Most Recent Proofs - Intuitionistic Logic Explorer

Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the iset.mm database for the Intuitionistic Logic Explorer. The iset.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from the commit given on the MPE Most Recent Proofs page. The database from that commit is also available here: iset.mm.

See the MPE Most Recent Proofs page for news and some useful links.

Last updated on 20-Aug-2026 at 6:58 AM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
14-Aug-2026	reaplog 15972	Apartness and the real natural logarithm. (Contributed by Jim Kingdon, 14-Aug-2026.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A # B <-> ( log ` A ) # ( log ` B ) ) )
13-Aug-2026	efap1p 15882	If the exponential of a number is apart from one plus that number, the number is apart from zero. To some extent can be thought of as the converse of efgt1p 12465. (Contributed by Jim Kingdon, 13-Aug-2026.)
|- ( ( A e. RR /\ ( 1 + A ) # ( exp ` A ) ) -> A # 0 )
6-Aug-2026	relndmfv 5728	The value of a relation outside its domain is the empty set. (Contributed by Jim Kingdon, 6-Aug-2026.)
|- ( ( Rel F /\ -. A e. dom F ) -> ( F ` A ) = (/) )
1-Aug-2026	wexmiddifxy 17058	Being able to subtract an arbitrary finite set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7198. (Contributed by Jim Kingdon, 1-Aug-2026.)
|- (WEXMID <-> A. x A. y ( ( x e. Fin /\ y e. Fin ) -> ( x \ y ) e. Fin ) )
1-Aug-2026	wexmiddifxylem 17057	Lemma for wexmiddifxylem 17057. Showing weak excluded middle given a suitable finite set. (Contributed by Jim Kingdon, 1-Aug-2026.)
|- ( ( { 1o } \ { { x e. 1o | ph } } ) e. Fin -> DECID -. ph )
31-Jul-2026	rabid1o 17046	Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)
|- ( { x e. 1o | ph } = 1o <-> ph )
30-Jul-2026	wexmiddc 17054	Weak excluded middle expressed using WEXMID implies decidability of a negated proposition. (Contributed by Jim Kingdon, 30-Jul-2026.)
|- (WEXMID -> DECID -. ph )
30-Jul-2026	df-wexmid 17053	Weak excluded middle is the principle that any negated proposition is decidable. (Contributed by Jim Kingdon, 30-Jul-2026.)
|- (WEXMID <-> A. p e. ~P 1o ( -. p = 1o \/ -. -. p = 1o ) )
29-Jul-2026	wexmiddiffi 17056	Being able to subtract an arbitrary set from a finite set and get a finite set is equivalent to weak excluded middle. By adding additional conditions we can get a theorem which does not need weak excluded middle, at diffifi 7198. (Contributed by Jim Kingdon, 29-Jul-2026.)
|- (WEXMID <-> A. x A. y ( x e. Fin -> ( x \ y ) e. Fin ) )
29-Jul-2026	wexmiddiffilem 17055	Lemma for wexmiddiffi 17056. The reverse direction, using different notation. (Contributed by Jim Kingdon, 29-Jul-2026.)
|- ( A. x A. y ( x e. Fin -> ( x \ y ) e. Fin ) -> ( -. ph \/ -. -. ph ) )
24-Jul-2026	stnot 17051	A proposition is double negation stable if and only if it is equivalent to a negated proposition. Here by "proposition" we mean a subset of a singleton (which is a choice which allows us to quantify over them). Posed as an exercise online by Yannick Forster. (Contributed by Jim Kingdon, 24-Jul-2026.)
|- ( A e. ~P 1o -> ( ( -. -. A = 1o -> A = 1o ) <-> E. y e. ~P 1o ( A = 1o <-> -. y = 1o ) ) )
24-Jul-2026	slotm 13417	A structure with an inhabited slot is inhabited. (Contributed by Jim Kingdon, 24-Jul-2026.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- ( A e. ( E ` G ) -> E. j j e. G )
22-Jul-2026	alseu-no-surprise 17191	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 17159 by alseuals 17177. See als-no-surprise 17159 for why ordinary "for all" with implication has no such property. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- -. ( A.E! x ( ph -> ps ) /\ A.E! x ( ph -> -. ps ) )
22-Jul-2026	alseueu 17190	"The ph is ps " implies that exactly one thing is both ph and ps. This is the half of dfalseu2 17189 that drops the universal conjunct; it does not reverse, so E! x ( ph /\ ps ) cannot be used in place of an "all some one" statement. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) -> E! x ( ph /\ ps ) )
22-Jul-2026	dfalseu2 17189	An "all some one" statement is equivalent to its universal part conjoined with the claim that exactly one x satisfies both ph and ps. In other words, given A. x ( ph -> ps ), requiring exactly one x to satisfy ph, which is what df-alseu 17174 requires, and requiring exactly one x to satisfy ( ph /\ ps ) come to the same thing. Read ph as "is a king" and ps 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. E! x ( ph /\ ps ) on its own is strictly weaker than A.E! x ( ph -> ps ), since it is satisfied when many things are ph and just one of those is ps, as in a region with five kings exactly one of whom is hungry; see alseueu 17190 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, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E! x ( ph /\ ps ) ) )
22-Jul-2026	nfralseu 17188	Bound-variable hypothesis builder for "all some one" restricted to a class. This is the "all some one" counterpart of nfrals 17157. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- F/_ x A   &    |- F/ x ph   &    |- F/ x ps   =>    |- F/ x A.E! y e. A ( ph -> ps )
22-Jul-2026	nfalseu 17187	Bound-variable hypothesis builder for "all some one". This is the "all some one" counterpart of nfals 17156. Unlike the set.mm version of this theorem, no disjoint variable condition is needed, because nfeu 2105 here does not require one. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- F/ x ph   &    |- F/ x ps   =>    |- F/ x A.E! y ( ph -> ps )
22-Jul-2026	ralseubii 17186	Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 17154. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( A.E! x e. A ( ph -> ps ) <-> A.E! x e. A ( ch -> th ) )
22-Jul-2026	alseubii 17185	Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 17153. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( A.E! x ( ph -> ps ) <-> A.E! x ( ch -> th ) )
22-Jul-2026	ralseu2d 17184	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 ps, not merely be a member of A. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x e. A ( ps -> ch ) )   =>    |- ( ph -> E! x e. A ps )
22-Jul-2026	ralseu1d 17183	Deduction rule: Given "all some one" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x e. A ( ps -> ch ) )   =>    |- ( ph -> A. x e. A ( ps -> ch ) )
22-Jul-2026	alseu2d 17182	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "exactly one" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x ( ps -> ch ) )   =>    |- ( ph -> E! x ps )
22-Jul-2026	alseu1d 17181	Deduction rule: Given "all some one" applied to a top-level inference, you can extract the "for all" part. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A.E! x ( ps -> ch ) )   =>    |- ( ph -> A. x ( ps -> ch ) )
22-Jul-2026	ralseud 17180	Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d 17183 and ralseu2d 17184 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A. x e. A ( ps -> ch ) )   &    |- ( ph -> E! x e. A ps )   =>    |- ( ph -> A.E! x e. A ( ps -> ch ) )
22-Jul-2026	alseud 17179	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 17181 and alseu2d 17182 taken together. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( ph -> A. x ( ps -> ch ) )   &    |- ( ph -> E! x ps )   =>    |- ( ph -> A.E! x ( ps -> ch ) )
22-Jul-2026	ralseurals 17178	"All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 17177. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x e. A ( ph -> ps ) -> A.E. x e. A ( ph -> ps ) )
22-Jul-2026	alseuals 17177	"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 17191 is proved. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) -> A.E. x ( ph -> ps ) )
22-Jul-2026	dfralseu2 17176	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 17142. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x e. A ( ph -> ps ) <-> A.E! x ( ( x e. A /\ ph ) -> ps ) )
22-Jul-2026	df-ralseu 17175	Define "all some one" applied to a class, which means ps is true whenever ph is true for x in A, and exactly one x in A satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E! x e. A ph ) )
22-Jul-2026	df-alseu 17174	Define "all some one" applied to a top-level implication, which means ps is true whenever ph is true and exactly one x satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
|- ( A.E! x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )
22-Jul-2026	wralseu 17173	Extend wff definition to include "all some one" applied to a class, which means ps is true whenever ph is true for x in A, and exactly one x in A satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff A.E! x e. A ( ph -> ps )
22-Jul-2026	walseu 17172	Extend wff definition to include "all some one" applied to a top-level implication, which means ps is true whenever ph is true, and exactly one x satisfies ph. (Contributed by David A. Wheeler, 22-Jul-2026.)
wff A.E! x ( ph -> ps )
22-Jul-2026	mptmex 5945	If a function given by maps-to notation is inhabited, then the class it is defined on is inhabited. (Contributed by Jim Kingdon, 22-Jul-2026.)
|- ( C e. ( x e. A |-> B ) -> E. y y e. A )
20-Jul-2026	2alsraln0idm 17171	Nested general "all some" quantifiers with class membership as their antecedents, for the same class A: ph holds for every x and every y in A, and A is inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> A.E. y ( y e. A -> ph ) ) <-> ( A. x e. A A. y e. A ph /\ E. x x e. A ) )
20-Jul-2026	2alsraln0m 17170	Nested general "all some" quantifiers with class membership as their antecedents: ph holds for every x in A and every y in B, and both A and B are inhabited. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> A.E. y ( y e. B -> ph ) ) <-> ( A. x e. A A. y e. B ph /\ ( E. x x e. A /\ E. y y e. B ) ) )
20-Jul-2026	n0alsm 17169	If A is inhabited, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that ph holds for every x in A. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( E. x x e. A -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )
20-Jul-2026	alsraln0m 17166	The general "all some" quantifier with class membership as its antecedent holds if and only if ph holds for every x in A and A is inhabited. This is the intuitionistic form of what set.mm states using A =/= (/); see the section comment. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x x e. A ) )
20-Jul-2026	alsralrex 17165	The general "all some" quantifier with class membership as its antecedent holds if and only if ph holds for every x in A and some x in A satisfies ph. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x e. A ph ) )
20-Jul-2026	ralsanmo 17164	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 17163. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( ( A.E. x e. A ( ph -> ps ) /\ E* x e. A ph ) <-> ( A. x e. A ( ph -> ps ) /\ E! x e. A ph ) )
20-Jul-2026	alsanmo 17163	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 E* x ph supplies the at-most-one part, so together they yield E! x ph. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( ( A.E. x ( ph -> ps ) /\ E* x ph ) <-> ( A. x ( ph -> ps ) /\ E! x ph ) )
20-Jul-2026	rexrals 17162	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 17168. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( E. x e. A ph -> ( A.E. x e. A ( ph -> ps ) <-> A. x e. A ( ph -> ps ) ) )
20-Jul-2026	ralrals 17161	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 17167. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
|- ( A. x e. A ( ph -> ps ) -> ( A.E. x e. A ( ph -> ps ) <-> E. x e. A ph ) )
20-Jul-2026	als-no-surprise 17159	Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-als 17140: the universal parts give A. x -. ph, which contradicts the witness that the allsome quantifier supplies. Ordinary "for all" with implication has no such property, since A. x ( ph -> ps ) and A. x ( ph -> -. ps ) can both hold when nothing satisfies ph. (Contributed by David A. Wheeler, 27-Oct-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)
|- -. ( A.E. x ( ph -> ps ) /\ A.E. x ( ph -> -. ps ) )
20-Jul-2026	ralsmd 17150	Deduction rule: Given "all some" applied to a class, the class is inhabited. This is stronger than ralsn0d 17149, which only concludes that the class is nonempty; see n0r 3535. (Contributed by David A. Wheeler, 20-Jul-2026.)
|- ( ph -> A.E. x e. A ( ps -> ch ) )   =>    |- ( ph -> E. x x e. A )
19-Jul-2026	disjdifg 3598	A class and its relative complement are disjoint. (Contributed by NM, 24-Mar-1998.) Generalize from disjdif 3599. (Revised by BJ, 19-Jul-2026.)
|- ( A C_ B -> ( A i^i ( C \ B ) ) = (/) )
19-Jul-2026	sseq0b 3564	The only subclass of the empty class is itself. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Strengthen sseq0 3565 to a biconditional. (Revised by BJ, 19-Jul-2026.)
|- ( A = (/) -> ( B C_ A <-> B = (/) ) )
18-Jul-2026	sepab 4278	Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 4279. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)
|- ( A e. V -> { x | ( x e. A /\ ph ) } e. _V )
18-Jul-2026	inssdif0im 3592	Intersection, subclass, and difference relationship. The converse holds in classical logic but not in intuitionistic logic. (Contributed by Jim Kingdon, 3-Aug-2018.) (Proof shortened by BJ, 18-Jul-2026.)
|- ( ( A i^i B ) C_ C -> ( A i^i ( B \ C ) ) = (/) )
15-Jul-2026	rexals 17168	If some x in A satisfies ph, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that ph holds for every x in A. See rexrals 17162 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
|- ( E. x e. A ph -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )
15-Jul-2026	ralals 17167	If ph 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 ph. See ralrals 17161 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
|- ( A. x e. A ph -> ( A.E. x ( x e. A -> ph ) <-> E. x e. A ph ) )
14-Jul-2026	uniex2 4581	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.)
|- E. y y = U. x
14-Jul-2026	sepgi 4252	Inference associated with sepg 4251. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)
|- A e. _V   =>    |- E. y A. x ( x e. y <-> ( x e. A /\ ph ) )
13-Jul-2026	f1setfi 7317	The set of injections between two finite sets is finite. (Contributed by Jim Kingdon, 13-Jul-2026.)
|- ( ( A e. Fin /\ B e. Fin ) -> { f | f : A -1-1-> B } e. Fin )
13-Jul-2026	fdcf1 7316	It is decidable whether a function from a finite set into another finite set is one-to-one. (Contributed by Jim Kingdon, 13-Jul-2026.)
|- ( ( A e. Fin /\ B e. Fin /\ F : A --> B ) -> DECID F : A -1-1-> B )
12-Jul-2026	rals-no-surprise 17160	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 17159, and follows from it by dfrals2 17142. Note that this holds without any assumption that A is inhabited; that is the point of allsome, since the corresponding claim for the ordinary restricted "for all" fails when nothing in A satisfies ph. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- -. ( A.E. x e. A ( ph -> ps ) /\ A.E. x e. A ( ph -> -. ps ) )
12-Jul-2026	cbvals 17158	Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- ( x = y -> ( ph <-> ch ) )   &    |- ( x = y -> ( ps <-> th ) )   =>    |- ( A.E. x ( ph -> ps ) <-> A.E. y ( ch -> th ) )
12-Jul-2026	nfrals 17157	Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- F/_ x A   &    |- F/ x ph   &    |- F/ x ps   =>    |- F/ x A.E. y e. A ( ph -> ps )
12-Jul-2026	nfals 17156	Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
|- F/ x ph   &    |- F/ x ps   =>    |- F/ x A.E. y ( ph -> ps )
12-Jul-2026	alsbid 17155	Deduction form of alsbii 17153. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- F/ x ph   &    |- ( ph -> ( ps <-> th ) )   &    |- ( ph -> ( ch <-> ta ) )   =>    |- ( ph -> ( A.E. x ( ps -> ch ) <-> A.E. x ( th -> ta ) ) )
12-Jul-2026	ralsbii 17154	Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( A.E. x e. A ( ph -> ps ) <-> A.E. x e. A ( ch -> th ) )
12-Jul-2026	alsbii 17153	Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
|- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( A.E. x ( ph -> ps ) <-> A.E. x ( ch -> th ) )
12-Jul-2026	ralsex 17152	The consequent of an "all some" restricted to a class is witnessed: some member of A satisfying ph also satisfies ps. Restricted counterpart of alsex 17151. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- ( A.E. x e. A ( ph -> ps ) -> E. x e. A ps )
12-Jul-2026	alsex 17151	The consequent of an "all some" is witnessed: if ps holds of every x satisfying ph, and some x satisfies ph, then some x satisfies ps. This is the positive counterpart of als-no-surprise 17159, and it is the property that ordinary "for all" with implication lacks: from A. x ( ph -> ps ) alone nothing whatever follows about ps, since nothing need satisfy ph. It is the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- ( A.E. x ( ph -> ps ) -> E. x ps )
12-Jul-2026	ralsn0d 17149	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.)
|- ( ph -> A.E. x e. A ( ps -> ch ) )   =>    |- ( ph -> A =/= (/) )
12-Jul-2026	rals2d 17148	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent ps, not merely be a member of A. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
|- ( ph -> A.E. x e. A ( ps -> ch ) )   =>    |- ( ph -> E. x e. A ps )
12-Jul-2026	rals1d 17147	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.)
|- ( ph -> A.E. x e. A ( ps -> ch ) )   =>    |- ( ph -> A. x e. A ( ps -> ch ) )
12-Jul-2026	ralsd 17144	Introduction rule for "all some" restricted to a class. This is the converse of rals1d 17147 and rals2d 17148 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)
|- ( ph -> A. x e. A ( ps -> ch ) )   &    |- ( ph -> E. x e. A ps )   =>    |- ( ph -> A.E. x e. A ( ps -> ch ) )
12-Jul-2026	alsd 17143	Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 17145 and als2d 17146 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.)
|- ( ph -> A. x ( ps -> ch ) )   &    |- ( ph -> E. x ps )   =>    |- ( ph -> A.E. x ( ps -> ch ) )
12-Jul-2026	dfrals2 17142	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.)
|- ( A.E. x e. A ( ph -> ps ) <-> A.E. x ( ( x e. A /\ ph ) -> ps ) )
12-Jul-2026	df-rals 17141	Define "all some" applied to a class, which means ps is true whenever ph is true for x in A, and there is at least one x in A where ph 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 ph to the members of a class, asserting only ( A. x e. A ph /\ E. x x e. 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 E. x x e. 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.)
|- ( A.E. x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
12-Jul-2026	wrals 17139	Extend wff definition to include "all some" applied to a class, which means ps is true whenever ph is true for x in A, and there is at least one x in A where ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff A.E. x e. A ( ph -> ps )
12-Jul-2026	wals 17138	Extend wff definition to include "all some" applied to a top-level implication, which means ps is true whenever ph is true, and there is at least one x where ph is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff A.E. x ( ph -> ps )
12-Jul-2026	vvin 3569	Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 3567. (Contributed by BJ, 12-Jul-2026.)
|- ( ( A = _V /\ B = _V ) <-> ( A i^i B ) = _V )
7-Jul-2026	cmnsubm 14114	A submonoid of a commutative monoid is commutative. (Contributed by Jim Kingdon, 7-Jul-2026.)
|- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> G e. CMnd )   &    |- H = ( G ↾s S )   =>    |- ( ph -> H e. CMnd )
29-Jun-2026	dichmul0or 16772	Real number dichotomy is equivalent to the zero product principle for complex numbers: if a product is zero, one of its factors must be zero. (Contributed by Matthew House, 29-Jun-2026.)
|- ( A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) <-> A. z e. CC A. w e. CC ( ( z x. w ) = 0 -> ( z = 0 \/ w = 0 ) ) )
29-Jun-2026	dichmul0orlem5 16769	Lemma for dichmul0or 16772. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ph -> A e. RR )   &    |- ( ph -> ( ( abs ` A ) + A ) = 0 )   =>    |- ( ph -> A <_ 0 )
29-Jun-2026	dichmul0orlem4 16768	Lemma for dichmul0or 16772. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( ( ( abs ` A ) + A ) x. ( ( abs ` A ) - A ) ) = 0 )
29-Jun-2026	dichmul0orlem3 16767	Lemma for dichmul0or 16772. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ph -> A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A x. B ) = 0 )   =>    |- ( ph -> ( A = 0 \/ B = 0 ) )
29-Jun-2026	dichmul0orlem2 16766	Lemma for dichmul0or 16772. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> ( A x. B ) = 0 )   &    |- ( ph -> ( abs ` A ) <_ ( abs ` B ) )   =>    |- ( ph -> A = 0 )
29-Jun-2026	dichmul0orlem1 16765	Lemma for dichmul0or 16772. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A x. B ) = 0 )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> A = 0 )
29-Jun-2026	lealltlt2 16764	Alternative definition for <_ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A. x e. RR ( B < x -> A < x ) ) )
29-Jun-2026	lealltlt1 16763	Alternative definition for <_ on real numbers. (Contributed by Matthew House, 29-Jun-2026.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A. x e. RR ( x < A -> x < B ) ) )
28-Jun-2026	dichmul0orlem7 16771	Lemma for dichmul0or 16772. (Contributed by Matthew House, 28-Jun-2026.)
