)
 
27-Jun-2026isidom2 49646 The predicate "is an integral domain": An integral domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ PrmRing ∧ 𝑅 ∈ CRing))
 
27-Jun-2026dfidom2 49645 Alternate definition of the class of integral domains. An integral domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
IDomn = (PrmRing ∩ CRing)
 
27-Jun-2026crngprmringdom 49644 A commutative ring is a prime ring if and only if it is a domain. (Contributed by AV, 27-Jun-2026.)
(𝑅 ∈ CRing → (𝑅 ∈ PrmRing ↔ 𝑅 ∈ Domn))
 
27-Jun-2026crngprmringidom 49643 A commutative ring is a prime ring if and only if it is an integral domain. (Contributed by AV, 27-Jun-2026.)
(𝑅 ∈ CRing → (𝑅 ∈ PrmRing ↔ 𝑅 ∈ IDomn))
 
27-Jun-2026df2idl2crng 21607 The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ CRing → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))
 
27-Jun-2026lidlbasel 21521 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 27-Jun-2026.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)    ⇒   ((𝑈 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝐵)
 
27-Jun-20260ring01eqbi2 20813 In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 27-Jun-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐵 = { 0 } ↔ 1 = 0 ))
 
26-Jun-2026prmrngring 49640 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.) (Proof shortened by AV, 26-Jun-2026.)
(𝑅 ∈ PrmRing → 𝑅 ∈ Ring)
 
26-Jun-2026prmringnzring 49639 A prime ring is a nonzero ring. (Contributed by AV, 26-Jun-2026.)
(𝑅 ∈ PrmRing → 𝑅 ∈ NzRing)
 
26-Jun-20261enum 36061 The Fundamental Theorem of Enumeration. According to Doron Zeilberger (in https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), this theorem was independently discovered by several anonymous cave-dwellers.

Zeilberger also states that "While this formula is still useful after all these years, enumerating specific finite sets is no longer considered mathematics. A genuine mathematical fact has to incorporate infinitely many facts". Fortunately, theorems in Metamath are actually theorem schemes that correspond to an infinite number of object-language theorems, so this concern does not apply to us.

See 1enumen 35807 for a version that uses equinumerosity , 1enumcard 35808 for a version that uses the card function, and 1enumkard 35940 for a version that uses the kard function.

(Contributed by BTernaryTau, 26-Jun-2026.)

(𝐴 ∈ Fin → (♯‘𝐴) = Σ𝑎 ∈ 𝐴 1)
 
26-Jun-20261enumcard 35808 The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets.

The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10273 for context on disjoint union as a representation of cardinal addition.

This theorem does not depend on AC, but it is only meaningful for numerable sets. See 1enumen 35807 and 1enumkard 35940 for versions that are meaningful for non-numerable sets, and see 1enum 36061 for a version that uses an explicit sum of complex number 1s.

(Contributed by BTernaryTau, 26-Jun-2026.)

(𝐴 ∈ V → (card‘𝐴) = (card‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o)))
 
26-Jun-20261enumen 35807 The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets.

The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10273 for context on disjoint union as a representation of cardinal addition.

This theorem is not limited to numerable sets, but it also does not depend on AC. See 1enumcard 35808 for a version that uses the card function, 1enumkard 35940 for a version that uses the kard function , and 1enum 36061 for a version that uses an explicit sum of complex number 1s.

(Contributed by BTernaryTau, 26-Jun-2026.)

(𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1o))
 
26-Jun-20264anpull2 1383 An equivalence of two four-terms conjunctions with the terms regrouped (here, the second sub-conjunct of the first term is pulled separately). (Contributed by Zhi Wang, 4-Sep-2024.) (Proof shortened by Garrett Katz, 26-Jun-2026.)
(((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
 
25-Jun-2026ifpdfbi 1086 Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.) (Proof shortened by Garrett Katz, 25-Jun-2026.)
((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
 
24-Jun-2026inv2 35781 The intersection of the universal class with a class is itself. A commuted form of inv1 4351. (Contributed by BTernaryTau, 24-Jun-2026.)
(V ∩ 𝐴) = 𝐴
 
23-Jun-2026rankfn 35818 The rank function is a function on the universe. (Contributed by BTernaryTau, 23-Jun-2026.)
rank Fn V
 
23-Jun-2026rankfo 35817 The rank function maps the universe onto the ordinals. (Contributed by BTernaryTau, 23-Jun-2026.)
rank:V⟶ontoOn
 
23-Jun-2026fnfvintima 35786 Condition for a function value to equal the intersection of an image that contains it. (Contributed by BTernaryTau, 23-Jun-2026.)
((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))
 
22-Jun-2026vonf1onprcf1ac 36044 If 𝐹 maps the universe one-to-one into the ordinals and 𝐴 is a proper class, then 𝐼 maps the ordinals one-to-one into 𝐴 and the Axiom of Choice holds. This is the ZFC version of (6 → 7) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the first hypothesis would be something like 𝜑 → ∀𝑋∃𝐹𝐹:𝑋⟶1-1On, but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 22-Jun-2026.)
(𝜑 → 𝐹:V⟶1-1On)    &   (𝜑 → ¬ 𝐴 ∈ V)    &   𝐼 = (◡(𝐹 ↾ 𝐴) ∘ 𝐻)    &   𝐻 = OrdIso( E , (𝐹 “ 𝐴))    ⇒   (𝜑 → (𝐼:On⟶1-1𝐴 ∧ CHOICE))
 
21-Jun-2026f1preimaex 35785 If the image under a one-to-one function exists, then the corresponding preimage also exists. (Contributed by BTernaryTau, 21-Jun-2026.)
((𝐹:𝐴⟶1-1𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 “ 𝐶) ∈ 𝑉) → 𝐶 ∈ V)
 
21-Jun-2026sgnrn 15280 The range of the signum function. (Contributed by AV, 16-Jun-2026.) (Proof shortened by TA, 21-Jun-2026.)
ran sgn = { -1, 0, 1}
 
20-Jun-2026reldmun 6035 Split a relation into two parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.) Remove requirement that 𝐴 and 𝐵 are disjoint. (Revised by Eric Schmidt, 20-Jun-2026.)
((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵)) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)))
 
19-Jun-2026ringen1zr0 21065 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 20436, and semirings, see srgen1zr0 20472. (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    &   𝑍 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
19-Jun-2026srg1zr 20471 The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
18-Jun-2026drngprmrng 49642 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 18-Jun-2026.)
(𝑅 ∈ DivRing → 𝑅 ∈ PrmRing)
 
18-Jun-2026smprngprmrng 49641 A simple ring (a nonzero ring whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    ⇒   ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → 𝑅 ∈ PrmRing)
 
18-Jun-2026isprmrng 49638 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)
0 = (0g‘𝑅)    &   𝑃 = (PrmIdeal‘𝑅)    ⇒   (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))
 
18-Jun-2026df-prmring 49637 Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)
PrmRing = {𝑟 ∈ Ring ∣ {(0g‘𝑟)} ∈ (PrmIdeal‘𝑟)}
 
18-Jun-2026prlngsym 29449 Parallelism is symmetric. Theorem 12.5 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∥ 𝐵)    ⇒   (𝜑 → 𝐵 ∥ 𝐴)
 
18-Jun-2026prlngref 29448 Parallelism is reflexive. Theorem 12.4 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    ⇒   (𝜑 → 𝐴 ∥ 𝐴)
 
18-Jun-2026prlngd 29447 Deduce parallelism between two lines 𝐴 and 𝐵. (Contributed by Thierry Arnoux, 18-Jun-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐵 ∈ ran 𝐿)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝐴 ⊆ 𝐻)    &   (𝜑 → 𝐵 ⊆ 𝐻)    &   (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   (𝜑 → 𝐴 ∥ 𝐵)
 