|- ( ph -> A. x e. CC A. y e. CC ( ( x x. y ) = 0 -> ( x = 0 \/ y = 0 ) ) )   &    |- ( ph -> A e. RR )   =>    |- ( ph -> ( A <_ 0 \/ 0 <_ A ) )
28-Jun-2026	dichmul0orlem6 16770	Lemma for dichmul0or 16772. (Contributed by Matthew House, 28-Jun-2026.)
|- ( ph -> A e. RR )   &    |- ( ph -> ( ( abs ` A ) - A ) = 0 )   =>    |- ( ph -> 0 <_ A )
28-Jun-2026	msq0 8998	A number is zero iff its square is zero. (Contributed by Matthew House, 28-Jun-2026.)
|- ( A e. CC -> ( ( A x. A ) = 0 <-> A = 0 ) )
28-Jun-2026	msqap0 8997	A number is apart from zero iff its square is apart from zero. (Contributed by Matthew House, 28-Jun-2026.)
|- ( A e. CC -> ( ( A x. A ) # 0 <-> A # 0 ) )
28-Jun-2026	letrid 8442	Tightness of real apartness. (Contributed by Matthew House, 28-Jun-2026.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A <_ B )   &    |- ( ph -> B <_ A )   =>    |- ( ph -> A = B )
19-Jun-2026	ringen1zr0 14624	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 14263, and semirings, see srgen1zr0 14294. (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 e. Ring /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
19-Jun-2026	srg1zr 14293	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 e. SRing /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18-Jun-2026	rngen1zr0 14263	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 e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( B ~~ 1o <-> ( .+ = { <. <. .0. , .0. >. , .0. >. } /\ .* = { <. <. .0. , .0. >. , .0. >. } ) ) )
18-Jun-2026	rngen1zr 14262	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 e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B ~~ 1o <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18-Jun-2026	rng1zr 14261	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 e. Rng /\ .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
18-Jun-2026	rng1zrlem 14260	Lemma for rng1zr 14261 and srg1zr 14293. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .* = ( .r ` R )   =>    |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
17-Jun-2026	ballotfi 13284	Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.) (Revised by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   =>    |- ( P ` E ) = ( ( M - N ) / ( M + N ) )
17-Jun-2026	ballotfilembfi 13241	The set of countings where B got the first vote is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   =>    |- { c e. ( O \ E ) | -. 1 e. c } e. Fin
17-Jun-2026	ballotfilemafi 13240	The set of countings where A got the first vote, but does not stay strictly ahead throughout, is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   =>    |- { c e. ( O \ E ) | 1 e. c } e. Fin
17-Jun-2026	ballotfilemefi 13239	E is finite. (Contributed by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   =>    |- E e. Fin
17-Jun-2026	rabxmdc 3554	Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Jim Kingdon, 17-Jun-2026.)
|- ( A. x e. A DECID ph -> A = ( { x e. A | ph } u. { x e. A | -. ph } ) )
15-Jun-2026	ballotfilemgun 13270	A property of the defined .^ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   &    |- ( ph -> U e. O )   &    |- ( ph -> L e. ( J ... K ) )   =>    |- ( ph -> ( U .^ ( J ... K ) ) = ( ( U .^ ( J ... ( L - 1 ) ) ) + ( U .^ ( L ... K ) ) ) )
15-Jun-2026	ballotfilemgval 13269	Expand the value of .^. (Contributed by Thierry Arnoux, 21-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   &    |- ( ph -> U e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> V = ( J ... K ) )   =>    |- ( ph -> ( U .^ V ) = ( (♯ ` ( V i^i U ) ) - (♯ ` ( V \ U ) ) ) )
15-Jun-2026	ballotfilemdifcfz 13229	Lemma for ballotfi . The portion of a counting representing votes for B within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> ( ( J ... K ) \ C ) e. Fin )
15-Jun-2026	ballotfilemcinfz 13228	Lemma for ballotfi . The portion of a counting representing votes for A within a specified integer range is finite. (Contributed by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> ( ( J ... K ) i^i C ) e. Fin )
12-Jun-2026	ballotfilemsle 13250	The infimum of the set of zeroes of F is a lower bound. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 }   =>    |- ( ( C e. ( O \ E ) /\ X e. S ) -> inf ( S , RR , < ) <_ X )
12-Jun-2026	ballotfilemscl 13249	The set of zeroes of F has an infimum. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 }   =>    |- ( C e. ( O \ E ) -> inf ( S , RR , < ) e. S )
12-Jun-2026	infssfzledc 10672	The infimum of a decidable inhabited subset of an integer range is a lower bound for that set. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- S = { n e. ( M ... N ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) <_ A )
12-Jun-2026	infssfzcldc 10671	The infimum of a decidable inhabited subset of an integer range is a member of the set. (Contributed by Jim Kingdon, 12-Jun-2026.)
|- S = { n e. ( M ... N ) | ps }   &    |- ( ph -> A e. S )   &    |- ( ( ph /\ n e. ( M ... A ) ) -> DECID ps )   =>    |- ( ph -> inf ( S , RR , < ) e. S )
10-Jun-2026	2t3e6 9463	2 times 3 equals 6. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
|- ( 2 x. 3 ) = 6
8-Jun-2026	ballotfilemdifcfi 13227	Lemma for ballotfi . The portion of a counting representing votes for B up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   =>    |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )
8-Jun-2026	ballotfilemcinfi 13226	Lemma for ballotfi . The portion of a counting representing votes for A up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> J e. ZZ )   =>    |- ( ph -> ( ( 1 ... J ) i^i C ) e. Fin )
8-Jun-2026	zfidc 9725	Whether an integer is an element of a finite set of integers is decidable. (Contributed by Jim Kingdon, 8-Jun-2026.)
|- ( ( S C_ ZZ /\ A e. ZZ /\ S e. Fin ) -> DECID A e. S )
7-Jun-2026	ballotfilemcdc 13225	Lemma for ballotfi . It is decidable whether a given integer is an element of a particular element of O. (Contributed by Jim Kingdon, 7-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- ( ph -> C e. O )   &    |- ( ph -> K e. ZZ )   =>    |- ( ph -> DECID K e. C )
5-Jun-2026	hashpwfi 11271	The number of finite subsets of a finite set is two raised to the power of the size of the set. For a similar theorem with set size expressed using equinumerosity, see 2omapfi 7320. For the number of subsets (which need not be finite) of a set, see pw1mapen 17038. (Contributed by Jim Kingdon, 5-Jun-2026.)
|- ( A e. Fin -> (♯ ` ( ~P A i^i Fin ) ) = ( 2 ^ (♯ ` A ) ) )
4-Jun-2026	ballotfilemonn 13223	The size of the universe is at least one. (Contributed by Jim Kingdon, 4-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   =>    |- (♯ ` O ) e. NN
3-Jun-2026	papeq2 7610	Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
|- ( A = B -> ( R Ap A <-> R Ap B ) )
3-Jun-2026	papeq1 7609	Equality theorem for apartness predicate. (Contributed by Jim Kingdon, 3-Jun-2026.)
|- ( R = S -> ( R Ap A <-> S Ap A ) )
2-Jun-2026	resq01 11097	If a real number equals its square, it must be 0 or 1. (Contributed by Jim Kingdon, 2-Jun-2026.)
|- ( A e. RR -> ( ( A ^ 2 ) = A <-> ( A = 0 \/ A = 1 ) ) )
31-May-2026	aprprop 14603	If two structures have the same ring components (properties), df-apr 14592 generates the same relation for both of them. (Contributed by Jim Kingdon, 31-May-2026.)
|- ( Base ` K ) = ( Base ` L )   &    |- ( +g ` K ) = ( +g ` L )   &    |- ( .r ` K ) = ( .r ` L )   =>    |- ( K e. Ring -> (#r ` K ) = (#r ` L ) )
31-May-2026	ringunitsap0 14596	The set of units of a ring. If R is a local ring, # is an apartness and this theorem states that the units of a ring are those elements apart from zero (see aprlring 14602). Given the definition of #r this theorem holds even if # is not an apartness, however. (Contributed by Jim Kingdon, 31-May-2026.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. Ring -> { x e. B | x # .0. } = (Unit ` R ) )
30-May-2026	ringunitap 14595	Elementhood in the set of units. (Contributed by Jim Kingdon, 30-May-2026.)
|- B = ( Base ` R )   &    |- U = (Unit ` R )   &    |- .0. = ( 0g ` R )   &    |- # = (#r ` R )   =>    |- ( R e. Ring -> ( X e. U <-> ( X e. B /\ X # .0. ) ) )
29-May-2026	drnglring 14609	A division ring is a local ring. (Contributed by Jim Kingdon, 29-May-2026.)
|- ( R e. DivRing -> R e. LRing )
29-May-2026	isdrngtap 14608	The predicate "is a division ring". (Contributed by Jim Kingdon, 29-May-2026.)
|- B = ( Base ` R )   &    |- # = (#r ` R )   =>    |- ( R e. DivRing <-> ( R e. Ring /\ # TAp B ) )
29-May-2026	df-drngap 14606	Define class of all division rings. A division ring is a ring in which the relation given by df-apr 14592 is a tight apartness. (Contributed by Jim Kingdon, 29-May-2026.)
|- DivRing = { r e. Ring | (#r ` r ) TAp ( Base ` r ) }
29-May-2026	aprunit 14594	The df-apr 14592 relation with zero expresses whether a ring element is a unit. That is, the difference of an element of a ring and zero is invertible iff the element is a unit. (Contributed by Jim Kingdon, 29-May-2026.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- U = (Unit ` R )   &    |- # = (#r ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X # .0. <-> X e. U ) )
29-May-2026	tapap 7616	A tight apartness is an apartness. (Contributed by Jim Kingdon, 29-May-2026.)
|- ( R TAp A -> R Ap A )
28-May-2026	aprlring 14602	A ring is a local ring if and only if the relation given by df-apr 14592 is an apartness relation. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( R e. Ring -> ( R e. LRing <-> (#r ` R ) Ap ( Base ` R ) ) )
28-May-2026	papcotr 7613	An apartness is cotransitive. (Contributed by Jim Kingdon, 28-May-2026.)
|- ( ph -> R Ap A )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. A )   &    |- ( ph -> X R Y )   &    |- ( ph -> Z e. A )   =>    |- ( ph -> ( X R Z \/ Y R Z ) )
27-May-2026	aprnzr 14601	If the relation given by df-apr 14592 on a ring is an apartness relation, then the ring is a nonzero ring. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ( R e. Ring /\ (#r ` R ) Ap ( Base ` R ) ) -> R e. NzRing )
27-May-2026	papsym 7612	An apartness is symmetric. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ph -> R Ap A )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. A )   &    |- ( ph -> X R Y )   =>    |- ( ph -> Y R X )
27-May-2026	papirr 7611	An apartness is irreflexive. (Contributed by Jim Kingdon, 27-May-2026.)
|- ( ( R Ap A /\ X e. A ) -> -. X R X )
24-May-2026	gsumzfi 14160	Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. CMnd /\ A e. Fin ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
22-May-2026	sshashneg 11283	Subsets of a class of a negative size (a degenerate case). Together with ssenneg 11282 this shows that sseqn 11281 could not be extended beyond N e. NN0. (Contributed by Jim Kingdon, 22-May-2026.)
|- ( ( N e. ZZ /\ N < 0 ) -> { x e. ( ~P A i^i Fin ) | (♯ ` x ) = N } = (/) )
22-May-2026	ssenneg 11282	Subsets of a class of a negative size (a degenerate case). Together with sshashneg 11283 this shows that sseqn 11281 could not be extended beyond N e. NN0. (Contributed by Jim Kingdon, 22-May-2026.)
|- ( ( N e. ZZ /\ N < 0 ) -> { x e. ~P A | x ~~ ( 1 ... N ) } = { (/) } )
22-May-2026	sseqn 11281	Two ways to express the subsets of a class of a given size. It might seem that { x e. ~P A | (♯ ` x ) = N } would suffice, but that would require the converse of hashcl 11222 or something similar. Although each side of the equality would be well defined if we changed N e. NN0 to N e. ZZ, they would give different results for the (degenerate) case of a negative size, as shown at ssenneg 11282 and sshashneg 11283. (Contributed by Jim Kingdon, 22-May-2026.)
|- ( N e. NN0 -> { x e. ~P A | x ~~ ( 1 ... N ) } = { x e. ( ~P A i^i Fin ) | (♯ ` x ) = N } )
22-May-2026	bilanri 389	Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
|- ( ph <-> ps )   =>    |- ( ( ch /\ ps ) -> ph )
22-May-2026	biranri 388	Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
|- ( ph <-> ps )   =>    |- ( ( ps /\ ch ) -> ph )
22-May-2026	bilani 387	Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
|- ( ph <-> ps )   =>    |- ( ( ch /\ ph ) -> ps )
22-May-2026	birani 386	Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
|- ( ph <-> ps )   =>    |- ( ( ph /\ ch ) -> ps )
20-May-2026	ballotfilemofi 13221	O is finite. (Contributed by Jim Kingdon, 20-May-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   =>    |- O e. Fin
19-May-2026	fipwfi 7321	The set of finite subsets of a finite set is finite. (Contributed by Jim Kingdon, 19-May-2026.)
|- ( A e. Fin -> ( ~P A i^i Fin ) e. Fin )
18-May-2026	2omapfi 7320	The number of finite subsets of a finite set. For a similar theorem with set size expressed using ♯ (df-ihash 11217), see hashpwfi 11271. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( A e. Fin -> ( 2o ^m A ) ~~ ( ~P A i^i Fin ) )
18-May-2026	fissfi 7263	A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A. x e. A DECID x e. S )
18-May-2026	fresaunres1disj 5571	From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.) (Revised by Jim Kingdon, 18-May-2026.)
|- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` A ) = F )
18-May-2026	fresaunres2disj 5570	From the union of two functions with disjoint domains, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Jim Kingdon, 18-May-2026.)
|- ( ( F : A --> C /\ G : B --> C /\ ( A i^i B ) = (/) ) -> ( ( F u. G ) |` B ) = G )
15-May-2026	fsuppcorn 7301	The composition of a 1-1 function with a finitely supported function is finitely supported. The purpose of the ( F supp Z ) C_ ran G condition is to ensure we don't subset the support of the function in such a way as to fun afoul of exmidssfi 7246. (Other alternative conditions might also be sufficient). (Contributed by AV, 28-May-2019.) (Revised by Jim Kingdon, 15-May-2026.)
|- ( ph -> F finSupp Z )   &    |- ( ph -> G : X -1-1-> Y )   &    |- ( ph -> Z e. W )   &    |- ( ph -> F e. V )   &    |- ( ph -> G e. U )   &    |- ( ph -> ( F supp Z ) C_ ran G )   =>    |- ( ph -> ( F o. G ) finSupp Z )
13-May-2026	lincmble 10408	A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10407 but generalized to require merely A <_ B not A < B. (Contributed by Jim Kingdon, 13-May-2026.)
|- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )
5-May-2026	fmelpw1o 7606	With a formula ph one can associate an element of ~P 1o, which can therefore be thought of as the set of "truth values" (but recall that there are no other genuine truth values than T. and F., by nndc 863, which translate to 1o and (/) respectively by iftrue 3645 and iffalse 3648, giving pwtrufal 17039). As proved in if0ab 3641, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.) (Proof shortened by BJ, 5-May-2026.)
|- if ( ph , 1o , (/) ) e. ~P 1o
5-May-2026	if0elpw 4295	A conditional class with the False alternative being sent to the empty class is an element of the powerset of the class corresponding to the True alternative when that class is a set. This statement requires fewer axioms than the general case ifelpwung 4627. (Contributed by BJ, 5-May-2026.)
|- ( A e. V -> if ( ph , A , (/) ) e. ~P A )
5-May-2026	if0ss 3642	A conditional class with the False alternative being sent to the empty class is included in the class corresponding to the True alternative. (Contributed by BJ, 5-May-2026.)
|- if ( ph , A , (/) ) C_ A
27-Apr-2026	repiecef 17089	Piecewise definition on the reals yields a function. The function agrees with F and G on their respective parts of the real line; see repiecele0 17087 and repiecege0 17088. From an online post by James E Hanson. The construction was published in Martín Hötzel Escardó, "Effective and sequential definition by cases on the reals via infinite signed-digit numerals", Electronic Notes in Theoretical Computer Science 10 (1998), page 2, https://martinescardo.github.io/papers/lexnew.pdf. 17088 (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ph -> H : RR --> RR )
27-Apr-2026	repiecege0 17088	Piecewise definition on the reals agrees with the nonnegative part of the definition. See repiecef 17089 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ( ph /\ A e. RR /\ 0 <_ A ) -> ( H ` A ) = ( G ` A ) )
27-Apr-2026	repiecele0 17087	Piecewise definition on the reals agrees with the nonpositive part of the definition. See repiecef 17089 for more on this construction. (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ( ph /\ A e. RR /\ A <_ 0 ) -> ( H ` A ) = ( F ` A ) )
27-Apr-2026	repiecelem 17086	Lemma for repiecele0 17087, repiecege0 17088, and repiecef 17089. The function H is defined everywhere. (Contributed by Jim Kingdon, 27-Apr-2026.)
|- ( ph -> F : ( -oo (,] 0 ) --> RR )   &    |- ( ph -> G : ( 0 [,) +oo ) --> RR )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   &    |- H = ( x e. RR |-> ( ( ( F ` inf ( { x , 0 } , RR , < ) ) + ( G ` sup ( { x , 0 } , RR , < ) ) ) - ( F ` 0 ) ) )   =>    |- ( ( ph /\ A e. RR ) -> ( ( ( F ` inf ( { A , 0 } , RR , < ) ) + ( G ` sup ( { A , 0 } , RR , < ) ) ) - ( F ` 0 ) ) e. RR )
24-Apr-2026	qdiff 17110	The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to apdiff 17109 but by stating the result positively we can completely sidestep the issue of not equal versus apart in the statement of the result. From an online post by Ingo Blechschmidt. (Contributed by Jim Kingdon, 24-Apr-2026.)
|- ( A e. RR -> ( A e. QQ <-> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) )
23-Apr-2026	exmidpeirce 17050	Excluded middle is equivalent to Peirce's law. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfrompeirce 1499. (Contributed by Jim Kingdon, 23-Apr-2026.)
|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( ( x = 1o -> y = 1o ) -> x = 1o ) -> x = 1o ) )
22-Apr-2026	exmidcon 17049	Excluded middle is equivalent to the form of contraposition which removes negation. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromcon 1498. (Contributed by Jim Kingdon, 22-Apr-2026.)
|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1o ( ( -. y = 1o -> -. x = 1o ) -> ( x = 1o -> y = 1o ) ) )
22-Apr-2026	exmidnotnotr 17048	Excluded middle is equivalent to double negation elimination. Read an element of ~P 1o as being a truth value and x = 1o being that x is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1o, see dcfromnotnotr 1497. (Contributed by Jim Kingdon, 22-Apr-2026.)
|- (EXMID <-> A. x e. ~P 1o ( -. -. x = 1o -> x = 1o ) )
18-Apr-2026	hashtpglem 11300	Lemma for hashtpg 11301. This is one of the three not-equal conclusions required for the reverse direction. (Contributed by Jim Kingdon, 18-Apr-2026.)
|- ( ph -> A e. U )   &    |- ( ph -> B e. V )   &    |- ( ph -> C e. W )   &    |- ( ph -> (♯ ` { A , B , C } ) = 3 )   =>    |- ( ph -> B =/= C )
17-Apr-2026	hashtpgim 11299	The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.) (Revised by Jim Kingdon, 17-Apr-2026.)
|- ( ( A e. U /\ B e. V /\ C e. W ) -> ( ( A =/= B /\ B =/= C /\ C =/= A ) -> (♯ ` { A , B , C } ) = 3 ) )
14-Apr-2026	depind 16762	Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   =>    |- ( ph -> E! f e. X_ n e. NN0 ( P ` n ) ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) )
14-Apr-2026	depindlem3 16761	Lemma for depind 16762. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> A. f e. X_ n e. NN0 ( P ` n ) ( ( ( f ` 0 ) = A /\ A. n e. NN0 ( f ` ( n + 1 ) ) = ( ( H ` n ) ` ( f ` n ) ) ) -> f = F ) )
14-Apr-2026	depindlem2 16760	Lemma for depind 16762. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> F e. X_ n e. NN0 ( P ` n ) )
14-Apr-2026	depindlem1 16759	Lemma for depind 16762. (Contributed by Matthew House, 14-Apr-2026.)
|- ( ph -> P : NN0 --> _V )   &    |- ( ph -> A e. ( P ` 0 ) )   &    |- ( ph -> A. n e. NN0 ( H ` n ) : ( P ` n ) --> ( P ` ( n + 1 ) ) )   &    |- F = seq 0 ( ( x e. _V , h e. _V |-> ( h ` x ) ) , ( m e. NN0 |-> if ( m = 0 , A , ( H ` ( m - 1 ) ) ) ) )   =>    |- ( ph -> ( F : NN0 --> _V /\ ( F ` 0 ) = A /\ A. n e. NN0 ( F ` ( n + 1 ) ) = ( ( H ` n ) ` ( F ` n ) ) ) )
11-Apr-2026	indval0 9298	The indicator function generator does not generate a (meaningful) indicator function for a class which is not a subset of the domain. (Contributed by AV, 11-Apr-2026.)
|- ( -. A C_ O -> ( (𝟭 ` O ) ` A ) = (/) )
8-Apr-2026	gsumclfi 14161	Closure of a finite group sum. (Contributed by Jim Kingdon, 8-Apr-2026.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( G gsumg F ) e. B )
4-Apr-2026	gzsumsplit0 14150	Splitting off the rightmost summand of a group sum (even if it is the only summand). Similar to gzsumsplit1r 13717 except that N can equal M - 1. (Contributed by Jim Kingdon, 4-Apr-2026.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M - 1 ) ) )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) = ( ( G gzsumgz ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
4-Apr-2026	fzf1o 12144	A finite set can be enumerated by integers starting at one. (Contributed by Jim Kingdon, 4-Apr-2026.)
|- ( A e. Fin -> E. f f : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )
3-Apr-2026	gsump1 14159	Splitting off one element from a finite group sum. This would typically used in a proof by induction. (Contributed by Jim Kingdon, 3-Apr-2026.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> F : ( Y u. { Z } ) --> B )   &    |- ( ph -> Y e. Fin )   &    |- ( ph -> Z e. V )   &    |- ( ph -> -. Z e. Y )   =>    |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` Y ) ) .+ ( F ` Z ) ) )
2-Apr-2026	gsumsncmn 14158	Group sum of a singleton. (Contributed by Jim Kingdon, 2-Apr-2026.)
|- B = ( Base ` G )   &    |- ( k = M -> A = C )   =>    |- ( ( G e. CMnd /\ M e. V /\ C e. B ) -> ( G gsumg ( k e. { M } |-> A ) ) = C )
31-Mar-2026	sspw1or2 7544	The set of subsets of a given set with one or two elements can be expressed as elements of the power set or as inhabited elements of the power set. (Contributed by Jim Kingdon, 31-Mar-2026.)
|- { x e. { s e. ~P V | E. j j e. s } | ( x ~~ 1o \/ x ~~ 2o ) } = { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) }
28-Mar-2026	imaf1fi 7240	The image of a finite set under a one-to-one mapping is finite. (Contributed by Jim Kingdon, 28-Mar-2026.)
|- ( ( F : A -1-1-> B /\ X C_ A /\ X e. Fin ) -> ( F " X ) e. Fin )
26-Mar-2026	gsum0cmn 14156	An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
|- ( G e. CMnd -> ( G gsumg (/) ) = ( 0g ` G ) )
26-Mar-2026	gzsumshift 14151	Shifting the indexes of a group sum indexed by consecutive integers. (Contributed by Jim Kingdon, 26-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... N ) --> B )   &    |- S = ( j e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( j - ( 1 - M ) ) )   =>    |- ( ph -> ( G gzsumgz F ) = ( G gzsumgz ( F o. S ) ) )
25-Mar-2026	gzsumgsum 14157	On an integer range, gzsumgz and gsumg agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) = ( G gsumg F ) )
25-Mar-2026	gzsumgsum1 14155	On an integer range starting at one, gzsumgz and gsumg agree. (Contributed by Jim Kingdon, 25-Mar-2026.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> F : ( 1 ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) = ( G gsumg F ) )
24-Mar-2026	gsumvalfi 14154	Value of the finite group sum over an unordered finite set. (Contributed by Jim Kingdon, 24-Mar-2026.)
|- B = ( Base ` W )   &    |- ( ph -> W e. CMnd )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> G : ( 1 ... (♯ ` A ) ) -1-1-onto-> A )   =>    |- ( ph -> ( W gsumg F ) = ( W gzsumgz ( F o. G ) ) )
23-Mar-2026	df-gsumfi 14153	Define the finite group sum (iterated sum) over an unordered finite set. Given G gsumg F where F : A --> ( Base ` G ), the set of indices is A and the values are given by F at each index. For this notation, A is a finite set and G is a commutative monoid, and the sum adds up these elements in some order (the sum does not depend on the order). For a sum indexed by consecutive integers (and thus defining an order for the sum), see df-gzsum 13615. (Contributed by Jim Kingdon, 23-Mar-2026.)
|- gsumg = ( w e. CMnd , f e. _V |-> ( iota x ( dom f e. Fin /\ E. g ( g : ( 1 ... (♯ ` dom f ) ) -1-1-onto-> dom f /\ x = ( w gzsumgz ( f o. g ) ) ) ) ) )
20-Mar-2026	exmidssfi 7246	Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)
|- (EXMID <-> A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) )
18-Mar-2026	umgr1een 16378	A graph with one non-loop edge is a multigraph. (Contributed by Jim Kingdon, 18-Mar-2026.)
|- ( ph -> K e. X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> <. V , { <. K , E >. } >. e. UMGraph )
18-Mar-2026	upgr1een 16377	A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 16374 for a different way of specifying a graph with one edge. (Contributed by Jim Kingdon, 18-Mar-2026.)
|- ( ph -> K e. X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. ~P V )   &    |- ( ph -> E ~~ 2o )   =>    |- ( ph -> <. V , { <. K , E >. } >. e. UPGraph )
14-Mar-2026	trlsex 16640	The class of trails on a graph is a set. (Contributed by Jim Kingdon, 14-Mar-2026.)
|- ( G e. V -> (Trails ` G ) e. _V )
13-Mar-2026	eupthv 16699	The classes involved in a Eulerian path are sets. (Contributed by Jim Kingdon, 13-Mar-2026.)
|- ( F (EulerPaths ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
13-Mar-2026	1hevtxdg0fi 16560	The vertex degree of vertex D in a finite pseudograph G with only one edge E is 0 if D is not incident with the edge E. (Contributed by AV, 2-Mar-2021.) (Revised by Jim Kingdon, 13-Mar-2026.)
|- ( ph -> (iEdg ` G ) = { <. A , E >. } )   &    |- ( ph -> (Vtx ` G ) = V )   &    |- ( ph -> A e. X )   &    |- ( ph -> D e. V )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   &    |- ( ph -> E e. Y )   &    |- ( ph -> D e/ E )   =>    |- ( ph -> ( (VtxDeg ` G ) ` D ) = 0 )
11-Mar-2026	en1hash 11241	A set equinumerous to the ordinal one has size 1 . (Contributed by Jim Kingdon, 11-Mar-2026.)
|- ( A ~~ 1o -> (♯ ` A ) = 1 )
4-Mar-2026	elmpom 6474	If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( D e. F -> E. z z e. A )
22-Feb-2026	isclwwlkni 16660	A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Jim Kingdon, 22-Feb-2026.)
|- ( W e. ( N ClWWalksN G ) -> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) )
21-Feb-2026	clwwlkex 16651	Existence of the set of closed walks (represented by words). (Contributed by Jim Kingdon, 21-Feb-2026.)
|- ( G e. V -> (ClWWalks ` G ) e. _V )
17-Feb-2026	vtxdgfif 16546	In a finite graph, the vertex degree function is a function from vertices to nonnegative integers. (Contributed by Jim Kingdon, 17-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> (VtxDeg ` G ) : V --> NN0 )
16-Feb-2026	vtxlpfi 16543	In a finite graph, the number of loops from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> { x e. A | ( I ` x ) = { U } } e. Fin )
16-Feb-2026	vtxedgfi 16542	In a finite graph, the number of edges from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> { x e. A | U e. ( I ` x ) } e. Fin )
15-Feb-2026	eqsndc 7210	Decidability of equality between a finite subset of a set with decidable equality, and a singleton whose element is an element of the larger set. (Contributed by Jim Kingdon, 15-Feb-2026.)
|- ( ph -> A. x e. B A. y e. B DECID x = y )   &    |- ( ph -> X e. B )   &    |- ( ph -> A C_ B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID A = { X } )
14-Feb-2026	pw1ninf 17033	The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7217), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7213. (Contributed by Jim Kingdon, 14-Feb-2026.)
|- -. om ~<_ ~P 1o
14-Feb-2026	pw1ndom3 17032	The powerset of 1o does not dominate 3o. This is another way of saying that ~P 1o does not have three elements (like pwntru 4336). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
|- -. 3o ~<_ ~P 1o
14-Feb-2026	pw1ndom3lem 17031	Lemma for pw1ndom3 17032. (Contributed by Jim Kingdon, 14-Feb-2026.)
|- ( ph -> X e. ~P 1o )   &    |- ( ph -> Y e. ~P 1o )   &    |- ( ph -> Z e. ~P 1o )   &    |- ( ph -> X =/= Y )   &    |- ( ph -> X =/= Z )   &    |- ( ph -> Y =/= Z )   =>    |- ( ph -> X = (/) )
12-Feb-2026	pw1dceq 17047	The powerset of 1o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)
|- (EXMID <-> A. x e. ~P 1o A. y e. ~P 1oDECID x = y )
12-Feb-2026	3dom 17030	A set that dominates ordinal 3 has at least 3 different members. (Contributed by Jim Kingdon, 12-Feb-2026.)
|- ( 3o ~<_ A -> E. x e. A E. y e. A E. z e. A ( x =/= y /\ x =/= z /\ y =/= z ) )
11-Feb-2026	elssdc 7209	Membership in a finite subset of a set with decidable equality is decidable. (Contributed by Jim Kingdon, 11-Feb-2026.)
|- ( ph -> A. x e. B A. y e. B DECID x = y )   &    |- ( ph -> X e. B )   &    |- ( ph -> A C_ B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID X e. A )
10-Feb-2026	vtxdgfifival 16544	The degree of a vertex for graphs with finite vertex and edge sets. (Contributed by Jim Kingdon, 10-Feb-2026.)
|- V = (Vtx ` G )   &    |- I = (iEdg ` G )   &    |- A = dom I   &    |- ( ph -> A e. Fin )   &    |- ( ph -> V e. Fin )   &    |- ( ph -> U e. V )   &    |- ( ph -> G e. UPGraph )   =>    |- ( ph -> ( (VtxDeg ` G ) ` U ) = ( (♯ ` { x e. A | U e. ( I ` x ) } ) + (♯ ` { x e. A | ( I ` x ) = { U } } ) ) )
10-Feb-2026	fidcen 7203	Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
|- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )
8-Feb-2026	wlkvtxm 16593	A graph with a walk has at least one vertex. (Contributed by Jim Kingdon, 8-Feb-2026.)
|- V = (Vtx ` G )   =>    |- ( F (Walks ` G ) P -> E. x x e. V )
7-Feb-2026	trlsv 16637	The classes involved in a trail are sets. (Contributed by Jim Kingdon, 7-Feb-2026.)
|- ( F (Trails ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
7-Feb-2026	wlkex 16578	The class of walks on a graph is a set. (Contributed by Jim Kingdon, 7-Feb-2026.)
|- ( G e. V -> (Walks ` G ) e. _V )
3-Feb-2026	dom1oi 7117	A set with an element dominates one. (Contributed by Jim Kingdon, 3-Feb-2026.)
|- ( ( A e. V /\ B e. A ) -> 1o ~<_ A )
2-Feb-2026	edginwlkd 16608	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) (Revised by Jim Kingdon, 2-Feb-2026.)
|- I = (iEdg ` G )   &    |- E = (Edg ` G )   &    |- ( ph -> Fun I )   &    |- ( ph -> F e. Word dom I )   &    |- ( ph -> K e. ( 0..^ (♯ ` F ) ) )   &    |- ( ph -> G e. V )   =>    |- ( ph -> ( I ` ( F ` K ) ) e. E )
2-Feb-2026	wlkelvv 16602	A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)
|- ( W e. (Walks ` G ) -> W e. ( _V X. _V ) )
1-Feb-2026	wlkcprim 16603	A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)
|- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
1-Feb-2026	wlkmex 16572	If there are walks on a graph, the graph is a set. (Contributed by Jim Kingdon, 1-Feb-2026.)
|- ( W e. (Walks ` G ) -> G e. _V )
31-Jan-2026	fvmbr 5731	If a function value is inhabited, the argument is related to the function value. (Contributed by Jim Kingdon, 31-Jan-2026.)
|- ( A e. ( F ` X ) -> X F ( F ` X ) )
30-Jan-2026	elfvfvex 5730	If a function value is inhabited, the function value is a set. (Contributed by Jim Kingdon, 30-Jan-2026.)
|- ( A e. ( F ` B ) -> ( F ` B ) e. _V )
30-Jan-2026	reldmm 5000	A relation is inhabited iff its domain is inhabited. (Contributed by Jim Kingdon, 30-Jan-2026.)
|- ( Rel A -> ( E. x x e. A <-> E. y y e. dom A ) )
25-Jan-2026	indconst1 9304	Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
|- ( O e. V -> ( (𝟭 ` O ) ` O ) = ( O X. { 1 } ) )
25-Jan-2026	indconst0 9303	Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
|- ( O e. V -> ( (𝟭 ` O ) ` (/) ) = ( O X. { 0 } ) )
25-Jan-2026	ifp2 993	Forward direction of dfifp2dc 994. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
|- (if- ( ph , ps , ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
25-Jan-2026	ifpdc 992	The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
|- (if- ( ph , ps , ch ) -> DECID ph )
24-Jan-2026	eqabcbw 2376	Version of eqabcb 2377 using implicit substitution, which requires fewer axioms. (Contributed by TM, 24-Jan-2026.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( { x | ph } = A <-> A. y ( ps <-> y e. A ) )
20-Jan-2026	cats1fvd 11540	A symbol other than the last in a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> (♯ ` S ) = M )   &    |- ( ph -> Y e. V )   &    |- ( ph -> X e. W )   &    |- ( ph -> ( S ` N ) = Y )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> N < M )   =>    |- ( ph -> ( T ` N ) = Y )
20-Jan-2026	cats1fvnd 11539	The last symbol of a concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 20-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. V )   &    |- ( ph -> (♯ ` S ) = M )   =>    |- ( ph -> ( T ` M ) = X )
19-Jan-2026	cats2catd 11543	Closure of concatenation of concatenations with singleton words. (Contributed by AV, 1-Mar-2021.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- ( ph -> B e. Word _V )   &    |- ( ph -> D e. Word _V )   &    |- ( ph -> X e. V )   &    |- ( ph -> Y e. W )   &    |- ( ph -> A = ( B ++ <" X "> ) )   &    |- ( ph -> C = ( <" Y "> ++ D ) )   =>    |- ( ph -> ( A ++ C ) = ( ( B ++ <" X Y "> ) ++ D ) )
19-Jan-2026	cats1catd 11542	Closure of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> A e. Word _V )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. W )   &    |- ( ph -> C = ( B ++ <" X "> ) )   &    |- ( ph -> B = ( A ++ S ) )   =>    |- ( ph -> C = ( A ++ T ) )
19-Jan-2026	cats1lend 11541	The length of concatenation with a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 19-Jan-2026.)
|- T = ( S ++ <" X "> )   &    |- ( ph -> S e. Word _V )   &    |- ( ph -> X e. W )   &    |- (♯ ` S ) = M   &    |- ( M + 1 ) = N   =>    |- ( ph -> (♯ ` T ) = N )
18-Jan-2026	rexanaliim 2656	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)
|- ( E. x e. A ( ph /\ -. ps ) -> -. A. x e. A ( ph -> ps ) )
15-Jan-2026	df-uspgren 16408	Define the class of all undirected simple pseudographs (which could have loops). An undirected simple pseudograph is a special undirected pseudograph or a special undirected simple hypergraph, consisting of a set v (of "vertices") and an injective (one-to-one) function e (representing (indexed) "edges") into subsets of v of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a pseudograph, there is at most one edge between two vertices resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by Jim Kingdon, 15-Jan-2026.)
|- USPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
11-Jan-2026	en2prde 7539	A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by Jim Kingdon, 11-Jan-2026.)
|- ( V ~~ 2o -> E. a E. b ( a =/= b /\ V = { a , b } ) )
10-Jan-2026	pw1mapen 17038	Equinumerosity of ( ~P 1o ^m A ) and the set of subsets of A. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( A e. V -> ( ~P 1o ^m A ) ~~ ~P A )
10-Jan-2026	pw1if 7584	Expressing a truth value in terms of an if expression. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( A e. ~P 1o -> if ( A = 1o , 1o , (/) ) = A )
10-Jan-2026	pw1m 7583	A truth value which is inhabited is equal to true. This is a variation of pwntru 4336 and pwtrufal 17039. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )
10-Jan-2026	1ndom2 7166	Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- -. 2o ~<_ 1o
9-Jan-2026	pw1map 17037	Mapping between ( ~P 1o ^m A ) and subsets of A. (Contributed by Jim Kingdon, 9-Jan-2026.)
|- F = ( s e. ( ~P 1o ^m A ) |-> { z e. A | ( s ` z ) = 1o } )   =>    |- ( A e. V -> F : ( ~P 1o ^m A ) -1-1-onto-> ~P A )
9-Jan-2026	iftrueb01 7582	Using an if expression to represent a truth value by (/) or 1o. Unlike some theorems using if, ph does not need to be decidable. (Contributed by Jim Kingdon, 9-Jan-2026.)
|- ( if ( ph , 1o , (/) ) = 1o <-> ph )
8-Jan-2026	pfxclz 11453	Closure of the prefix extractor. This extends pfxclg 11452 from NN0 to ZZ (negative lengths are trivial, resulting in the empty word). (Contributed by Jim Kingdon, 8-Jan-2026.)
|- ( ( S e. Word A /\ L e. ZZ ) -> ( S prefix L ) e. Word A )
8-Jan-2026	fnpfx 11451	The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
|- prefix Fn ( _V X. NN0 )
7-Jan-2026	pr1or2 7540	An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
|- ( ( A e. C /\ B e. D /\ DECID A = B ) -> ( { A , B } ~~ 1o \/ { A , B } ~~ 2o ) )
6-Jan-2026	upgr1elem1 16373	Lemma for upgr1edc 16374. (Contributed by AV, 16-Oct-2020.) (Revised by Jim Kingdon, 6-Jan-2026.)
|- ( ph -> { B , C } e. S )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> DECID B = C )   =>    |- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )
3-Jan-2026	df-umgren 16347	Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set v (of "vertices") and a function e (representing indexed "edges") into subsets of v of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." (Contributed by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
|- UMGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | x ~~ 2o } }
3-Jan-2026	df-upgren 16346	Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set v (of "vertices") and a function e (representing indexed "edges") into subsets of v of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgren 16347). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.) (Revised by Jim Kingdon, 3-Jan-2026.)
|- UPGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { x e. ~P v | ( x ~~ 1o \/ x ~~ 2o ) } }
3-Jan-2026	dom1o 7116	Two ways of saying that a set is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A e. V -> ( 1o ~<_ A <-> E. j j e. A ) )
3-Jan-2026	en2m 7113	A set with two elements is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 2o -> E. x x e. A )
3-Jan-2026	en1m 7092	A set with one element is inhabited. (Contributed by Jim Kingdon, 3-Jan-2026.)
|- ( A ~~ 1o -> E. x x e. A )
31-Dec-2025	pw0ss 16336	There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)
|- { s e. ~P (/) | E. j j e. s } = (/)
31-Dec-2025	df-ushgrm 16323	Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function e is an injective (one-to-one) function into subsets of the set of vertices v, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by Jim Kingdon, 31-Dec-2025.)
|- USHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e -1-1-> { s e. ~P v | E. j j e. s } }
29-Dec-2025	df-uhgrm 16322	Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set v (of "vertices") and a function e (representing indexed "edges") into the set of inhabited subsets of this set. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by Jim Kingdon, 29-Dec-2025.)
|- UHGraph = { g | [. (Vtx ` g ) / v ]. [. (iEdg ` g ) / e ]. e : dom e --> { s e. ~P v | E. j j e. s } }
29-Dec-2025	iedgex 16272	Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (iEdg ` G ) e. _V )
29-Dec-2025	vtxex 16271	Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (Vtx ` G ) e. _V )
29-Dec-2025	snmb 3834	A singleton is inhabited iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.) (Revised by Jim Kingdon, 29-Dec-2025.)
|- ( A e. _V <-> E. x x e. { A } )
27-Dec-2025	lswex 11358	Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11355 or lswcl 11357 if you want more specific results for empty or nonempty words, respectively. (Contributed by Jim Kingdon, 27-Dec-2025.)
|- ( W e. Word V -> (lastS ` W ) e. _V )
23-Dec-2025	fzowrddc 11421	Decidability of whether a range of integers is a subset of a word's domain. (Contributed by Jim Kingdon, 23-Dec-2025.)