18-Jun-2026brprlng 29446 Property of two lines 𝐴 and 𝐵 to be parallel. (Contributed by Thierry Arnoux, 18-Jun-2026.)
𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &    ∥ = (parlnG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))
 
18-Jun-2026rngen1zr0 20436 The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨ 0 , 0 ⟩, 0 ⟩} ∧ ∗ = {⟨⟨ 0 , 0 ⟩, 0 ⟩})))
 
18-Jun-2026rngen1zr 20435 The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
18-Jun-2026rng1zr 20434 The only ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
18-Jun-2026rng1zrlem 20433 Lemma for rng1zr 20434 and srg1zr 20471. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    ∗ = (.r‘𝑅)    ⇒   (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
17-Jun-2026df-prlng 29445 Define the parallel relation for lines. Definition 12.2 of [Schwabhauser] p. 121. Note that the textbook first defines a "strict" parallelism where equal lines are not considered parallel in the strict sense: here we jump directly to the more common definition which allows equality. (Contributed by Thierry Arnoux, 17-Jun-2026.)
parlnG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})
 
17-Jun-2026plng3p 29305 If 𝐻 is a plane containing a line 𝐴 and a point 𝑅 not on 𝐴, then 𝐻 is the plane defined by 𝐴 and 𝑅. Theorem 9.26 of [Schwabhauser] p. 76. See tglinethru 29134 for the 2-point line equivalent. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝐻 ∖ 𝐴))    &   (𝜑 → 𝐴 ⊆ 𝐻)    ⇒   (𝜑 → 𝐻 = (𝐴𝐸𝑅))
 
17-Jun-2026lnssplng1 29301 A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. First part of Theorem 9.25 of [Schwabhauser] p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝑋 ∈ 𝐻)    &   (𝜑 → 𝑌 ∈ 𝐻)    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → (𝑋𝐿𝑌) ⊆ 𝐻)
 
17-Jun-2026lnssplng 29300 A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. Theorem 9.25 of [Schwabhauser] p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝑋 ∈ 𝐻)    &   (𝜑 → 𝑌 ∈ 𝐻)    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))
 
17-Jun-2026lnssplnglem 29299 Lemma for lnssplng 29300. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ (𝐴𝐸𝑅))    &   (𝜑 → 𝑌 ∈ (𝐴𝐸𝑅))    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌))    &   (𝜑 → ¬ 𝑌 ∈ 𝐴)    ⇒   (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝐴𝐸𝑅) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝐴𝐸𝑅) = ((𝑋𝐿𝑌)𝐸𝑠)))
 
17-Jun-2026plngrot 29298 The plane defined by a line (𝑋𝐿𝑌) and a point 𝑍 is also defined by the line (𝑍𝐿𝑌) and the point 𝑋. See first part of Theorem 9.24 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) = ((𝑍𝐿𝑌)𝐸𝑋))
 
17-Jun-2026plngrotlem3 29297 Lemma for plngrot 29298. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   (𝜑 → 𝑋 ≠ 𝑌)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}    ⇒   (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))
 
17-Jun-2026plngrotlem2 29296 Lemma for plngrot 29298. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   (𝜑 → 𝑋 ≠ 𝑌)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))    &   (𝜑 → 𝑌 ≠ 𝑊)    ⇒   (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))
 
17-Jun-2026plngrotlem1 29295 Lemma for plngrot 29298. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌)))    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌)))    &   (𝜑 → 𝑋 ≠ 𝑌)    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝑊 ∈ 𝑃)    &   (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊))    &   (𝜑 → 𝑌 ≠ 𝑊)    &   (𝜑 → 𝑆 ∈ ((𝑋𝐿𝑌)𝐸𝑍))    &   (𝜑 → (𝑆 ∈ (𝑋𝐿𝑌) ∨ 𝑆((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍))    ⇒   (𝜑 → 𝑆 ∈ ((𝑍𝐿𝑌)𝐸𝑋))
 