|- ( ( S e. Word A /\ F e. ZZ /\ L e. ZZ ) -> DECID ( F..^ L ) C_ dom S )
19-Dec-2025	ccatclab 11364	The concatenation of words over two sets is a word over the union of those sets. (Contributed by Jim Kingdon, 19-Dec-2025.)
|- ( ( S e. Word A /\ T e. Word B ) -> ( S ++ T ) e. Word ( A u. B ) )
18-Dec-2025	lswwrd 11353	Extract the last symbol of a word. (Contributed by Alexander van der Vekens, 18-Mar-2018.) (Revised by Jim Kingdon, 18-Dec-2025.)
|- ( W e. Word V -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )
14-Dec-2025	2strstrndx 13474	A constructed two-slot structure not depending on the hard-coded index value of the base set. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 14-Dec-2025.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. ( Base ` ndx ) , N >. )
12-Dec-2025	funiedgdm2vald 16285	The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 12-Dec-2025.)
|- A e. _V   &    |- B e. _V   &    |- ( ph -> G e. X )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { A , B } C_ dom G )   =>    |- ( ph -> (iEdg ` G ) = (.ef ` G ) )
11-Dec-2025	funvtxdm2vald 16284	The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
|- A e. _V   &    |- B e. _V   &    |- ( ph -> G e. X )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { A , B } C_ dom G )   =>    |- ( ph -> (Vtx ` G ) = ( Base ` G ) )
11-Dec-2025	funiedgdm2domval 16283	The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
11-Dec-2025	funvtxdm2domval 16282	The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )
4-Dec-2025	hash2en 11297	Two equivalent ways to say a set has two elements. (Contributed by Jim Kingdon, 4-Dec-2025.)
|- ( V ~~ 2o <-> ( V e. Fin /\ (♯ ` V ) = 2 ) )
30-Nov-2025	nninfnfiinf 17078	An element of ℕ∞ which is not finite is infinite. (Contributed by Jim Kingdon, 30-Nov-2025.)
|- ( ( A e. ℕ∞ /\ -. E. n e. om A = ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) -> A = ( i e. om |-> 1o ) )
30-Nov-2025	eluz3nn 9969	An integer greater than or equal to 3 is a positive integer. (Contributed by Alexander van der Vekens, 17-Sep-2018.) (Proof shortened by AV, 30-Nov-2025.)
|- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
27-Nov-2025	psrelbasfi 15069	Simpler form of psrelbas 15068 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- ( ph -> I e. Fin )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : ( NN0 ^m I ) --> K )
26-Nov-2025	mplsubgfileminv 15093	Lemma for mplsubgfi 15094. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- N = ( invg ` S )   =>    |- ( ph -> ( N ` X ) e. U )
26-Nov-2025	mplsubgfilemcl 15092	Lemma for mplsubgfi 15094. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( X .+ Y ) e. U )
25-Nov-2025	nninfinfwlpo 7520	The point at infinity in ℕ∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ∞ corresponding to natural numbers are isolated (nninfisol 7473). (Contributed by Jim Kingdon, 25-Nov-2025.)
|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )
23-Nov-2025	psrbagfi 15061	A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. Fin -> D = ( NN0 ^m I ) )
22-Nov-2025	df-acnm 7525	Define a local and length-limited version of the axiom of choice. The definition of the predicate X e. AC A is that for all families of inhabited subsets of X indexed on A (i.e. functions A --> { z e. ~P X | E. j j e. z }), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) Change nonempty to inhabited. (Revised by Jim Kingdon, 22-Nov-2025.)
|- AC A = { x | ( A e. _V /\ A. f e. ( { z e. ~P x | E. j j e. z } ^m A ) E. g A. y e. A ( g ` y ) e. ( f ` y ) ) }
21-Nov-2025	mplsubgfilemm 15091	Lemma for mplsubgfi 15094. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> E. j j e. U )
15-Nov-2025	uzuzle35 9967	An integer greater than or equal to 5 is an integer greater than or equal to 3. (Contributed by AV, 15-Nov-2025.)
|- ( A e. ( ZZ>= ` 5 ) -> A e. ( ZZ>= ` 3 ) )
14-Nov-2025	2omapen 7319	Equinumerosity of ( 2o ^m A ) and the set of decidable subsets of A. (Contributed by Jim Kingdon, 14-Nov-2025.)
|- ( A e. V -> ( 2o ^m A ) ~~ { x e. ~P A | A. y e. A DECID y e. x } )
12-Nov-2025	2omap 7318	Mapping between ( 2o ^m A ) and decidable subsets of A. (Contributed by Jim Kingdon, 12-Nov-2025.)
|- F = ( s e. ( 2o ^m A ) |-> { z e. A | ( s ` z ) = 1o } )   =>    |- ( A e. V -> F : ( 2o ^m A ) -1-1-onto-> { x e. ~P A | A. y e. A DECID y e. x } )
11-Nov-2025	domomsubct 17043	A set dominated by om is subcountable. (Contributed by Jim Kingdon, 11-Nov-2025.)
|- ( A ~<_ om -> E. s ( s C_ om /\ E. f f : s -onto-> A ) )
10-Nov-2025	prdsbaslemss 14176	Lemma for prdsbas 14178 and similar theorems. (Contributed by Jim Kingdon, 10-Nov-2025.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- A = ( E ` P )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( ph -> T e. X )   &    |- ( ph -> { <. ( E ` ndx ) , T >. } C_ P )   =>    |- ( ph -> A = T )
5-Nov-2025	fnmpl 15086	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
4-Nov-2025	mplelbascoe 15085	Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> ( X e. U <-> ( X e. B /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = .0. ) ) ) )
4-Nov-2025	mplbascoe 15084	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
4-Nov-2025	mplvalcoe 15083	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
1-Nov-2025	ficardon 7534	The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
|- ( A e. Fin -> ( card ` A ) e. On )
31-Oct-2025	bitsdc 12716	Whether a bit is set is decidable. (Contributed by Jim Kingdon, 31-Oct-2025.)
|- ( ( N e. ZZ /\ M e. NN0 ) -> DECID M e. (bits ` N ) )
28-Oct-2025	nn0maxcl 11993	The maximum of two nonnegative integers is a nonnegative integer. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> sup ( { A , B } , RR , < ) e. NN0 )
28-Oct-2025	qdcle 10683	Rational <_ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A <_ B )
17-Oct-2025	plycoeid3 15860	Reconstruct a polynomial as an explicit sum of the coefficient function up to an index no smaller than the degree of the polynomial. (Contributed by Jim Kingdon, 17-Oct-2025.)
|- ( ph -> D e. NN0 )   &    |- ( ph -> A : NN0 --> CC )   &    |- ( ph -> ( A " ( ZZ>= ` ( D + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... D ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> M e. ( ZZ>= ` D ) )   &    |- ( ph -> X e. CC )   =>    |- ( ph -> ( F ` X ) = sum_ j e. ( 0 ... M ) ( ( A ` j ) x. ( X ^ j ) ) )
13-Oct-2025	tpfidceq 7237	A triple is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. D )   &    |- ( ph -> B e. D )   &    |- ( ph -> C e. D )   &    |- ( ph -> A. x e. D A. y e. D DECID x = y )   =>    |- ( ph -> { A , B , C } e. Fin )
13-Oct-2025	prfidceq 7235	A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> A. x e. C A. y e. C DECID x = y )   =>    |- ( ph -> { A , B } e. Fin )
13-Oct-2025	dcun 3637	The union of two decidable classes is decidable. (Contributed by Jim Kingdon, 5-Oct-2022.) (Revised by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> DECID C e. A )   &    |- ( ph -> DECID C e. B )   =>    |- ( ph -> DECID C e. ( A u. B ) )
9-Oct-2025	dvdsfi 13019	A natural number has finitely many divisors. (Contributed by Jim Kingdon, 9-Oct-2025.)
|- ( N e. NN -> { x e. NN | x || N } e. Fin )
7-Oct-2025	df-mplcoe 15050	Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). The index set (which has an element for each variable) is i, the coefficients are in ring r, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for r). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)
|- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / w ]_ ( w ↾s { f e. ( Base ` w ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) )
6-Oct-2025	dvconstss 15801	Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)
|- ( ph -> S e. { RR , CC } )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> X e. J )   &    |- ( ph -> A e. CC )   =>    |- ( ph -> ( S _D ( X X. { A } ) ) = ( X X. { 0 } ) )
6-Oct-2025	dcfrompeirce 1499	The decidability of a proposition ch follows from a suitable instance of Peirce's law. Therefore, if we were to introduce Peirce's law as a general principle (without the decidability condition in peircedc 926), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since Perice's law is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
|- ( ph <-> ( ch \/ -. ch ) )   &    |- ( ps <-> F. )   &    |- ( ( ( ph -> ps ) -> ph ) -> ph )   =>    |- DECID ch
6-Oct-2025	dcfromcon 1498	The decidability of a proposition ch follows from a suitable instance of the principle of contraposition. Therefore, if we were to introduce contraposition as a general principle (without the decidability condition in condc 865), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since the principle of contraposition is itself classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
|- ( ph <-> ( ch \/ -. ch ) )   &    |- ( ps <-> T. )   &    |- ( ( -. ph -> -. ps ) -> ( ps -> ph ) )   =>    |- DECID ch
6-Oct-2025	dcfromnotnotr 1497	The decidability of a proposition ps follows from a suitable instance of double negation elimination (DNE). Therefore, if we were to introduce DNE as a general principle (without the decidability condition in notnotrdc 855), then we could prove that every proposition is decidable, giving us the classical system of propositional calculus (since DNE itself is classically valid). (Contributed by Adrian Ducourtial, 6-Oct-2025.)
|- ( ph <-> ( ps \/ -. ps ) )   &    |- ( -. -. ph -> ph )   =>    |- DECID ps
3-Oct-2025	dvidre 15800	Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( RR _D ( _I |` RR ) ) = ( RR X. { 1 } )
3-Oct-2025	dvconstre 15799	Real derivative of a constant function. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( A e. CC -> ( RR _D ( RR X. { A } ) ) = ( RR X. { 0 } ) )
3-Oct-2025	dvidsslem 15796	Lemma for dvconstss 15801. Analogue of dvidlemap 15794 where F is defined on an open subset of the real or complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( ph -> S e. { RR , CC } )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> F : X --> CC )   &    |- ( ph -> X e. J )   &    |- ( ( ph /\ ( x e. X /\ z e. X /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )   &    |- B e. CC   =>    |- ( ph -> ( S _D F ) = ( X X. { B } ) )
3-Oct-2025	dvidrelem 15795	Lemma for dvidre 15800 and dvconstre 15799. Analogue of dvidlemap 15794 for real numbers rather than complex numbers. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( ph -> F : RR --> CC )   &    |- ( ( ph /\ ( x e. RR /\ z e. RR /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )   &    |- B e. CC   =>    |- ( ph -> ( RR _D F ) = ( RR X. { B } ) )
28-Sep-2025	metuex 14894	Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
|- ( A e. V -> (metUnif ` A ) e. _V )
28-Sep-2025	cndsex 14892	The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
|- ( abs o. - ) e. _V
25-Sep-2025	cntopex 14893	The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
|- ( MetOpen ` ( abs o. - ) ) e. _V
24-Sep-2025	mopnset 14891	Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
|- ( D e. V -> ( MetOpen ` D ) e. _V )
24-Sep-2025	blfn 14890	The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
|- ball Fn _V
23-Sep-2025	elfzoext 10612	Membership of an integer in an extended open range of integers, extension added to the right. (Contributed by AV, 30-Apr-2020.) (Proof shortened by AV, 23-Sep-2025.)
|- ( ( Z e. ( M..^ N ) /\ I e. NN0 ) -> Z e. ( M..^ ( N + I ) ) )
22-Sep-2025	plycjlemc 15863	Lemma for plycj 15864. (Contributed by Mario Carneiro, 24-Jul-2014.) (Revised by Jim Kingdon, 22-Sep-2025.)
|- ( ph -> N e. NN0 )   &    |- G = ( ( * o. F ) o. * )   &    |- ( ph -> A : NN0 --> ( S u. { 0 } ) )   &    |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )   &    |- ( ph -> F e. (Poly ` S ) )   =>    |- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( * o. A ) ` k ) x. ( z ^ k ) ) ) )
22-Sep-2025	asclelbas 15028	Lifted scalars are in the base set of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof shortened by Thierry Arnoux, 22-Sep-2025.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- ( ph -> W e. AssAlg )   &    |- ( ph -> C e. B )   =>    |- ( ph -> ( A ` C ) e. ( Base ` W ) )
20-Sep-2025	plycolemc 15861	Lemma for plyco 15862. The result expressed as a sum, with a degree and coefficients for F specified as hypotheses. (Contributed by Jim Kingdon, 20-Sep-2025.)
|- ( ph -> F e. (Poly ` S ) )   &    |- ( ph -> G e. (Poly ` S ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x + y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A : NN0 --> ( S u. { 0 } ) )   &    |- ( ph -> ( A " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )   &    |- ( ph -> F = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( x ^ k ) ) ) )   =>    |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( ( G ` z ) ^ k ) ) ) e. (Poly ` S ) )
18-Sep-2025	elfzoextl 10611	Membership of an integer in an extended open range of integers, extension added to the left. (Contributed by AV, 31-Aug-2025.) Generalized by replacing the left border of the ranges. (Revised by SN, 18-Sep-2025.)
|- ( ( Z e. ( M..^ N ) /\ I e. NN0 ) -> Z e. ( M..^ ( I + N ) ) )
16-Sep-2025	lgsquadlemofi 16207	Lemma for lgsquad 16211. There are finitely many members of S with odd first part. (Contributed by Jim Kingdon, 16-Sep-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- ( ph -> Q e. ( Prime \ { 2 } ) )   &    |- ( ph -> P =/= Q )   &    |- M = ( ( P - 1 ) / 2 )   &    |- N = ( ( Q - 1 ) / 2 )   &    |- S = { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) }   =>    |- ( ph -> { z e. S | -. 2 || ( 1st ` z ) } e. Fin )
16-Sep-2025	lgsquadlemsfi 16206	Lemma for lgsquad 16211. S is finite. (Contributed by Jim Kingdon, 16-Sep-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- ( ph -> Q e. ( Prime \ { 2 } ) )   &    |- ( ph -> P =/= Q )   &    |- M = ( ( P - 1 ) / 2 )   &    |- N = ( ( Q - 1 ) / 2 )   &    |- S = { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) }   =>    |- ( ph -> S e. Fin )
16-Sep-2025	opabfi 7247	Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
|- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> A. x e. A A. y e. B DECID ps )   =>    |- ( ph -> S e. Fin )
13-Sep-2025	uchoice 6371	Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7563 (with the key difference being the change of E. to E!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
|- ( ( A e. V /\ A. x e. A E! y ph ) -> E. f ( f Fn A /\ A. x e. A [. ( f ` x ) / y ]. ph ) )
11-Sep-2025	assa2ass2 15012	Left- and right-associative property of an associative algebra. Notice that the scalars are not commuted! (Contributed by Zhi Wang, 11-Sep-2025.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .* = ( .r ` F )   &    |- .x. = ( .s ` W )   &    |- .X. = ( .r ` W )   =>    |- ( ( W e. AssAlg /\ ( A e. B /\ C e. B ) /\ ( X e. V /\ Y e. V ) ) -> ( ( A .x. X ) .X. ( C .x. Y ) ) = ( ( A .* C ) .x. ( X .X. Y ) ) )
11-Sep-2025	assa2ass 15011	Left- and right-associative property of an associative algebra. Notice that the scalars are commuted! (Contributed by AV, 14-Aug-2019.) (Proof shortened by Zhi Wang, 11-Sep-2025.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .* = ( .r ` F )   &    |- .x. = ( .s ` W )   &    |- .X. = ( .r ` W )   =>    |- ( ( W e. AssAlg /\ ( A e. B /\ C e. B ) /\ ( X e. V /\ Y e. V ) ) -> ( ( A .x. X ) .X. ( C .x. Y ) ) = ( ( C .* A ) .x. ( X .X. Y ) ) )
11-Sep-2025	expghmap 14944	Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)
|- M = (mulGrp ` ℂfld )   &    |- U = ( M ↾s { z e. CC | z # 0 } )   =>    |- ( ( A e. CC /\ A # 0 ) -> ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) )
11-Sep-2025	cnfldui 14926	The invertible complex numbers are exactly those apart from zero. This is recapb 9002 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
|- { z e. CC | z # 0 } = (Unit ` ℂfld )
9-Sep-2025	gzsummhm2 14148	Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 9-Sep-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> ( x e. B |-> C ) e. ( G MndHom H ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> X e. B )   &    |- ( x = X -> C = D )   &    |- ( x = ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) -> C = E )   =>    |- ( ph -> ( H gzsumgz ( k e. ( M ... N ) |-> D ) ) = E )
8-Sep-2025	gzsummhm 14147	Apply a monoid homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 6-Jun-2019.) (Revised by Jim Kingdon, 8-Sep-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> H e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K e. ( G MndHom H ) )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( H gzsumgz ( K o. F ) ) = ( K ` ( G gzsumgz F ) ) )
8-Sep-2025	5ndvds6 12704	5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 6
8-Sep-2025	5ndvds3 12703	5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 3
7-Sep-2025	5eluz3 9963	5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
|- 5 e. ( ZZ>= ` 3 )
6-Sep-2025	gzsumconst 14145	Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Jim Kingdon, 6-Sep-2025.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   =>    |- ( ( G e. Mnd /\ N e. ( ZZ>= ` M ) /\ X e. B ) -> ( G gzsumgz ( k e. ( M ... N ) |-> X ) ) = ( ( ( N - M ) + 1 ) .x. X ) )
5-Sep-2025	uzuzle34 9966	An integer greater than or equal to 4 is an integer greater than or equal to 3. (Contributed by AV, 5-Sep-2025.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 3 ) )
30-Aug-2025	gzsumsubmcl 14144	Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 30-Aug-2025.)
|- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : ( M ... N ) --> S )   =>    |- ( ph -> ( G gzsumgz F ) e. S )
30-Aug-2025	seqm1g 10913	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 30-Aug-2025.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F e. W )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
29-Aug-2025	seqf1og 10960	Rearrange a sum via an arbitrary bijection on ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 29-Aug-2025.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> C C_ S )   &    |- ( ph -> .+ e. V )   &    |- ( ph -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( G ` x ) e. C )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( G ` ( F ` k ) ) )   &    |- ( ph -> G e. W )   &    |- ( ph -> H e. X )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( seq M ( .+ , G ) ` N ) )
25-Aug-2025	irrmulap 10050	The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 10049. (Contributed by Jim Kingdon, 25-Aug-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> A. q e. QQ A # q )   &    |- ( ph -> B e. QQ )   &    |- ( ph -> B =/= 0 )   &    |- ( ph -> Q e. QQ )   =>    |- ( ph -> ( A x. B ) # Q )
19-Aug-2025	seqp1g 10905	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
|- ( ( N e. ( ZZ>= ` M ) /\ F e. V /\ .+ e. W ) -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
19-Aug-2025	seq1g 10902	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 19-Aug-2025.)
|- ( ( M e. ZZ /\ F e. V /\ .+ e. W ) -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
18-Aug-2025	iswrdiz 11313	A zero-based sequence is a word. In iswrdinn0 11311 we can specify a length as an nonnegative integer. However, it will occasionally be helpful to allow a negative length, as well as zero, to specify an empty sequence. (Contributed by Jim Kingdon, 18-Aug-2025.)
|- ( ( W : ( 0..^ L ) --> S /\ L e. ZZ ) -> W e. Word S )
16-Aug-2025	gzsumcl 13806	Closure of an ordered group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 16-Aug-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) e. B )
16-Aug-2025	iswrdinn0 11311	A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Revised by Jim Kingdon, 16-Aug-2025.)
|- ( ( W : ( 0..^ L ) --> S /\ L e. NN0 ) -> W e. Word S )
14-Aug-2025	gzsumfzval 13713	An expression for gzsumgz when summing over a finite set of sequential integers. (Contributed by Jim Kingdon, 14-Aug-2025.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   =>    |- ( ph -> ( G gzsumgz F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )
13-Aug-2025	znidom 14994	The ℤ/nℤ structure is an integral domain when n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Jim Kingdon, 13-Aug-2025.)