17-Jun-2026plngcp 29294 The plane defined by a line 𝐴 and a point 𝑅 can also be defined using a different point 𝑅 on the same plane: changes the point used to define the plane. Theorem 9.21 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝑆 ∈ ((𝐴𝐸𝑅) ∖ 𝐴))    ⇒   (𝜑 → (𝐴𝐸𝑅) = (𝐴𝐸𝑆))
 
17-Jun-2026plngcplem 29293 Lemma for plngcp 29294. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝑆 ∈ ((𝐴𝐸𝑅) ∖ 𝐴))    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    ⇒   (𝜑 → (𝐴𝐸𝑅) = (𝐴𝐸𝑆))
 
17-Jun-2026lnincplng 29292 If two lines 𝐴 and 𝐵 intersect, then 𝐵 is in a plane defined by 𝐴 and any point of 𝐵. Lemma 9.22 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐵 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝑃)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   (𝜑 → (𝐴 ∩ 𝐵) = {𝑌})    ⇒   (𝜑 → 𝐵 ⊆ (𝐴𝐸𝑋))
 
17-Jun-2026elplnglnid 29291 The line 𝐴 itself is a subset of a plane defined by the line 𝐴 and a point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))
 
17-Jun-2026elplngid 29290 The point 𝑅 is itself an element of a plane defined by a line 𝐴 and the point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   (𝜑 → 𝑅 ∈ (𝐴𝐸𝑅))
 
17-Jun-2026plngssp 29289 Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   (𝜑 → 𝑋 ∈ (𝐴𝐸𝑅))    ⇒   (𝜑 → 𝑋 ∈ 𝑃)
 
17-Jun-2026elplng 29288 Elementhood in the plane defined by a line 𝐴 and a point 𝑅. Definition 9.20 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    &   𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))}    &   (𝜑 → 𝑋 ∈ 𝑃)    ⇒   (𝜑 → (𝑋 ∈ (𝐴𝐸𝑅) ↔ (𝑋 ∈ 𝐴 ∨ 𝑋((hpG‘𝐺)‘𝐴)𝑅 ∨ 𝑋𝑂𝑅)))
 
17-Jun-2026plngrnssp 29287 Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    &   (𝜑 → 𝑋 ∈ 𝐻)    ⇒   (𝜑 → 𝑋 ∈ 𝑃)
 
17-Jun-2026isplng 29286 The property of being a plane. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐻 ∈ ran 𝐸)    ⇒   (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))
 
17-Jun-2026plngval 29285 The plane defined by a line 𝐴 and a point 𝑅 outside of 𝐴. This is defined as the union of 3 parts: the line itself, the open half-plane containing 𝑅, and the points opposite to 𝑅 (see islnopp 29245). (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   (𝜑 → (𝐴𝐸𝑅) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))})
 
17-Jun-2026tgelrnpln 29284 The property of being a plane, generated by a line and a point. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴))    ⇒   (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)
 
17-Jun-2026tgplnfn 29283 The plane generating function as a function. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   𝐸 = (hlG‘𝐺)    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → 𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ◡ E ))
 
17-Jun-2026df-plng 29282 Define the function building a plane from a line and a point not on that line. Definition 9.20 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
hlG = (𝑔 ∈ V ↦ (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}))
 
17-Jun-2026tglnpt4 29153 Find a second point on a line, outside of a second line. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐵 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝐴 ≠ 𝐵)    ⇒   (𝜑 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)
 
17-Jun-2026tglnpt3 29152 Find a third point on a line. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋 ≠ 𝑌)    ⇒   (𝜑 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑌))
 
17-Jun-2026tglineinsn 29142 If two distinct lines intersect, it is at a single point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝑃 = (Base‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐵 ∈ ran 𝐿)    &   (𝜑 → 𝐴 ≠ 𝐵)    &   (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵))    ⇒   (𝜑 → (𝐴 ∩ 𝐵) = {𝑋})
 
17-Jun-2026tglinesseq 29138 If a line is a subset of another line, they are equal. (Contributed by Thierry Arnoux, 17-Jun-2026.)
𝐿 = (LineG‘𝐺)    &   (𝜑 → 𝐺 ∈ TarskiG)    &   (𝜑 → 𝐴 ∈ ran 𝐿)    &   (𝜑 → 𝐵 ∈ ran 𝐿)    &   (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝐴 = 𝐵)
 
17-Jun-2026mpoexd 8118 Existence of an operation class abstraction. (Contributed by Thierry Arnoux, 17-Jun-2026.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)
 
17-Jun-2026xpdifcnvepel 6172 The set of couples in a Cartesian product, where the second is not an element of the first. (Contributed by Thierry Arnoux, 17-Jun-2026.)
∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐵 ∖ 𝑥)) = ((𝐴 × 𝐵) ∖ ◡ E )
 
16-Jun-2026sgnfo 15281 The signum function as onto function. (Contributed by AV, 16-Jun-2026.)
sgn:ℝ*⟶onto{ -1, 0, 1}
 
16-Jun-2026sgndm 15278 The domain of the signum function. (Contributed by AV, 16-Jun-2026.)
dom sgn = ℝ*
 
16-Jun-2026resdmdfsn 6033 Restricting a class to its domain without a set is the same as restricting the class to the universe without this set. (Contributed by AV, 2-Dec-2018.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)
(𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
 
16-Jun-2026resindm 6031 When restricting a class, intersecting with the domain of the class has no effect. (Contributed by FL, 6-Oct-2008.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)
(𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)
 
16-Jun-20263jaao 1461 Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Garrett Katz, 16-Jun-2026.)
(𝜑 → (𝜓 → 𝜒))    &   (𝜃 → (𝜏 → 𝜒))    &   (𝜂 → (𝜁 → 𝜒))    ⇒   ((𝜑 ∧ 𝜃 ∧ 𝜂) → ((𝜓 ∨ 𝜏 ∨ 𝜁) → 𝜒))
 
16-Jun-20263jaoi 1455 Disjunction of three antecedents (inference). (Contributed by NM, 12-Sep-1995.) (Proof shortened by Garrett Katz, 16-Jun-2026.)
(𝜑 → 𝜓)    &   (𝜒 → 𝜓)    &   (𝜃 → 𝜓)    ⇒   ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)
 
15-Jun-2026anbiim 653 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)
(𝜑 → (𝜒 → 𝜃))    &   (𝜓 → (𝜃 → 𝜒))    ⇒   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
 
15-Jun-2026imbibi 396 The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)
(((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))
 
13-Jun-2026ad5ant135 1395 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)
 
13-Jun-2026ad5ant134 1393 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)
 
13-Jun-2026ad5ant125 1391 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃)
 
13-Jun-2026ad5ant124 1389 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)    ⇒   (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜒) ∧ 𝜂) → 𝜃)
 
12-Jun-2026nmulcld 37120 Closure law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 12-Jun-2026.)
(𝜑 → 𝐴 ∈ On)    &   (𝜑 → 𝐵 ∈ On)    ⇒   (𝜑 → (𝐴 ·no 𝐵) ∈ On)
 
12-Jun-2026onvfowev 36050 If 𝐹 maps the ordinals onto the universe, then 𝑅 well-orders the universe. This is the ZFC version of (8 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋(𝑋 ≠ ∅ → ∃𝐹𝐹:On⟶onto𝑋), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 12-Jun-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐻‘𝑥) ∈ (𝐻‘𝑦)}    &   𝐻 = (𝑧 ∈ V ↦ ∩ (◡𝐹 “ {𝑧}))    ⇒   (𝐹:On⟶ontoV → 𝑅 We V)
 