|- Y = (ℤ/nℤ ` N )   =>    |- ( N e. Prime -> Y e. IDomn )
12-Aug-2025	rrgmex 14571	A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
|- E = (RLReg ` R )   =>    |- ( A e. E -> R e. _V )
10-Aug-2025	gausslemma2dlem1cl 16190	Lemma for gausslemma2dlem1 16192. Closure of the body of the definition of R. (Contributed by Jim Kingdon, 10-Aug-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   &    |- ( ph -> A e. ( 1 ... H ) )   =>    |- ( ph -> if ( ( A x. 2 ) < ( P / 2 ) , ( A x. 2 ) , ( P - ( A x. 2 ) ) ) e. ZZ )
9-Aug-2025	gausslemma2dlem1f1o 16191	Lemma for gausslemma2dlem1 16192. (Contributed by Jim Kingdon, 9-Aug-2025.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- H = ( ( P - 1 ) / 2 )   &    |- R = ( x e. ( 1 ... H ) |-> if ( ( x x. 2 ) < ( P / 2 ) , ( x x. 2 ) , ( P - ( x x. 2 ) ) ) )   =>    |- ( ph -> R : ( 1 ... H ) -1-1-onto-> ( 1 ... H ) )
7-Aug-2025	qdclt 10682	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A < B )
22-Jul-2025	ivthdich 15756	The intermediate value theorem implies real number dichotomy. Because real number dichotomy (also known as analytic LLPO) is a constructive taboo, this means we will be unable to prove the intermediate value theorem as stated here (although versions with additional conditions, such as ivthinc 15746 for strictly monotonic functions, can be proved). The proof is via a function which we call the hover function and which is also described in Section 5.1 of [Bauer], p. 493. Consider any real number z. We want to show that z <_ 0 \/ 0 <_ z. Because of hovercncf 15749, hovera 15750, and hoverb 15751, we are able to apply the intermediate value theorem to get a value c such that the hover function at c equals z. By axltwlin 8393, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15752 or 0 <_ z by hovergt0 15753. (Contributed by Jim Kingdon and Mario Carneiro, 22-Jul-2025.)
|- ( A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) -> A. r e. RR A. s e. RR ( r <_ s \/ s <_ r ) )
22-Jul-2025	dich0 15755	Real number dichotomy stated in terms of two real numbers or a real number and zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- ( A. z e. RR ( z <_ 0 \/ 0 <_ z ) <-> A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) )
22-Jul-2025	ivthdichlem 15754	Lemma for ivthdich 15756. The result, with a few notational conveniences. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   &    |- ( ph -> Z e. RR )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> ( Z <_ 0 \/ 0 <_ Z ) )
22-Jul-2025	hovergt0 15753	The hover function evaluated at a point greater than zero. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( ( C e. RR /\ 0 < C ) -> 0 <_ ( F ` C ) )
22-Jul-2025	hoverlt1 15752	The hover function evaluated at a point less than one. (Contributed by Jim Kingdon, 22-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( ( C e. RR /\ C < 1 ) -> ( F ` C ) <_ 0 )
21-Jul-2025	hoverb 15751	A point at which the hover function is greater than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( Z e. RR -> Z < ( F ` ( Z + 2 ) ) )
21-Jul-2025	hovera 15750	A point at which the hover function is less than a given value. (Contributed by Jim Kingdon, 21-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- ( Z e. RR -> ( F ` ( Z - 1 ) ) < Z )
21-Jul-2025	rexeqtrrdv 2760	Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> E. x e. A ps )   &    |- ( ph -> B = A )   =>    |- ( ph -> E. x e. B ps )
21-Jul-2025	raleqtrrdv 2759	Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> A. x e. A ps )   &    |- ( ph -> B = A )   =>    |- ( ph -> A. x e. B ps )
21-Jul-2025	rexeqtrdv 2758	Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> E. x e. A ps )   &    |- ( ph -> A = B )   =>    |- ( ph -> E. x e. B ps )
21-Jul-2025	raleqtrdv 2757	Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
|- ( ph -> A. x e. A ps )   &    |- ( ph -> A = B )   =>    |- ( ph -> A. x e. B ps )
20-Jul-2025	hovercncf 15749	The hover function is continuous. By hover function, we mean a a function which starts out as a line of slope one, is constant at zero from zero to one, and then resumes as a slope of one. (Contributed by Jim Kingdon, 20-Jul-2025.)
|- F = ( x e. RR |-> sup ( {inf ( { x , 0 } , RR , < ) , ( x - 1 ) } , RR , < ) )   =>    |- F e. ( RR -cn-> RR )
19-Jul-2025	mincncf 15719	The minimum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 19-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> inf ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
18-Jul-2025	maxcncf 15718	The maximum of two continuous real functions is continuous. (Contributed by Jim Kingdon, 18-Jul-2025.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> RR ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> RR ) )   =>    |- ( ph -> ( x e. X |-> sup ( { A , B } , RR , < ) ) e. ( X -cn-> RR ) )
14-Jul-2025	xnn0nnen 10876	The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)
|- NN0* ~~ NN
12-Jul-2025	nninfninc 7463	All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
|- ( ph -> A e. ℕ∞ )   &    |- ( ph -> X e. om )   &    |- ( ph -> Y e. om )   &    |- ( ph -> X C_ Y )   &    |- ( ph -> ( A ` X ) = (/) )   =>    |- ( ph -> ( A ` Y ) = (/) )
10-Jul-2025	nninfctlemfo 12819	Lemma for nninfct 12820. (Contributed by Jim Kingdon, 10-Jul-2025.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- F = ( n e. om |-> ( i e. om |-> if ( i e. n , 1o , (/) ) ) )   &    |- I = ( ( F o. `' G ) u. { <. +oo , ( om X. { 1o } ) >. } )   =>    |- ( om e. Omni -> I :NN0* -onto->ℕ∞ )
8-Jul-2025	nnnninfen 17076	Equinumerosity of the natural numbers and ℕ∞ is equivalent to the Limited Principle of Omniscience (LPO). Remark in Section 1.1 of [Pradic2025], p. 2. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om ~~ ℕ∞ <-> om e. Omni )
8-Jul-2025	nninfct 12820	The limited principle of omniscience (LPO) implies that ℕ∞ is countable. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- ( om e. Omni -> E. f f : om -onto-> (ℕ∞ ⊔ 1o ) )
8-Jul-2025	nninfinf 10882	ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- om ~<_ ℕ∞
7-Jul-2025	ivthreinc 15748	Restating the intermediate value theorem. Given a hypothesis stating the intermediate value theorem (in a strong form which is not provable given our axioms alone), provide a conclusion similar to the theorem as stated in the Metamath Proof Explorer (which is also similar to how we state the theorem for a strictly monotonic function at ivthinc 15746). Being able to have a hypothesis stating the intermediate value theorem will be helpful when it comes time to show that it implies a constructive taboo. This version of the theorem requires that the function F is continuous on the entire real line, not just ( A [,] B ) which may be an unnecessary condition but which is sufficient for the way we want to use it. (Contributed by Jim Kingdon, 7-Jul-2025.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> F e. ( RR -cn-> RR ) )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ph -> A. f ( f e. ( RR -cn-> RR ) -> A. a e. RR A. b e. RR ( ( a < b /\ ( f ` a ) < 0 /\ 0 < ( f ` b ) ) -> E. x e. RR ( a < x /\ x < b /\ ( f ` x ) = 0 ) ) ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
28-Jun-2025	fngzsum 13710	Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
|- gzsumgz Fn ( _V X. _V )
28-Jun-2025	iotaexel 6043	Set existence of an iota expression in which all values are contained within a set. (Contributed by Jim Kingdon, 28-Jun-2025.)
|- ( ( A e. V /\ A. x ( ph -> x e. A ) ) -> ( iota x ph ) e. _V )
27-Jun-2025	df-gzsum 13615	Define a finite group sum (also called "iterated sum") of a structure. Given G gzsumgz F where F : A --> ( Base ` G ), the set of indices is A and the values are given by F at each index. A group sum over a multiplicative group may be viewed as a product. The definition is meaningful in different contexts, depending on the size of the index set A and each demanding different properties of G. 1. If A = (/) and G has an identity element, then the sum equals this identity. 2. If A = ( M ... N ) and G is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e., ( ( F ` 1 ) + ( F ` 2 ) ) + ( F ` 3 ), etc. 3. This definition does not handle other cases. But see df-gsumfi 14153 for the case where A is a finite set (which need not specify an order) and G is a commutative monoid. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)
|- gzsumgz = ( w e. _V , f e. _V |-> ( iota x ( ( dom f = (/) /\ x = ( 0g ` w ) ) \/ E. m E. n e. ( ZZ>= ` m ) ( dom f = ( m ... n ) /\ x = ( seq m ( ( +g ` w ) , f ) ` n ) ) ) ) )
20-Jun-2025	opprnzrbg 14494	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 14495. (Contributed by SN, 20-Jun-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. NzRing <-> O e. NzRing ) )
16-Jun-2025	fnpsr 15053	The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
|- mPwSer Fn ( _V X. _V )
14-Jun-2025	basm 13416	A structure whose base is inhabited is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
|- B = ( Base ` G )   =>    |- ( A e. B -> E. j j e. G )
14-Jun-2025	elfvm 5729	If a function value has a member, the function is inhabited. (Contributed by Jim Kingdon, 14-Jun-2025.)
|- ( A e. ( F ` B ) -> E. j j e. F )
6-Jun-2025	pcxqcl 13093	The general prime count function is an integer or infinite. (Contributed by Jim Kingdon, 6-Jun-2025.)
|- ( ( P e. Prime /\ N e. QQ ) -> ( ( P pCnt N ) e. ZZ \/ ( P pCnt N ) = +oo ) )
5-Jun-2025	xqltnle 10704	"Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or +oo. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in NN0* or RR*, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)
|- ( ( ( A e. QQ \/ A = +oo ) /\ ( B e. QQ \/ B = +oo ) ) -> ( A < B <-> -. B <_ A ) )
5-Jun-2025	ceqsexv2d 2862	Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025.) (Proof shortened by SN, 5-Jun-2025.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ps   =>    |- E. x ph
31-May-2025	vtocl4ga 2895	Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) (Proof shortened by Wolf Lammen, 31-May-2025.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( y = B -> ( ps <-> ch ) )   &    |- ( z = C -> ( ch <-> rh ) )   &    |- ( w = D -> ( rh <-> th ) )   &    |- ( ( ( x e. Q /\ y e. R ) /\ ( z e. S /\ w e. T ) ) -> ph )   =>    |- ( ( ( A e. Q /\ B e. R ) /\ ( C e. S /\ D e. T ) ) -> th )
30-May-2025	4sqexercise2 13180	Exercise which may help in understanding the proof of 4sqlemsdc 13181. (Contributed by Jim Kingdon, 30-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ n = ( ( x ^ 2 ) + ( y ^ 2 ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )
27-May-2025	iotaexab 5356	Existence of the iota class when all the possible values are contained in a set. (Contributed by Jim Kingdon, 27-May-2025.)
|- ( { x | ph } e. V -> ( iota x ph ) e. _V )
25-May-2025	4sqlemsdc 13181	Lemma for 4sq 13191. The property of being the sum of four squares is decidable. The proof involves showing that (for a particular A) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 13179 and 4sqexercise2 13180 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ E. y e. ZZ E. z e. ZZ E. w e. ZZ n = ( ( ( x ^ 2 ) + ( y ^ 2 ) ) + ( ( z ^ 2 ) + ( w ^ 2 ) ) ) }   =>    |- ( A e. NN0 -> DECID A e. S )
25-May-2025	4sqexercise1 13179	Exercise which may help in understanding the proof of 4sqlemsdc 13181. (Contributed by Jim Kingdon, 25-May-2025.)
|- S = { n | E. x e. ZZ n = ( x ^ 2 ) }   =>    |- ( A e. NN0 -> DECID A e. S )
24-May-2025	4sqleminfi 13178	Lemma for 4sq 13191. A i^i ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ( A i^i ran F ) e. Fin )
24-May-2025	4sqlemffi 13177	Lemma for 4sq 13191. ran F is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   &    |- F = ( v e. A |-> ( ( P - 1 ) - v ) )   =>    |- ( ph -> ran F e. Fin )
24-May-2025	4sqlemafi 13176	Lemma for 4sq 13191. A is finite. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ph -> N e. NN )   &    |- ( ph -> P e. NN )   &    |- A = { u | E. m e. ( 0 ... N ) u = ( ( m ^ 2 ) mod P ) }   =>    |- ( ph -> A e. Fin )
24-May-2025	infidc 7248	The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ( A e. Fin /\ A. x e. A DECID x e. B ) -> ( A i^i B ) e. Fin )
19-May-2025	zrhex 14958	Set existence for ZRHom. (Contributed by Jim Kingdon, 19-May-2025.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L e. _V )
16-May-2025	rhmex 14466	Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
|- ( ( R e. V /\ S e. W ) -> ( R RingHom S ) e. _V )
15-May-2025	ghmex 14060	The set of group homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
|- ( ( S e. Grp /\ T e. Grp ) -> ( S GrpHom T ) e. _V )
15-May-2025	mhmex 13771	The set of monoid homomorphisms exists. (Contributed by Jim Kingdon, 15-May-2025.)
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( S MndHom T ) e. _V )
14-May-2025	idomcringd 14589	An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) (Proof shortened by SN, 14-May-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. CRing )
6-May-2025	rrgnz 14579	In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
|- E = (RLReg ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. NzRing -> -. .0. e. E )
5-May-2025	rngressid 14255	A non-unital ring restricted to its base set is a non-unital ring. It will usually be the original non-unital ring exactly, of course, but to show that needs additional conditions such as those in strressid 13427. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Rng -> ( G ↾s B ) e. Rng )
5-May-2025	ablressid 14141	A commutative group restricted to its base set is a commutative group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13427. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Abel -> ( G ↾s B ) e. Abel )
30-Apr-2025	dvply2g 15869	The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) (Revised by GG, 30-Apr-2025.)
|- ( ( S e. (SubRing ` ℂfld ) /\ F e. (Poly ` S ) ) -> ( CC _D F ) e. (Poly ` S ) )
29-Apr-2025	rlmscabas 14799	Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- ( R e. X -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
29-Apr-2025	ressbasid 13426	The trivial structure restriction leaves the base set unchanged. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- B = ( Base ` W )   =>    |- ( W e. V -> ( Base ` ( W ↾s B ) ) = B )
28-Apr-2025	lssmex 14694	If a linear subspace is inhabited, the class it is built from is a set. (Contributed by Jim Kingdon, 28-Apr-2025.)
|- S = ( LSubSp ` W )   =>    |- ( U e. S -> W e. _V )
27-Apr-2025	cnfldmul 14903	The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
|- x. = ( .r ` ℂfld )
27-Apr-2025	cnfldadd 14901	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
|- + = ( +g ` ℂfld )
27-Apr-2025	lidlex 14812	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> (LIdeal ` W ) e. _V )
27-Apr-2025	lssex 14693	Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> ( LSubSp ` W ) e. _V )
25-Apr-2025	rspex 14813	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. V -> (RSpan ` W ) e. _V )
25-Apr-2025	lspex 14734	Existence of the span of a set of vectors. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. X -> ( LSpan ` W ) e. _V )
25-Apr-2025	eqgex 14026	The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )
25-Apr-2025	qusex 13648	Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )
23-Apr-2025	1dom1el 7107	If a set is dominated by one, then any two of its elements are equal. (Contributed by Jim Kingdon, 23-Apr-2025.)
|- ( ( A ~<_ 1o /\ B e. A /\ C e. A ) -> B = C )
22-Apr-2025	mulgex 13928	Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
|- ( G e. V -> (.g ` G ) e. _V )
21-Apr-2025	uspgruhgr 16440	An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
|- ( G e. USPGraph -> G e. UHGraph )
20-Apr-2025	uspgriedgedg 16432	In a simple pseudograph, for each indexed edge there is exactly one edge. (Contributed by AV, 20-Apr-2025.)
|- E = (Edg ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( G e. USPGraph /\ X e. dom I ) -> E! k e. E k = ( I ` X ) )
20-Apr-2025	uspgredgiedg 16431	In a simple pseudograph, for each edge there is exactly one indexed edge. (Contributed by AV, 20-Apr-2025.)
|- E = (Edg ` G )   &    |- I = (iEdg ` G )   =>    |- ( ( G e. USPGraph /\ K e. E ) -> E! x e. dom I K = ( I ` x ) )
20-Apr-2025	elovmpod 6287	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6288 in deduction form. (Revised by AV, 20-Apr-2025.)
|- O = ( a e. A , b e. B |-> C )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. B )   &    |- ( ph -> D e. V )   &    |- ( ( a = X /\ b = Y ) -> C = D )   =>    |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )
20-Apr-2025	fdmeu 5746	There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025.)
|- ( ( F : A --> B /\ X e. A ) -> E! y e. B ( F ` X ) = y )
18-Apr-2025	fsumdvdsmul 16111	Product of two divisor sums. (This is also the main part of the proof that " sum_ k || N F ( k ) is a multiplicative function if F is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 8302. (Revised by GG, 18-Apr-2025.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. NN )   &    |- ( ph -> ( M gcd N ) = 1 )   &    |- X = { x e. NN | x || M }   &    |- Y = { x e. NN | x || N }   &    |- Z = { x e. NN | x || ( M x. N ) }   &    |- ( ( ph /\ j e. X ) -> A e. CC )   &    |- ( ( ph /\ k e. Y ) -> B e. CC )   &    |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> ( A x. B ) = D )   &    |- ( i = ( j x. k ) -> C = D )   =>    |- ( ph -> ( sum_ j e. X A x. sum_ k e. Y B ) = sum_ i e. Z C )
18-Apr-2025	mpodvdsmulf1o 16110	If M and N are two coprime integers, multiplication forms a bijection from the set of pairs <. j , k >. where j || M and k || N, to the set of divisors of M x. N. (Contributed by GG, 18-Apr-2025.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. NN )   &    |- ( ph -> ( M gcd N ) = 1 )   &    |- X = { x e. NN | x || M }   &    |- Y = { x e. NN | x || N }   &    |- Z = { x e. NN | x || ( M x. N ) }   =>    |- ( ph -> ( ( x e. CC , y e. CC |-> ( x x. y ) ) |` ( X X. Y ) ) : ( X X. Y ) -1-1-onto-> Z )
18-Apr-2025	df2idl2 14848	Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (2Ideal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) ) )
18-Apr-2025	2idlmex 14840	Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- T = (2Ideal ` W )   =>    |- ( U e. T -> W e. _V )
18-Apr-2025	dflidl2 14827	Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) ) )
18-Apr-2025	lidlmex 14814	Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- I = (LIdeal ` W )   =>    |- ( U e. I -> W e. _V )
18-Apr-2025	lsslsp 14768	Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.)
|- X = ( W ↾s U )   &    |- M = ( LSpan ` W )   &    |- N = ( LSpan ` X )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. LMod /\ U e. L /\ G C_ U ) -> ( N ` G ) = ( M ` G ) )
16-Apr-2025	sraex 14785	Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> A e. _V )
14-Apr-2025	grpmgmd 13833	A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mgm )
12-Apr-2025	psraddcl 15073	Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> R e. Mgm )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
10-Apr-2025	cndcap 17121	Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. z e. CC A. w e. CC DECID z # w )
4-Apr-2025	ghmf1 14078	Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 4-Apr-2025.)
|- A = ( Base ` R )   &    |- B = ( Base ` S )   &    |- N = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( F e. ( R GrpHom S ) -> ( F : A -1-1-> B <-> A. x e. A ( ( F ` x ) = .0. -> x = N ) ) )
3-Apr-2025	quscrng 14872	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (LIdeal ` R )   =>    |- ( ( R e. CRing /\ S e. I ) -> U e. CRing )
31-Mar-2025	cnfldds 14907	The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14896. (Revised by GG, 31-Mar-2025.)
|- ( abs o. - ) = ( dist ` ℂfld )
31-Mar-2025	cnfldle 14906	The ordering of the field of complex numbers. Note that this is not actually an ordering on CC, but we put it in the structure anyway because restricting to RR does not affect this component, so that (ℂfld ↾s RR ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14896. (Revised by GG, 31-Mar-2025.)
|- <_ = ( le ` ℂfld )
31-Mar-2025	cnfldtset 14905	The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 31-Mar-2025.)