11-Jun-2026vonf1oonfo 36049 If 𝐹 is a bijection from the ordinals to the universe and 𝐴 is non-empty, then 𝐻 maps the ordinals onto 𝐴. This is the ZFC version of (5 → 8) in https://tinyurl.com/hamkins-gblac, though it neglects to specify that 𝐴 must be non-empty. Note that in NBG set theory the antecedent would be something like ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:𝑋⟶1-1ontoOn), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. This theorem can also be viewed as (2 → 8). (Contributed by BTernaryTau, 11-Jun-2026.)
𝐻 = (𝑥 ∈ On ↦ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷))    &   𝐷 = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴})    ⇒   ((𝐹:On⟶1-1ontoV ∧ 𝐴 ≠ ∅) → 𝐻:On⟶onto𝐴)
 
11-Jun-2026vonf1owev 36037 If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (2 → 3) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 6-Dec-2025.) (Proof shortened by BTernaryTau, 11-Jun-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}    ⇒   (𝐹:V⟶1-1ontoOn → 𝑅 We V)
 
11-Jun-2026vonf1wev 36036 If 𝐹 maps the universe one-to-one into the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (6 → 3) which can be used in place of (7 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋∃𝐹𝐹:𝑋⟶1-1On, but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 11-Jun-2026.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)}    ⇒   (𝐹:V⟶1-1On → 𝑅 We V)
 
10-Jun-2026nmull0 37123 Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
(𝐴 ∈ On → (∅ ·no 𝐴) = ∅)
 
10-Jun-2026nmulr0 37122 Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
(𝐴 ∈ On → (𝐴 ·no ∅) = ∅)
 
10-Jun-2026nmulcom 37121 Natural multiplication is commutative. (Contributed by Scott Fenton, 10-Jun-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = (𝐵 ·no 𝐴))
 
10-Jun-2026nmulval 37119 Show the value of natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·no 𝐵) +no (𝐴 ·no 𝑏)) ∈ (𝑥 +no (𝑎 ·no 𝑏))})
 
10-Jun-2026nmulcl 37118 Closure law for natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·no 𝐵) ∈ On)
 
10-Jun-2026vonf1oonf1 36043 If 𝐹 is a bijection from the universe to the ordinals, then 𝐻 maps 𝐴 one-to-one into the ordinals. This is the ZFC version of (5 → 6) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:𝑋⟶1-1ontoOn), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. This theorem can also be viewed as (2 → 6). (Contributed by BTernaryTau, 10-Jun-2026.)
𝐻 = (𝐹 ↾ 𝐴)    ⇒   (𝐹:V⟶1-1ontoOn → 𝐻:𝐴⟶1-1On)
 
10-Jun-2026bdaydm 28165 The birthday function's domain is No. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
dom bday = No
 
10-Jun-2026pige0 26818 π is nonnegative. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
0 ≤ π
 
10-Jun-20262picn 26816 (2 · π) is a complex number. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
(2 · π) ∈ 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 https://us.metamath.org/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.

(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.

(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).

(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."

(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.

(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).

FrontBackDetail
ZFC T-shirt front ZFC T-shirt back ZFC T-shirt detail

(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.

(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101) release: https://us.metamath.org/ocat/mmj2/mmj2.zip https://us.metamath.org/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)! See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems. See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if there are errors in the command line arguments -- and the last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes. Good luck. And thanks for all of your help!

(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.

(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),

(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.

(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)

(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.

(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.

(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.

To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.

To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.

I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.

Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.

(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.

(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.

(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.

(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.

OldNewDescription
C. _C binomial coefficient
E _E epsilon relation
e _e Euler's constant
I _I identity relation
i _i imaginary unit
V _V universal class
(_ C_ subset
(. C. proper subset
P~ ~P power class
H~ ~H Hilbert space

(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.

(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.

(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)

(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.

(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.

(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.

(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.

(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.

(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.

(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)

(28-May-2009) AsteroidMeta is back on-line. Note the URL change.

(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.

(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)

(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.

(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."

(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.

(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.

(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.

(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.

(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).

(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.

(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.