|- ( MetOpen ` ( abs o. - ) ) = (TopSet ` ℂfld )
31-Mar-2025	mpocnfldmul 14902	The multiplication operation of the field of complex numbers. Version of cnfldmul 14903 using maps-to notation, which does not require ax-mulf 8302. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` ℂfld )
31-Mar-2025	mpocnfldadd 14900	The addition operation of the field of complex numbers. Version of cnfldadd 14901 using maps-to notation, which does not require ax-addf 8301. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x + y ) ) = ( +g ` ℂfld )
31-Mar-2025	df-cnfld 14896	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator. The contract of this set is defined entirely by cnfldex 14898, cnfldadd 14901, cnfldmul 14903, cnfldcj 14904, cnfldtset 14905, cnfldle 14906, cnfldds 14907, and cnfldbas 14899. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)
|- ℂfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. (TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , (metUnif ` ( abs o. - ) ) >. } ) )
31-Mar-2025	2idlcpbl 14863	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
|- X = ( Base ` R )   &    |- E = ( R ~QG S )   &    |- I = (2Ideal ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ S e. I ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )
22-Mar-2025	idomringd 14590	An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Ring )
22-Mar-2025	idomdomd 14588	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
|- ( ph -> R e. IDomn )   =>    |- ( ph -> R e. Domn )
21-Mar-2025	df2idl2rng 14847	Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (2Ideal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) )
21-Mar-2025	isridlrng 14821	A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
21-Mar-2025	dflidl2rng 14820	Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( x .x. y ) e. I ) )
20-Mar-2025	ccoslid 13594	Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- (comp = Slot (comp ` ndx ) /\ (comp ` ndx ) e. NN )
20-Mar-2025	homslid 13591	Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- ( Hom = Slot ( Hom ` ndx ) /\ ( Hom ` ndx ) e. NN )
19-Mar-2025	ptex 13620	Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
|- ( F e. V -> ( Xt_ ` F ) e. _V )
18-Mar-2025	prdsex 14174	Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
|- ( ( S e. V /\ R e. W ) -> ( S X_s R ) e. _V )
16-Mar-2025	plycn 15865	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8302. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
16-Mar-2025	expcn 15672	The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 8302. (Revised by GG, 16-Mar-2025.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )
16-Mar-2025	mpomulcn 15669	Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)
|- J = ( TopOpen ` ℂfld )   =>    |- ( x e. CC , y e. CC |-> ( x x. y ) ) e. ( ( J tX J ) Cn J )
16-Mar-2025	mpomulf 8316	Multiplication is an operation on complex numbers. Version of ax-mulf 8302 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8277. (Contributed by GG, 16-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC
15-Mar-2025	lmodvscld 14644	Closure of scalar product for a left module. (Contributed by SN, 15-Mar-2025.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- .x. = ( .s ` W )   &    |- K = ( Base ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> R e. K )   &    |- ( ph -> X e. V )   =>    |- ( ph -> ( R .x. X ) e. V )
13-Mar-2025	2idlss 14853	A two-sided ideal is a subset of the base set. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
|- B = ( Base ` W )   &    |- I = (2Ideal ` W )   =>    |- ( U e. I -> U C_ B )
13-Mar-2025	imasex 13628	Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
|- ( ( F e. V /\ R e. W ) -> ( F "s R ) e. _V )
11-Mar-2025	rng2idlsubgsubrng 14859	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> I e. (SubGrp ` R ) )   =>    |- ( ph -> I e. (SubRng ` R ) )
11-Mar-2025	rng2idlsubrng 14856	A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> ( R ↾s I ) e. Rng )   =>    |- ( ph -> I e. (SubRng ` R ) )
11-Mar-2025	rnglidlrng 14837	A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption U e. (SubGrp ` R ) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )
11-Mar-2025	rnglidlmsgrp 14836	The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Smgrp )
11-Mar-2025	rnglidlmmgm 14835	The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
11-Mar-2025	imasival 13629	Value of an image structure. The is a lemma for the theorems imasbas 13630, imasplusg 13631, and imasmulr 13632 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- .X. = ( .r ` R )   &    |- .x. = ( .s ` R )   &    |- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )   &    |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )
9-Mar-2025	2idlridld 14846	A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   &    |- O = (oppr ` R )   =>    |- ( ph -> I e. (LIdeal ` O ) )
9-Mar-2025	2idllidld 14845	A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   =>    |- ( ph -> I e. (LIdeal ` R ) )
9-Mar-2025	quseccl 14038	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 9-Mar-2025.)
|- H = ( G /.s ( G ~QG S ) )   &    |- V = ( Base ` G )   &    |- B = ( Base ` H )   =>    |- ( ( S e. (NrmSGrp ` G ) /\ X e. V ) -> [ X ] ( G ~QG S ) e. B )
9-Mar-2025	fovcl 6194	Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.)
|- F : ( R X. S ) --> C   =>    |- ( ( A e. R /\ B e. S ) -> ( A F B ) e. C )
8-Mar-2025	subgex 13981	The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
|- ( G e. Grp -> (SubGrp ` G ) e. _V )
8-Mar-2025	fsuppfund 7294	A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
|- ( ph -> F finSupp Z )   =>    |- ( ph -> Fun F )
7-Mar-2025	ringrzd 14353	The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .x. .0. ) = .0. )
7-Mar-2025	ringlzd 14352	The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .0. .x. X ) = .0. )
7-Mar-2025	qusecsub 14137	Two subgroup cosets are equal if and only if the difference of their representatives is a member of the subgroup. (Contributed by AV, 7-Mar-2025.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- .~ = ( G ~QG S )   =>    |- ( ( ( G e. Abel /\ S e. (SubGrp ` G ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ = [ Y ] .~ <-> ( Y .- X ) e. S ) )
2-Mar-2025	isassad 15013	Sufficient condition for being an associative algebra. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by SN, 2-Mar-2025.)
|- ( ph -> V = ( Base ` W ) )   &    |- ( ph -> F = (Scalar ` W ) )   &    |- ( ph -> B = ( Base ` F ) )   &    |- ( ph -> .x. = ( .s ` W ) )   &    |- ( ph -> .X. = ( .r ` W ) )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   &    |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) )   &    |- ( ( ph /\ ( r e. B /\ x e. V /\ y e. V ) ) -> ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) )   =>    |- ( ph -> W e. AssAlg )
2-Mar-2025	assasca 15010	The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)
|- F = (Scalar ` W )   =>    |- ( W e. AssAlg -> F e. Ring )
2-Mar-2025	isassa 15004	The properties of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- .X. = ( .r ` W )   =>    |- ( W e. AssAlg <-> ( ( W e. LMod /\ W e. Ring ) /\ A. r e. B A. x e. V A. y e. V ( ( ( r .x. x ) .X. y ) = ( r .x. ( x .X. y ) ) /\ ( x .X. ( r .x. y ) ) = ( r .x. ( x .X. y ) ) ) ) )
2-Mar-2025	df-assa 15001	Definition of an associative algebra. An associative algebra is a set equipped with a left-module structure on a ring, coupled with a multiplicative internal operation on the vectors of the module that is associative and distributive for the additive structure of the left-module (so giving the vectors a ring structure) and that is also bilinear under the scalar product. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by SN, 2-Mar-2025.)
|- AssAlg = { w e. ( LMod i^i Ring ) | [. (Scalar ` w ) / f ]. A. r e. ( Base ` f ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) }
1-Mar-2025	quselbasg 14035	Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)
|- .~ = ( G ~QG S )   &    |- U = ( G /.s .~ )   &    |- B = ( Base ` G )   =>    |- ( ( G e. V /\ X e. W /\ S e. Z ) -> ( X e. ( Base ` U ) <-> E. x e. B X = [ x ] .~ ) )
28-Feb-2025	qusmulrng 14871	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14872. Similar to qusmul2 14868. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
|- .~ = ( R ~QG S )   &    |- H = ( R /.s .~ )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` H )   =>    |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )
28-Feb-2025	ringressid 14370	A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 13427. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Ring -> ( G ↾s B ) e. Ring )
28-Feb-2025	grpressid 13868	A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 13427. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
27-Feb-2025	imasringf1 14372	The image of a ring under an injection is a ring. (Contributed by AV, 27-Feb-2025.)
|- U = ( F "s R )   &    |- V = ( Base ` R )   =>    |- ( ( F : V -1-1-> B /\ R e. Ring ) -> U e. Ring )
26-Feb-2025	strext 13461	Extending the upper range of a structure. This works because when we say that a structure has components in A ... C we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)
|- ( ph -> F Struct <. A , B >. )   &    |- ( ph -> C e. ( ZZ>= ` B ) )   =>    |- ( ph -> F Struct <. A , C >. )
25-Feb-2025	subrngringnsg 14515	A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
|- ( A e. (SubRng ` R ) -> A e. (NrmSGrp ` R ) )
25-Feb-2025	rngansg 14251	Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
|- ( R e. Rng -> (NrmSGrp ` R ) = (SubGrp ` R ) )
25-Feb-2025	ecqusaddd 14043	Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)
|- ( ph -> I e. (NrmSGrp ` R ) )   &    |- B = ( Base ` R )   &    |- .~ = ( R ~QG I )   &    |- Q = ( R /.s .~ )   =>    |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> [ ( A ( +g ` R ) C ) ] .~ = ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) )
24-Feb-2025	ecqusaddcl 14044	Closure of the addition in a quotient group. (Contributed by AV, 24-Feb-2025.)
|- ( ph -> I e. (NrmSGrp ` R ) )   &    |- B = ( Base ` R )   &    |- .~ = ( R ~QG I )   &    |- Q = ( R /.s .~ )   =>    |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) e. ( Base ` Q ) )
24-Feb-2025	quseccl0g 14036	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 14038 for arbitrary sets G. (Revised by AV, 24-Feb-2025.)
|- .~ = ( G ~QG S )   &    |- H = ( G /.s .~ )   &    |- C = ( Base ` G )   &    |- B = ( Base ` H )   =>    |- ( ( G e. V /\ X e. C /\ S e. Z ) -> [ X ] .~ e. B )
23-Feb-2025	ltlenmkv 17132	If < can be expressed as holding exactly when <_ holds and the values are not equal, then the analytic Markov's Principle applies. (To get the regular Markov's Principle, combine with neapmkv 17130). (Contributed by Jim Kingdon, 23-Feb-2025.)
|- ( A. x e. RR A. y e. RR ( x < y <-> ( x <_ y /\ y =/= x ) ) -> A. x e. RR A. y e. RR ( x =/= y -> x # y ) )
23-Feb-2025	neap0mkv 17131	The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.)
|- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) <-> A. x e. RR ( x =/= 0 -> x # 0 ) )
23-Feb-2025	qus2idrng 14864	The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14866 analog). (Contributed by AV, 23-Feb-2025.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (2Ideal ` R )   =>    |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )
23-Feb-2025	2idlcpblrng 14862	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
|- X = ( Base ` R )   &    |- E = ( R ~QG S )   &    |- I = (2Ideal ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )
23-Feb-2025	lringuplu 14505	If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> R e. LRing )   &    |- ( ph -> ( X .+ Y ) e. U )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X e. U \/ Y e. U ) )
23-Feb-2025	lringnz 14504	A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. LRing -> .1. =/= .0. )
23-Feb-2025	lringring 14503	A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- ( R e. LRing -> R e. Ring )
23-Feb-2025	lringnzr 14502	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
|- ( R e. LRing -> R e. NzRing )
23-Feb-2025	islring 14501	The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .1. = ( 1r ` R )   &    |- U = (Unit ` R )   =>    |- ( R e. LRing <-> ( R e. NzRing /\ A. x e. B A. y e. B ( ( x .+ y ) = .1. -> ( x e. U \/ y e. U ) ) ) )
23-Feb-2025	df-lring 14500	A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
|- LRing = { r e. NzRing | A. x e. ( Base ` r ) A. y e. ( Base ` r ) ( ( x ( +g ` r ) y ) = ( 1r ` r ) -> ( x e. (Unit ` r ) \/ y e. (Unit ` r ) ) ) }
23-Feb-2025	01eq0ring 14498	If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( ( R e. Ring /\ .0. = .1. ) -> B = { .0. } )
23-Feb-2025	nzrring 14492	A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
|- ( R e. NzRing -> R e. Ring )
23-Feb-2025	qusrng 14259	The quotient structure of a non-unital ring is a non-unital ring (qusring2 14373 analog). (Contributed by AV, 23-Feb-2025.)
|- ( ph -> U = ( R /.s .~ ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   &    |- ( ph -> .~ Er V )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )   &    |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )   &    |- ( ph -> R e. Rng )   =>    |- ( ph -> U e. Rng )
23-Feb-2025	rngsubdir 14253	Ring multiplication distributes over subtraction. (subdir 8713 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14364. (Revised by AV, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )
23-Feb-2025	rngsubdi 14252	Ring multiplication distributes over subtraction. (subdi 8712 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14363. (Revised by AV, 23-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )
23-Feb-2025	abbib 2356	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 1557 and df-clel 2234 (by avoiding use of cleqh 2338). (Revised by BJ, 23-Jun-2019.) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025.)
|- ( { x | ph } = { x | ps } <-> A. x ( ph <-> ps ) )
22-Feb-2025	imasrngf1 14258	The image of a non-unital ring under an injection is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
|- U = ( F "s R )   &    |- V = ( Base ` R )   =>    |- ( ( F : V -1-1-> B /\ R e. Rng ) -> U e. Rng )
22-Feb-2025	imasrng 14257	The image structure of a non-unital ring is a non-unital ring (imasring 14371 analog). (Contributed by AV, 22-Feb-2025.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .x. b ) ) = ( F ` ( p .x. q ) ) ) )   &    |- ( ph -> R e. Rng )   =>    |- ( ph -> U e. Rng )
22-Feb-2025	rngmgpf 14238	Restricted functionality of the multiplicative group on non-unital rings (mgpf 14317 analog). (Contributed by AV, 22-Feb-2025.)
|- (mulGrp |` Rng ) :Rng -->Smgrp
22-Feb-2025	imasabl 14142	The image structure of an abelian group is an abelian group (imasgrp 13916 analog). (Contributed by AV, 22-Feb-2025.)
|- ( ph -> U = ( F "s R ) )   &    |- ( ph -> V = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> F : V -onto-> B )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )   &    |- ( ph -> R e. Abel )   &    |- .0. = ( 0g ` R )   =>    |- ( ph -> ( U e. Abel /\ ( F ` .0. ) = ( 0g ` U ) ) )
21-Feb-2025	prdssgrpd 14193	The product of a family of semigroups is a semigroup. (Contributed by AV, 21-Feb-2025.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I -->Smgrp )   =>    |- ( ph -> Y e. Smgrp )
21-Feb-2025	prdsplusgsgrpcl 14192	Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025.)
|- Y = ( S X_s R )   &    |- B = ( Base ` Y )   &    |- .+ = ( +g ` Y )   &    |- ( ph -> S e. V )   &    |- ( ph -> I e. W )   &    |- ( ph -> R : I -->Smgrp )   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F .+ G ) e. B )
21-Feb-2025	dftap2 7617	Tight apartness with the apartness properties from df-pap 7608 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.)
|- ( R TAp A <-> ( R C_ ( A X. A ) /\ ( A. x e. A -. x R x /\ A. x e. A A. y e. A ( x R y -> y R x ) ) /\ ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) )
20-Feb-2025	rng2idlsubg0 14861	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> I e. (SubGrp ` R ) )   =>    |- ( ph -> ( 0g ` R ) e. I )
20-Feb-2025	rng2idlsubgnsg 14860	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> I e. (SubGrp ` R ) )   =>    |- ( ph -> I e. (NrmSGrp ` R ) )
20-Feb-2025	rng2idl0 14858	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> ( R ↾s I ) e. Rng )   =>    |- ( ph -> ( 0g ` R ) e. I )
20-Feb-2025	rng2idlnsg 14857	A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> ( R ↾s I ) e. Rng )   =>    |- ( ph -> I e. (NrmSGrp ` R ) )
20-Feb-2025	2idlelbas 14855	The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   &    |- J = ( R ↾s I )   &    |- B = ( Base ` J )   =>    |- ( ph -> ( B e. (LIdeal ` R ) /\ B e. (LIdeal ` (oppr ` R ) ) ) )
20-Feb-2025	2idlbas 14854	The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   &    |- J = ( R ↾s I )   &    |- B = ( Base ` J )   =>    |- ( ph -> B = I )
20-Feb-2025	2idlelb 14844	Membership in a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
|- I = (LIdeal ` R )   &    |- O = (oppr ` R )   &    |- J = (LIdeal ` O )   &    |- T = (2Ideal ` R )   =>    |- ( U e. T <-> ( U e. I /\ U e. J ) )
20-Feb-2025	aprap 14600	The relation given by df-apr 14592 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.)
|- ( R e. LRing -> (#r ` R ) Ap ( Base ` R ) )
20-Feb-2025	setscomd 13395	Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.)
|- ( ph -> A e. Y )   &    |- ( ph -> B e. Z )   &    |- ( ph -> S e. V )   &    |- ( ph -> A =/= B )   &    |- ( ph -> C e. W )   &    |- ( ph -> D e. X )   =>    |- ( ph -> ( ( S sSet <. A , C >. ) sSet <. B , D >. ) = ( ( S sSet <. B , D >. ) sSet <. A , C >. ) )
20-Feb-2025	ifnebibdc 3686	The converse of ifbi 3661 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) )
20-Feb-2025	ifnefals 3685	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )
20-Feb-2025	ifnetruedc 3684	Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )
18-Feb-2025	rnglidlmcl 14819	A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven. (Contributed by AV, 18-Feb-2025.)
|- .0. = ( 0g ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( ( R e. Rng /\ I e. U /\ .0. e. I ) /\ ( X e. B /\ Y e. I ) ) -> ( X .x. Y ) e. I )
17-Feb-2025	aprcotr 14599	The apartness relation given by df-apr 14592 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> R e. LRing )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X # Y -> ( X # Z \/ Y # Z ) ) )
17-Feb-2025	aprsym 14598	The apartness relation given by df-apr 14592 for a ring is symmetric. (Contributed by Jim Kingdon, 17-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X # Y -> Y # X ) )
17-Feb-2025	aprval 14593	Expand Definition df-apr 14592. (Contributed by Jim Kingdon, 17-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> .- = ( -g ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X # Y <-> ( X .- Y ) e. U ) )
17-Feb-2025	subrngpropd 14526	If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )   =>    |- ( ph -> (SubRng ` K ) = (SubRng ` L ) )
17-Feb-2025	rngm2neg 14250	Double negation of a product in a non-unital ring (mul2neg 8725 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14362. (Revised by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) )
17-Feb-2025	rngmneg2 14249	Negation of a product in a non-unital ring (mulneg2 8723 analog). In contrast to ringmneg2 14361, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )
17-Feb-2025	rngmneg1 14248	Negation of a product in a non-unital ring (mulneg1 8722 analog). In contrast to ringmneg1 14360, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Rng )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )
16-Feb-2025	aprirr 14597	The apartness relation given by df-apr 14592 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> # = (#r ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> ( 1r ` R ) =/= ( 0g ` R ) )   =>    |- ( ph -> -. X # X )
16-Feb-2025	rngrz 14247	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14351. (Revised by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ X e. B ) -> ( X .x. .0. ) = .0. )
16-Feb-2025	rng0cl 14244	The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Rng -> .0. e. B )
16-Feb-2025	rngacl 14243	Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   =>    |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
16-Feb-2025	rnggrp 14239	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
|- ( R e. Rng -> R e. Grp )
16-Feb-2025	aptap 8979	Complex apartness (as defined at df-ap 8911) is a tight apartness (as defined at df-tap 7615). (Contributed by Jim Kingdon, 16-Feb-2025.)
|- # TAp CC
15-Feb-2025	subsubrng2 14525	The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> (SubRng ` S ) = ( (SubRng ` R ) i^i ~P A ) )
15-Feb-2025	subsubrng 14524	A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> ( B e. (SubRng ` S ) <-> ( B e. (SubRng ` R ) /\ B C_ A ) ) )
15-Feb-2025	subrngin 14523	The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.)
|- ( ( A e. (SubRng ` R ) /\ B e. (SubRng ` R ) ) -> ( A i^i B ) e. (SubRng ` R ) )
15-Feb-2025	subrngintm 14522	The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
|- ( ( S C_ (SubRng ` R ) /\ E. j j e. S ) -> |^| S e. (SubRng ` R ) )
15-Feb-2025	opprsubrngg 14521	Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )
15-Feb-2025	issubrng2 14520	Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Rng -> ( A e. (SubRng ` R ) <-> ( A e. (SubGrp ` R ) /\ A. x e. A A. y e. A ( x .x. y ) e. A ) ) )
15-Feb-2025	opprrngbg 14385	A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14384. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Rng <-> O e. Rng ) )
15-Feb-2025	opprrng 14384	An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. Rng -> O e. Rng )
15-Feb-2025	rngpropd 14256	If two structures have the same base set, and the values of their group (addition) and ring (multiplication) operations are equal for all pairs of elements of the base set, one is a non-unital ring iff the other one is. (Contributed by AV, 15-Feb-2025.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )   =>    |- ( ph -> ( K e. Rng <-> L e. Rng ) )
15-Feb-2025	sgrppropd 13730	If two structures are sets, have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a semigroup iff the other one is. (Contributed by AV, 15-Feb-2025.)
|- ( ph -> K e. V )   &    |- ( ph -> L e. W )   &    |- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   =>    |- ( ph -> ( K e. Smgrp <-> L e. Smgrp ) )
15-Feb-2025	sgrpcl 13726	Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025.)
|- B = ( Base ` G )   &    |- .o. = ( +g ` G )   =>    |- ( ( G e. Smgrp /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B )
15-Feb-2025	tapeq2 7619	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
|- ( A = B -> ( R TAp A <-> R TAp B ) )
14-Feb-2025	subrngmcl 14519	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14543. (Revised by AV, 14-Feb-2025.)
|- .x. = ( .r ` R )   =>    |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .x. Y ) e. A )
14-Feb-2025	subrngacl 14518	A subring is closed under addition. (Contributed by AV, 14-Feb-2025.)
|- .+ = ( +g ` R )   =>    |- ( ( A e. (SubRng ` R ) /\ X e. A /\ Y e. A ) -> ( X .+ Y ) e. A )
14-Feb-2025	subrng0 14517	A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.)
|- S = ( R ↾s A )   &    |- .0. = ( 0g ` R )   =>    |- ( A e. (SubRng ` R ) -> .0. = ( 0g ` S ) )
14-Feb-2025	subrngbas 14516	Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> A = ( Base ` S ) )
14-Feb-2025	subrngsubg 14514	A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
|- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
14-Feb-2025	subrngrcl 14513	Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
|- ( A e. (SubRng ` R ) -> R e. Rng )
14-Feb-2025	subrngrng 14512	A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
|- S = ( R ↾s A )   =>    |- ( A e. (SubRng ` R ) -> S e. Rng )
14-Feb-2025	subrngid 14511	Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   =>    |- ( R e. Rng -> B e. (SubRng ` R ) )
14-Feb-2025	subrngss 14510	A subring is a subset. (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   =>    |- ( A e. (SubRng ` R ) -> A C_ B )
14-Feb-2025	issubrng 14509	The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   =>    |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ B ) )
14-Feb-2025	df-subrng 14508	Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
|- SubRng = ( w e. Rng |-> { s e. ~P ( Base ` w ) | ( w ↾s s ) e. Rng } )
14-Feb-2025	isrngd 14254	Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .+ = ( +g ` R ) )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> R e. Abel )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )   =>    |- ( ph -> R e. Rng )
14-Feb-2025	rngdi 14241	Distributive law for the multiplication operation of a non-unital ring (left-distributivity). (Contributed by AV, 14-Feb-2025.)
|- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .x. ( Y .+ Z ) ) = ( ( X .x. Y ) .+ ( X .x. Z ) ) )
14-Feb-2025	exmidmotap 7627	The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- (EXMID <-> A. x E* r r TAp x )
14-Feb-2025	exmidapne 7626	Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- (EXMID -> ( R TAp A <-> R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) )
14-Feb-2025	df-pap 7608	Apartness predicate. A relation R is an apartness if it is irreflexive, symmetric, and cotransitive. (Contributed by Jim Kingdon, 14-Feb-2025.)
|- ( R Ap A <-> ( ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) /\ ( A. x e. A A. y e. A ( x R y -> y R x ) /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) ) ) )
13-Feb-2025	2idl1 14852	Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
|- I = (2Ideal ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. I )
13-Feb-2025	2idl0 14851	Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
|- I = (2Ideal ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> { .0. } e. I )
13-Feb-2025	ridl1 14850	Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. U )
13-Feb-2025	ridl0 14849	Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> { .0. } e. U )
13-Feb-2025	isridl 14843	A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( y .x. x ) e. I ) ) )
13-Feb-2025	df-apr 14592	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14600. (Contributed by Jim Kingdon, 13-Feb-2025.)
|- #r = ( w e. _V |-> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( x ( -g ` w ) y ) e. (Unit ` w ) ) } )
13-Feb-2025	rngass 14240	Associative law for the multiplication operation of a non-unital ring. (Contributed by NM, 27-Aug-2011.) (Revised by AV, 13-Feb-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .x. Y ) .x. Z ) = ( X .x. ( Y .x. Z ) ) )
13-Feb-2025	issgrpd 13729	Deduce a semigroup from its properties. (Contributed by AV, 13-Feb-2025.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> .+ = ( +g ` G ) )   &    |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )   &    |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> G e. V )   =>    |- ( ph -> G e. Smgrp )
13-Feb-2025	eqab 2373	One direction of eqabb 2374. (Contributed by Wolf Lammen, 13-Feb-2025.)
|- ( A. x ( x e. A <-> ph ) -> A = { x | ph } )
12-Feb-2025	eqabb 2374	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbib 2356 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable ph (that has a free variable x) to a theorem with a class variable A, we substitute x e. A for ph throughout and simplify, where A is a new class variable not already in the wff. An example is the conversion of sepgi 4252 to inex1 4267 (look at the instance of sepgi 4252 that occurs in the proof of inex1 4267). Conversely, to convert a theorem with a class variable A to one with ph, we substitute { x | ph } for A throughout and simplify, where x and ph are new setvar and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 12-Feb-2025.)
|- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) )
8-Feb-2025	2oneel 7622	(/) and 1o are two unequal elements of 2o. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- <. (/) , 1o >. e. { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) }
8-Feb-2025	tapeq1 7618	Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- ( R = S -> ( R TAp A <-> S TAp A ) )
7-Feb-2025	psrgrp 15078	The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> S e. Grp )
7-Feb-2025	resrhm2b 14559	Restriction of the codomain of a (ring) homomorphism. resghm2b 14067 analog. (Contributed by SN, 7-Feb-2025.)
|- U = ( T ↾s X )   =>    |- ( ( X e. (SubRing ` T ) /\ ran F C_ X ) -> ( F e. ( S RingHom T ) <-> F e. ( S RingHom U ) ) )
6-Feb-2025	zzlesq 11148	An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025.)
|- ( N e. ZZ -> N <_ ( N ^ 2 ) )
6-Feb-2025	2omotap 7625	If there is at most one tight apartness on 2o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( E* r r TAp 2o -> EXMID )
6-Feb-2025	2omotaplemst 7624	Lemma for 2omotap 7625. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
6-Feb-2025	2omotaplemap 7623	Lemma for 2omotap 7625. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
6-Feb-2025	2onetap 7621	Negated equality is a tight apartness on 2o. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ u =/= v ) } TAp 2o
5-Feb-2025	netap 7620	Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.)
|- ( A. x e. A A. y e. A DECID x = y -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A )
5-Feb-2025	df-tap 7615	Tight apartness predicate. A relation R is a tight apartness if it is irreflexive, symmetric, cotransitive, and tight. (Contributed by Jim Kingdon, 5-Feb-2025.)
|- ( R TAp A <-> ( R Ap A /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) )
1-Feb-2025	mulgnn0cld 13948	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13943. (Contributed by SN, 1-Feb-2025.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N .x. X ) e. B )
31-Jan-2025	0subg 14004	The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Grp -> { .0. } e. (SubGrp ` G ) )
29-Jan-2025	grprinvd 13863	The right inverse of a group element. Deduction associated with grprinv 13858. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .+ ( N ` X ) ) = .0. )
29-Jan-2025	grplinvd 13862	The left inverse of a group element. Deduction associated with grplinv 13857. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( N ` X ) .+ X ) = .0. )
29-Jan-2025	grpinvcld 13856	A group element's inverse is a group element. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- N = ( invg ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N ` X ) e. B )
29-Jan-2025	grpridd 13841	The identity element of a group is a right identity. Deduction associated with grprid 13839. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .+ .0. ) = X )
29-Jan-2025	grplidd 13840	The identity element of a group is a left identity. Deduction associated with grplid 13838. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .0. .+ X ) = X )
29-Jan-2025	grpassd 13819	A group operation is associative. (Contributed by SN, 29-Jan-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
28-Jan-2025	dvdsrex 14407	Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
|- ( R e. SRing -> ( ||r ` R ) e. _V )
24-Jan-2025	reldvdsrsrg 14401	The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
|- ( R e. SRing -> Rel ( ||r ` R ) )
18-Jan-2025	rerecapb 9174	A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.)
|- ( A e. RR -> ( A # 0 <-> E. x e. RR ( A x. x ) = 1 ) )
18-Jan-2025	recapb 9002	A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.)
|- ( A e. CC -> ( A # 0 <-> E. x e. CC ( A x. x ) = 1 ) )
17-Jan-2025	flddrngd 14617	A field is a division ring. (Contributed by SN, 17-Jan-2025.)
|- ( ph -> R e. Field )   =>    |- ( ph -> R e. DivRing )
17-Jan-2025	ressval3d 13428	Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
|- R = ( S ↾s A )   &    |- B = ( Base ` S )   &    |- E = ( Base ` ndx )   &    |- ( ph -> S e. V )   &    |- ( ph -> Fun S )   &    |- ( ph -> E e. dom S )   &    |- ( ph -> A C_ B )   =>    |- ( ph -> R = ( S sSet <. E , A >. ) )
17-Jan-2025	strressid 13427	Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.)
|- ( ph -> B = ( Base ` W ) )   &    |- ( ph -> W Struct <. M , N >. )   &    |- ( ph -> Fun W )   &    |- ( ph -> ( Base ` ndx ) e. dom W )   =>    |- ( ph -> ( W ↾s B ) = W )
17-Jan-2025	snelpwg 4350	A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 4259. (Revised by BJ, 17-Jan-2025.)
|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )
16-Jan-2025	ressex 13421	Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
|- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) e. _V )
16-Jan-2025	ressvalsets 13420	Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
|- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
15-Jan-2025	vsnex 4348	A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
|- { x } e. _V
12-Jan-2025	isrim 14478	An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
|- B = ( Base ` R )   &    |- C = ( Base ` S )   =>    |- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) )
10-Jan-2025	rimrhm 14480	A ring isomorphism is a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
|- ( F e. ( R RingIso S ) -> F e. ( R RingHom S ) )
10-Jan-2025	isrim0 14470	A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
|- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )
10-Jan-2025	opprex 14380	Existence of the opposite ring. If you know that R is a ring, see opprring 14386. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> O e. _V )
10-Jan-2025	mgpex 14225	Existence of the multiplication group. If R is known to be a semiring, see srgmgp 14274. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
6-Jan-2025	ord3 6699	Ordinal 3 is an ordinal class. (Contributed by BTernaryTau, 6-Jan-2025.)
|- Ord 3o
5-Jan-2025	imbibi 252	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.)
|- ( ( ( ph -> ps ) <-> ch ) -> ( ph -> ( ps <-> ch ) ) )
1-Jan-2025	snss 3850	The singleton of an element of a class is a subset of the class (inference form of snssg 3849). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)
|- A e. _V   =>    |- ( A e. B <-> { A } C_ B )
1-Jan-2025	snssg 3849	The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)
|- ( A e. V -> ( A e. B <-> { A } C_ B ) )
1-Jan-2025	snssb 3848	Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)
|- ( { A } C_ B <-> ( A e. _V -> A e. B ) )
30-Dec-2024	rex2dom 7110	A set that has at least 2 different members dominates ordinal 2. (Contributed by BTernaryTau, 30-Dec-2024.)
|- ( ( A e. V /\ E. x e. A E. y e. A x =/= y ) -> 2o ~<_ A )
23-Dec-2024	en2prd 7106	Two proper unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> D e. Y )   &    |- ( ph -> A =/= B )   &    |- ( ph -> C =/= D )   =>    |- ( ph -> { A , B } ~~ { C , D } )
11-Dec-2024	elopabr 4425	Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)
|- ( A e. { <. x , y >. | x R y } -> A e. R )
10-Dec-2024	cbvreuw 2781	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2784 with a disjoint variable condition. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Dec-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
9-Dec-2024	nninfwlpoim 7519	Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
|- ( A. x e. ℕ∞ A. y e. ℕ∞ DECID x = y -> om e. WOmni )
8-Dec-2024	nninfinfwlpolem 7518	Lemma for nninfinfwlpo 7520. (Contributed by Jim Kingdon, 8-Dec-2024.)
|- ( ph -> F : om --> 2o )   &    |- G = ( i e. om |-> if ( E. x e. suc i ( F ` x ) = (/) , (/) , 1o ) )   &    |- ( ph -> A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) )   =>    |- ( ph -> DECID A. n e. om ( F ` n ) = 1o )
8-Dec-2024	nninfwlpoimlemdc 7517	Lemma for nninfwlpoim 7519. (Contributed by Jim Kingdon, 8-Dec-2024.)
|- ( ph -> F : om --> 2o )   &    |- G = ( i e. om |-> if ( E. x e. suc i ( F ` x ) = (/) , (/) , 1o ) )   &    |- ( ph -> A. x e. ℕ∞ A. y e. ℕ∞ DECID x = y )   =>    |- ( ph -> DECID A. n e. om ( F ` n ) = 1o )
8-Dec-2024	nninfwlpoimlemginf 7516	Lemma for nninfwlpoim 7519. (Contributed by Jim Kingdon, 8-Dec-2024.)
|- ( ph -> F : om --> 2o )   &    |- G = ( i e. om |-> if ( E. x e. suc i ( F ` x ) = (/) , (/) , 1o ) )   =>    |- ( ph -> ( G = ( i e. om |-> 1o ) <-> A. n e. om ( F ` n ) = 1o ) )
8-Dec-2024	nninfwlpoimlemg 7515	Lemma for nninfwlpoim 7519. (Contributed by Jim Kingdon, 8-Dec-2024.)
|- ( ph -> F : om --> 2o )   &    |- G = ( i e. om |-> if ( E. x e. suc i ( F ` x ) = (/) , (/) , 1o ) )   =>    |- ( ph -> G e. ℕ∞ )
7-Dec-2024	nninfwlpor 7514	The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
|- ( om e. WOmni -> A. x e. ℕ∞ A. y e. ℕ∞ DECID x = y )
7-Dec-2024	nninfwlporlem 7513	Lemma for nninfwlpor 7514. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
|- ( ph -> X : om --> 2o )   &    |- ( ph -> Y : om --> 2o )   &    |- D = ( i e. om |-> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) )   &    |- ( ph -> om e. WOmni )   =>    |- ( ph -> DECID X = Y )
7-Dec-2024	domssr 7064	If C is a superset of B and B dominates A, then C also dominates A. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( C e. V /\ B C_ C /\ A ~<_ B ) -> A ~<_ C )
7-Dec-2024	f1dom4g 7039	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 7044 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-> B ) -> A ~<_ B )
7-Dec-2024	f1oen4g 7038	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 7043 does not require the Axiom of Collection nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024.)
|- ( ( ( F e. V /\ A e. W /\ B e. X ) /\ F : A -1-1-onto-> B ) -> A ~~ B )
6-Dec-2024	nninfwlporlemd 7512	Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)
|- ( ph -> X : om --> 2o )   &    |- ( ph -> Y : om --> 2o )   &    |- D = ( i e. om |-> if ( ( X ` i ) = ( Y ` i ) , 1o , (/) ) )   =>    |- ( ph -> ( X = Y <-> D = ( i e. om |-> 1o ) ) )
3-Dec-2024	nninfwlpo 7521	Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
|- ( A. x e. ℕ∞ A. y e. ℕ∞ DECID x = y <-> om e. WOmni )
3-Dec-2024	nninfdcinf 7511	The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
|- ( ph -> om e. WOmni )   &    |- ( ph -> N e. ℕ∞ )   =>    |- ( ph -> DECID N = ( i e. om |-> 1o ) )
29-Nov-2024	brdom2g 7031	Dominance relation. This variation of brdomg 7032 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 7032. (Revised by BTernaryTau, 29-Nov-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
28-Nov-2024	basmexd 13415	A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.)
|- ( ph -> B = ( Base ` G ) )   &    |- ( ph -> A e. B )   =>    |- ( ph -> G e. _V )
23-Nov-2024	fldcrngd 14618	A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
|- ( ph -> R e. Field )   =>    |- ( ph -> R e. CRing )
22-Nov-2024	eliotaeu 5366	An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)
|- ( A e. ( iota x ph ) -> E! x ph )
22-Nov-2024	eliota 5365	An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)
|- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
18-Nov-2024	basmex 13414	A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.)
|- B = ( Base ` G )   =>    |- ( A e. B -> G e. _V )
14-Nov-2024	dcand 945	A conjunction of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.) (Revised by BJ, 14-Nov-2024.)
|- ( ph -> DECID ps )   &    |- ( ph -> DECID ch )   =>    |- ( ph -> DECID ( ps /\ ch ) )
12-Nov-2024	sravscag 14782	The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( .r ` W ) = ( .s ` A ) )
12-Nov-2024	srascag 14781	The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( W ↾s S ) = (Scalar ` A ) )
12-Nov-2024	slotsdifipndx 13531	The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.)
|- ( ( .s ` ndx ) =/= ( .i ` ndx ) /\ (Scalar ` ndx ) =/= ( .i ` ndx ) )
11-Nov-2024	bj-con1st 16791	Contraposition when the antecedent is a negated stable proposition. See con1dc 868. (Contributed by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
11-Nov-2024	slotsdifdsndx 13581	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
|- ( ( *r ` ndx ) =/= ( dist ` ndx ) /\ ( le ` ndx ) =/= ( dist ` ndx ) )
11-Nov-2024	plendxnocndx 13570	The slot for the orthocomplementation is not the slot for the order in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- ( le ` ndx ) =/= ( oc ` ndx )
11-Nov-2024	basendxnocndx 13569	The slot for the orthocomplementation is not the slot for the base set in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- ( Base ` ndx ) =/= ( oc ` ndx )
11-Nov-2024	slotsdifplendx 13566	The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.)
|- ( ( *r ` ndx ) =/= ( le ` ndx ) /\ (TopSet ` ndx ) =/= ( le ` ndx ) )
11-Nov-2024	tsetndxnstarvndx 13550	The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.)
|- (TopSet ` ndx ) =/= ( *r ` ndx )
11-Nov-2024	ofeqd 6304	Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
|- ( ph -> R = S )   =>    |- ( ph -> oF R = oF S )
11-Nov-2024	const 864	Contraposition when the antecedent is a negated stable proposition. See comment of condc 865. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
10-Nov-2024	slotsdifunifndx 13588	The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.)
|- ( ( ( +g ` ndx ) =/= ( UnifSet ` ndx ) /\ ( .r ` ndx ) =/= ( UnifSet ` ndx ) /\ ( *r ` ndx ) =/= ( UnifSet ` ndx ) ) /\ ( ( le ` ndx ) =/= ( UnifSet ` ndx ) /\ ( dist ` ndx ) =/= ( UnifSet ` ndx ) ) )
7-Nov-2024	ressbasd 13423	Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)
|- ( ph -> R = ( W ↾s A ) )   &    |- ( ph -> B = ( Base ` W ) )   &    |- ( ph -> W e. X )   &    |- ( ph -> A e. V )   =>    |- ( ph -> ( A i^i B ) = ( Base ` R ) )
6-Nov-2024	oppraddg 14383	Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
|- O = (oppr ` R )   &    |- .+ = ( +g ` R )   =>    |- ( R e. V -> .+ = ( +g ` O ) )
6-Nov-2024	opprbasg 14382	Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
|- O = (oppr ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. V -> B = ( Base ` O ) )
6-Nov-2024	opprsllem 14381	Lemma for opprbasg 14382 and oppraddg 14383. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
|- O = (oppr ` R )   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( E ` ndx ) =/= ( .r ` ndx )   =>    |- ( R e. V -> ( E ` R ) = ( E ` O ) )
4-Nov-2024	lgsfvalg 16136	Value of the function F which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by Jim Kingdon, 4-Nov-2024.)
|- F = ( n e. NN |-> if ( n e. Prime , ( if ( n = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( n - 1 ) / 2 ) ) + 1 ) mod n ) - 1 ) ) ^ ( n pCnt N ) ) , 1 ) )   =>    |- ( ( A e. ZZ /\ N e. NN /\ M e. NN ) -> ( F ` M ) = if ( M e. Prime , ( if ( M = 2 , if ( 2 || A , 0 , if ( ( A mod 8 ) e. { 1 , 7 } , 1 , -u 1 ) ) , ( ( ( ( A ^ ( ( M - 1 ) / 2 ) ) + 1 ) mod M ) - 1 ) ) ^ ( M pCnt N ) ) , 1 ) )
3-Nov-2024	znmul 14979	The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( .r ` U ) = ( .r ` Y ) )
3-Nov-2024	znadd 14978	The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( +g ` U ) = ( +g ` Y ) )
3-Nov-2024	znbas2 14977	The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   =>    |- ( N e. NN0 -> ( Base ` U ) = ( Base ` Y ) )
3-Nov-2024	znbaslemnn 14976	Lemma for znbas 14981. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) (Revised by AV, 3-Nov-2024.)