(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.

(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.

(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.

(30-Dec-2007) "How to build a library of formalized mathematics".

(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.

(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)

(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.

(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.

(5-Nov-2007) A Usenet challenge. :)

(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.

(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.

(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.

(29-Apr-2007) Blog mentions of Metamath: here and here.

(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.

(15-Mar-2007) A picture of Penny the cat guarding the us.metamath.org server and making the rounds.

(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.

(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.

(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.

(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.

(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.

(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.

(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.

(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.

(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.

(29-Oct-2006) ocat has announced that the long-awaited 1-Nov-2006 release of mmj2 is available now.
     The new "Unify+Get Hints" is quite useful, and any proof can be generated as follows. With "?" in the Hyp field and Ref field blank, select "Unify+Get Hints". Select a hint from the list and put it in the Ref field. Edit any $n dummy variables to become the desired wffs. Rinse and repeat for the new proof steps generated, until the proof is done.
     The new tutorial, mmj2PATutorial.bat, explains this in detail. One way to reduce or avoid dummy $n's is to fill in the Hyp field with a comma-separated list of any known hypothesis matches to earlier proof steps, keeping a "?" in the list to indicate that the remaining hypotheses are unknown. Then "Unify+Get Hints" can be applied. The tutorial page \mmj2\data\mmp\PATutorial\Page405.mmp has an example.
     Don't forget that the eimm export/import program lets you go back and forth between the mmj2 and the metamath program proof assistants, without exiting from either one, to exploit the best features of each as required.

(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.

(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.

(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.

(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.

(28-Sep-2006) A blog entry, Metamath responds.

(26-Sep-2006) A blog entry, Metamath isn't hygienic.

(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.

(26-Jul-2006) A new open problem in predicate calculus was added.

(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.

(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C program used to create the MIDI files on the Metamath Music Page, so that it works on MacOS X. This is a nice accomplishment, since the original program was written before C was standardized by ANSI and will not compile on modern compilers.
      Unfortunately, the original program source states no copyright terms. The main author, Tim Thompson, has kindly agreed to release his code to public domain, but two other authors have also contributed to the code, and so far I have been unable to contact them for copyright clearance. Therefore I cannot offer the MacOS X version for public download on this site until this is resolved. Update 10-May-2006: Another author, M. Czeiszperger, has released his contribution to public domain.
      If you are interested in Luca's modified source code, please contact me directly.

(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.

(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.

(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java applet to Java 1.5: (1) QSort has been stripped out: its functionality is in the Collections class that Sun ships; (2) all Vectors have been replaced by ArrayLists; (3) generic types have been tossed in wherever they fit: this cuts back drastically on casting; and (4) any warnings Eclipse spouted out have been dealt with. I haven't yet updated it officially, because I don't know if it will work with Microsoft's JVM in older versions of Internet Explorer. The current official version is compiled with Java 1.3, because it won't work with Microsoft's JVM if it is compiled with Java 1.4. (As distasteful as that seems, I will get complaints from users if it doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new version runs on Microsoft's JVM, I would be grateful. Scott's new version is mm.java-1.5.gz; after uncompressing it, rename it to mm.java, use it to replace the existing mm.java file in the Metamath Solitaire download, and recompile according to instructions in the mm.java comments.
      Scott has also created a second version, mmsol-0.2.tar.gz, that reads the axioms from ASCII files, instead of having the axioms hard-coded in the program. This can be very useful if you want to play with custom axioms, and you can also add a collection of starting theorems as "axioms" to work from. However, it must be run from the local directory with appletviewer, since the default Java security model doesn't allow reading files from a browser. It works with the JDK 5 Update 6 Java download.
To compile (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html

(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!

(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.

(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).

(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)

(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.

(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.

(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."

(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.

(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.

(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).

(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.

(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .

(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.

(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.

(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.

(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.

(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.

(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.

(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).

(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).

(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.

(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.

(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.

(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.

(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)

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