|- S = (RSpan ` ℤring )   &    |- U = (ℤring /.s (ℤring ~QG ( S ` { N } ) ) )   &    |- Y = (ℤ/nℤ ` N )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( E ` ndx ) =/= ( le ` ndx )   =>    |- ( N e. NN0 -> ( E ` U ) = ( E ` Y ) )
3-Nov-2024	zlmmulrg 14968	Ring operation of a ZZ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- .x. = ( .r ` G )   =>    |- ( G e. V -> .x. = ( .r ` W ) )
3-Nov-2024	zlmplusgg 14967	Group operation of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- .+ = ( +g ` G )   =>    |- ( G e. V -> .+ = ( +g ` W ) )
3-Nov-2024	zlmbasg 14966	Base set of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- B = ( Base ` G )   =>    |- ( G e. V -> B = ( Base ` W ) )
3-Nov-2024	zlmlemg 14965	Lemma for zlmbasg 14966 and zlmplusgg 14967. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
|- W = ( ZMod ` G )   &    |- E = Slot ( E ` ndx )   &    |- ( E ` ndx ) e. NN   &    |- ( E ` ndx ) =/= (Scalar ` ndx )   &    |- ( E ` ndx ) =/= ( .s ` ndx )   =>    |- ( G e. V -> ( E ` G ) = ( E ` W ) )
2-Nov-2024	zlmsca 14969	Scalar ring of a ZZ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
|- W = ( ZMod ` G )   =>    |- ( G e. V -> ℤring = (Scalar ` W ) )
1-Nov-2024	plendxnvscandx 13565	The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= ( .s ` ndx )
1-Nov-2024	plendxnscandx 13564	The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= (Scalar ` ndx )
1-Nov-2024	plendxnmulrndx 13563	The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.)
|- ( le ` ndx ) =/= ( .r ` ndx )
1-Nov-2024	qsqeqor 11089	The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)
|- ( ( A e. QQ /\ B e. QQ ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( A = B \/ A = -u B ) ) )
31-Oct-2024	dsndxnmulrndx 13578	The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( dist ` ndx ) =/= ( .r ` ndx )
31-Oct-2024	tsetndxnmulrndx 13549	The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) =/= ( .r ` ndx )
31-Oct-2024	tsetndxnbasendx 13547	The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) =/= ( Base ` ndx )
31-Oct-2024	basendxlttsetndx 13546	The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( Base ` ndx ) < (TopSet ` ndx )
31-Oct-2024	tsetndxnn 13545	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) e. NN
30-Oct-2024	basendxltedgfndx 16263	The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)
|- ( Base ` ndx ) < (.ef ` ndx )
30-Oct-2024	plendxnbasendx 13561	The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
|- ( le ` ndx ) =/= ( Base ` ndx )
30-Oct-2024	basendxltplendx 13560	The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
|- ( Base ` ndx ) < ( le ` ndx )
30-Oct-2024	plendxnn 13559	The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.)
|- ( le ` ndx ) e. NN
29-Oct-2024	sradsg 14787	Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( dist ` W ) = ( dist ` A ) )
29-Oct-2024	sratsetg 14784	Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> (TopSet ` W ) = (TopSet ` A ) )
29-Oct-2024	sramulrg 14780	Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( .r ` W ) = ( .r ` A ) )
29-Oct-2024	sraaddgg 14779	Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( +g ` W ) = ( +g ` A ) )
29-Oct-2024	srabaseg 14778	Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   =>    |- ( ph -> ( Base ` W ) = ( Base ` A ) )
29-Oct-2024	sralemg 14777	Lemma for srabaseg 14778 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.)
|- ( ph -> A = ( (subringAlg ` W ) ` S ) )   &    |- ( ph -> S C_ ( Base ` W ) )   &    |- ( ph -> W e. X )   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- (Scalar ` ndx ) =/= ( E ` ndx )   &    |- ( .s ` ndx ) =/= ( E ` ndx )   &    |- ( .i ` ndx ) =/= ( E ` ndx )   =>    |- ( ph -> ( E ` W ) = ( E ` A ) )
29-Oct-2024	dsndxntsetndx 13580	The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( dist ` ndx ) =/= (TopSet ` ndx )
29-Oct-2024	slotsdnscsi 13579	The slots Scalar, .s and .i are different from the slot dist. (Contributed by AV, 29-Oct-2024.)
|- ( ( dist ` ndx ) =/= (Scalar ` ndx ) /\ ( dist ` ndx ) =/= ( .s ` ndx ) /\ ( dist ` ndx ) =/= ( .i ` ndx ) )
29-Oct-2024	slotstnscsi 13551	The slots Scalar, .s and .i are different from the slot TopSet. (Contributed by AV, 29-Oct-2024.)
|- ( (TopSet ` ndx ) =/= (Scalar ` ndx ) /\ (TopSet ` ndx ) =/= ( .s ` ndx ) /\ (TopSet ` ndx ) =/= ( .i ` ndx ) )
29-Oct-2024	ipndxnmulrndx 13530	The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .i ` ndx ) =/= ( .r ` ndx )
29-Oct-2024	ipndxnplusgndx 13529	The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .i ` ndx ) =/= ( +g ` ndx )
29-Oct-2024	vscandxnmulrndx 13517	The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .s ` ndx ) =/= ( .r ` ndx )
29-Oct-2024	scandxnmulrndx 13512	The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- (Scalar ` ndx ) =/= ( .r ` ndx )
29-Oct-2024	fiubnn 11275	A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.)
|- ( ( A C_ NN /\ A e. Fin ) -> E. x e. NN A. y e. A y <_ x )
29-Oct-2024	fiubz 11274	A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.)
|- ( ( A C_ ZZ /\ A e. Fin ) -> E. x e. ZZ A. y e. A y <_ x )
29-Oct-2024	fiubm 11273	Lemma for fiubz 11274 and fiubnn 11275. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.)
|- ( ph -> A C_ B )   &    |- ( ph -> B C_ QQ )   &    |- ( ph -> C e. B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> E. x e. B A. y e. A y <_ x )
28-Oct-2024	edgfndxid 16262	The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)
|- ( G e. V -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
28-Oct-2024	unifndxntsetndx 13587	The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( UnifSet ` ndx ) =/= (TopSet ` ndx )
28-Oct-2024	basendxltunifndx 13585	The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( Base ` ndx ) < ( UnifSet ` ndx )
28-Oct-2024	unifndxnn 13584	The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
|- ( UnifSet ` ndx ) e. NN
28-Oct-2024	dsndxnbasendx 13576	The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
|- ( dist ` ndx ) =/= ( Base ` ndx )
28-Oct-2024	basendxltdsndx 13575	The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.)
|- ( Base ` ndx ) < ( dist ` ndx )
28-Oct-2024	dsndxnn 13574	The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.)
|- ( dist ` ndx ) e. NN
27-Oct-2024	bj-nnst 16783	Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 17028 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( -> , -. ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( -> , <-> , -. )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)
|- -. -. STAB ph
27-Oct-2024	bj-imnimnn 16778	If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 16777 as its last step. (Contributed by BJ, 27-Oct-2024.)
|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps
25-Oct-2024	nnwosdc 12818	Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( ( E. x e. NN ph /\ A. x e. NN DECID ph ) -> E. x e. NN ( ph /\ A. y e. NN ( ps -> x <_ y ) ) )
23-Oct-2024	nnwodc 12815	Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.)
|- ( ( A C_ NN /\ E. w w e. A /\ A. j e. NN DECID j e. A ) -> E. x e. A A. y e. A x <_ y )
22-Oct-2024	uzwodc 12816	Well-ordering principle: any inhabited decidable subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) (Revised by Jim Kingdon, 22-Oct-2024.)
|- ( ( S C_ ( ZZ>= ` M ) /\ E. x x e. S /\ A. x e. ( ZZ>= ` M )DECID x e. S ) -> E. j e. S A. k e. S j <_ k )
21-Oct-2024	nnnotnotr 17028	Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 862, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.)
|- -. -. ( -. -. ph -> ph )
21-Oct-2024	unifndxnbasendx 13586	The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( UnifSet ` ndx ) =/= ( Base ` ndx )
21-Oct-2024	ipndxnbasendx 13528	The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( .i ` ndx ) =/= ( Base ` ndx )
21-Oct-2024	scandxnbasendx 13510	The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- (Scalar ` ndx ) =/= ( Base ` ndx )
20-Oct-2024	isprm5lem 12921	Lemma for isprm5 12922. The interesting direction (showing that one only needs to check prime divisors up to the square root of P). (Contributed by Jim Kingdon, 20-Oct-2024.)
|- ( ph -> P e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A. z e. Prime ( ( z ^ 2 ) <_ P -> -. z || P ) )   &    |- ( ph -> X e. ( 2 ... ( P - 1 ) ) )   =>    |- ( ph -> -. X || P )
19-Oct-2024	resseqnbasd 13429	The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
|- R = ( W ↾s A )   &    |- C = ( E ` W )   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( E ` ndx ) =/= ( Base ` ndx )   &    |- ( ph -> W e. X )   &    |- ( ph -> A e. V )   =>    |- ( ph -> C = ( E ` R ) )
18-Oct-2024	rmodislmod 14690	The right module R induces a left module L by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to Definition df-lmod 14627 of a left module, see also islmod 14629. (Contributed by AV, 3-Dec-2021.) (Proof shortened by AV, 18-Oct-2024.)
|- V = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .s ` R )   &    |- F = (Scalar ` R )   &    |- K = ( Base ` F )   &    |- .+^ = ( +g ` F )   &    |- .X. = ( .r ` F )   &    |- .1. = ( 1r ` F )   &    |- ( R e. Grp /\ F e. Ring /\ A. q e. K A. r e. K A. x e. V A. w e. V ( ( ( w .x. r ) e. V /\ ( ( w .+ x ) .x. r ) = ( ( w .x. r ) .+ ( x .x. r ) ) /\ ( w .x. ( q .+^ r ) ) = ( ( w .x. q ) .+ ( w .x. r ) ) ) /\ ( ( w .x. ( q .X. r ) ) = ( ( w .x. q ) .x. r ) /\ ( w .x. .1. ) = w ) ) )   &    |- .* = ( s e. K , v e. V |-> ( v .x. s ) )   &    |- L = ( R sSet <. ( .s ` ndx ) , .* >. )   =>    |- ( F e. CRing -> L e. LMod )
18-Oct-2024	mgpress 14232	Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.)
|- S = ( R ↾s A )   &    |- M = (mulGrp ` R )   =>    |- ( ( R e. V /\ A e. W ) -> ( M ↾s A ) = (mulGrp ` S ) )
18-Oct-2024	dsndxnplusgndx 13577	The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( dist ` ndx ) =/= ( +g ` ndx )
18-Oct-2024	plendxnplusgndx 13562	The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( le ` ndx ) =/= ( +g ` ndx )
18-Oct-2024	tsetndxnplusgndx 13548	The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- (TopSet ` ndx ) =/= ( +g ` ndx )
18-Oct-2024	vscandxnscandx 13518	The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( .s ` ndx ) =/= (Scalar ` ndx )
18-Oct-2024	vscandxnplusgndx 13516	The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( .s ` ndx ) =/= ( +g ` ndx )
18-Oct-2024	vscandxnbasendx 13515	The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( .s ` ndx ) =/= ( Base ` ndx )
18-Oct-2024	scandxnplusgndx 13511	The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- (Scalar ` ndx ) =/= ( +g ` ndx )
18-Oct-2024	starvndxnmulrndx 13500	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( *r ` ndx ) =/= ( .r ` ndx )
18-Oct-2024	starvndxnplusgndx 13499	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( *r ` ndx ) =/= ( +g ` ndx )
18-Oct-2024	starvndxnbasendx 13498	The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.)
|- ( *r ` ndx ) =/= ( Base ` ndx )
17-Oct-2024	basendxltplusgndx 13469	The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.)
|- ( Base ` ndx ) < ( +g ` ndx )
17-Oct-2024	plusgndxnn 13467	The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.)
|- ( +g ` ndx ) e. NN
17-Oct-2024	elnndc 10014	Membership of an integer in NN is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN )
14-Oct-2024	2zinfmin 12011	Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> inf ( { A , B } , RR , < ) = if ( A <_ B , A , B ) )
14-Oct-2024	mingeb 12010	Equivalence of <_ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> inf ( { A , B } , RR , < ) = A ) )
13-Oct-2024	edgfndxnn 16261	The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
|- (.ef ` ndx ) e. NN
13-Oct-2024	edgfndx 16260	Index value of the df-edgf 16258 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
|- (.ef ` ndx ) = ; 1 8
13-Oct-2024	prdsvallem 13623	Lemma for prdsval 14175. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 14175, dependency on df-hom 13457 removed. (Revised by AV, 13-Oct-2024.)
|- ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) e. _V
13-Oct-2024	pcxnn0cl 13091	Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( P e. Prime /\ N e. ZZ ) -> ( P pCnt N ) e. NN0* )
13-Oct-2024	xnn0letri 10207	Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) )
13-Oct-2024	xnn0dcle 10206	Decidability of <_ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
9-Oct-2024	nn0leexp2 11150	Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.)
|- ( ( ( A e. RR /\ M e. NN0 /\ N e. NN0 ) /\ 1 < A ) -> ( M <_ N <-> ( A ^ M ) <_ ( A ^ N ) ) )
8-Oct-2024	pclemdc 13069	Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- A = { n e. NN0 | ( P ^ n ) || N }   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. A )
8-Oct-2024	elnn0dc 10013	Membership of an integer in NN0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN0 )
7-Oct-2024	pclemub 13068	Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
|- A = { n e. NN0 | ( P ^ n ) || N }   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. A y <_ x )
7-Oct-2024	pclem0 13067	Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.)
|- A = { n e. NN0 | ( P ^ n ) || N }   =>    |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> 0 e. A )
7-Oct-2024	nn0ltexp2 11149	Special case of ltexp2 16049 which we use here because we haven't yet defined df-rpcxp 15963 which is used in the current proof of ltexp2 16049. (Contributed by Jim Kingdon, 7-Oct-2024.)
|- ( ( ( A e. RR /\ M e. NN0 /\ N e. NN0 ) /\ 1 < A ) -> ( M < N <-> ( A ^ M ) < ( A ^ N ) ) )
6-Oct-2024	suprzcl2dc 10676	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8300.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
|- ( ph -> A C_ ZZ )   &    |- ( ph -> A. x e. ZZ DECID x e. A )   &    |- ( ph -> E. x e. ZZ A. y e. A y <_ x )   &    |- ( ph -> E. x x e. A )   =>    |- ( ph -> sup ( A , RR , < ) e. A )
5-Oct-2024	zsupssdc 10675	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8300.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
|- ( ph -> A C_ ZZ )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ZZ DECID x e. A )   &    |- ( ph -> E. x e. ZZ A. y e. A y <_ x )   =>    |- ( ph -> E. x e. A ( A. y e. A -. x < y /\ A. y e. B ( y < x -> E. z e. A y < z ) ) )
5-Oct-2024	suprzubdc 10673	The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
|- ( ph -> A C_ ZZ )   &    |- ( ph -> A. x e. ZZ DECID x e. A )   &    |- ( ph -> E. x e. ZZ A. y e. A y <_ x )   &    |- ( ph -> B e. A )   =>    |- ( ph -> B <_ sup ( A , RR , < ) )
3-Oct-2024	ab0w 3550	The class of sets verifying a property is the empty class if and only if that property is a contradiction. (Contributed by GG, 3-Oct-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( { x | ph } = (/) <-> A. y -. ps )
1-Oct-2024	infex2g 7374	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
|- ( A e. C -> inf ( B , A , R ) e. _V )
30-Sep-2024	unbendc 13347	An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ A. m e. NN E. n e. A m < n ) -> A ~~ NN )
30-Sep-2024	prmdc 12910	Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( N e. NN -> DECID N e. Prime )
30-Sep-2024	dcfi 7315	Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( ( A e. Fin /\ A. x e. A DECID ph ) -> DECID A. x e. A ph )
30-Sep-2024	cbvriotavw 6049	Change bound variable in a restricted description binder. Version of cbvriotav 6051 with a disjoint variable condition. (Contributed by NM, 18-Mar-2013.) (Revised by GG, 30-Sep-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )
30-Sep-2024	cbviotavw 5343	Change bound variables in a description binder. Version of cbviotav 5344 with a disjoint variable condition. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by GG, 30-Sep-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( iota x ph ) = ( iota y ps )
29-Sep-2024	ssnnct 13340	A decidable subset of NN is countable. (Contributed by Jim Kingdon, 29-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
29-Sep-2024	ssnnctlemct 13339	Lemma for ssnnct 13340. The result. (Contributed by Jim Kingdon, 29-Sep-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 1 )   =>    |- ( ( A C_ NN /\ A. x e. NN DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
28-Sep-2024	nninfdcex 10674	A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> E. y y e. A )   =>    |- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )
27-Sep-2024	infregelbex 10000	Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.)
|- ( ph -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( B <_ inf ( A , RR , < ) <-> A. z e. A B <_ z ) )
26-Sep-2024	nninfdclemp1 13343	Lemma for nninfdc 13346. Each element of the sequence F is greater than the previous element. (Contributed by Jim Kingdon, 26-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   &    |- ( ph -> U e. NN )   =>    |- ( ph -> ( F ` U ) < ( F ` ( U + 1 ) ) )
26-Sep-2024	nnminle 12814	The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12813. (Contributed by Jim Kingdon, 26-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ B e. A ) -> inf ( A , RR , < ) <_ B )
25-Sep-2024	nninfdclemcl 13341	Lemma for nninfdc 13346. (Contributed by Jim Kingdon, 25-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> P e. A )   &    |- ( ph -> Q e. A )   =>    |- ( ph -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) e. A )
24-Sep-2024	nninfdclemlt 13344	Lemma for nninfdc 13346. The function from nninfdclemf 13342 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   &    |- ( ph -> U e. NN )   &    |- ( ph -> V e. NN )   &    |- ( ph -> U < V )   =>    |- ( ph -> ( F ` U ) < ( F ` V ) )
23-Sep-2024	nninfdc 13346	An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ A. m e. NN E. n e. A m < n ) -> om ~<_ A )
23-Sep-2024	nninfdclemf1 13345	Lemma for nninfdc 13346. The function from nninfdclemf 13342 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   =>    |- ( ph -> F : NN -1-1-> A )
23-Sep-2024	nninfdclemf 13342	Lemma for nninfdc 13346. A function from the natural numbers into A. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ph -> A C_ NN )   &    |- ( ph -> A. x e. NN DECID x e. A )   &    |- ( ph -> A. m e. NN E. n e. A m < n )   &    |- ( ph -> ( J e. A /\ 1 < J ) )   &    |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )   =>    |- ( ph -> F : NN --> A )
23-Sep-2024	nnmindc 12813	An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.)
|- ( ( A C_ NN /\ A. x e. NN DECID x e. A /\ E. y y e. A ) -> inf ( A , RR , < ) e. A )
23-Sep-2024	breng 7029	Equinumerosity relation. This variation of bren 7030 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 7030. (Revised by BTernaryTau, 23-Sep-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
23-Sep-2024	dfnul3 3524	Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
|- (/) = { x e. A | -. x e. A }
23-Sep-2024	dfnul2 3523	Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Reduce axiom dependencies. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
|- (/) = { x | -. x = x }
23-Sep-2024	dfnul4 3522	Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-13 2211, df-clel 2234. (Revised by GG, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 3523. (Revised by BJ, 23-Sep-2024.)
|- (/) = { x | F. }
20-Sep-2024	ineqcomi 3423	Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 3422. Disjointness inference when C = (/). (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)
|- ( A i^i B ) = C   =>    |- ( B i^i A ) = C
19-Sep-2024	ssomct 13338	A decidable subset of om is countable. (Contributed by Jim Kingdon, 19-Sep-2024.)
|- ( ( A C_ om /\ A. x e. om DECID x e. A ) -> E. f f : om -onto-> ( A ⊔ 1o ) )
19-Sep-2024	2oex 6704	2o is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
|- 2o e. _V
19-Sep-2024	ecase2d 1392	Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)
|- ( ph -> ps )   &    |- ( ph -> -. ( ps /\ ch ) )   &    |- ( ph -> -. ( ps /\ th ) )   &    |- ( ph -> ( ta \/ ( ch \/ th ) ) )   =>    |- ( ph -> ta )
18-Sep-2024	fcof 5894	Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5552. (Contributed by AV, 18-Sep-2024.)
|- ( ( F : A --> B /\ Fun G ) -> ( F o. G ) : ( `' G " A ) --> B )
18-Sep-2024	eqabbw 2375	Version of eqabb 2374 using implicit substitution. (Contributed by GG and AV, 18-Sep-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) )
17-Sep-2024	fncofn 5893	Composition of a function with domain and a function as a function with domain. Generalization of fnco 5491. (Contributed by AV, 17-Sep-2024.)
|- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) )
14-Sep-2024	nnpredlt 4771	The predecessor (see nnpredcl 4770) of a nonzero natural number is less than (see df-iord 4511) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
13-Sep-2024	nninfisollemeq 7472	Lemma for nninfisol 7473. The case where N is a successor and N and X are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
|- ( ph -> X e. ℕ∞ )   &    |- ( ph -> ( X ` N ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> N =/= (/) )   &    |- ( ph -> ( X ` U. N ) = 1o )   =>    |- ( ph -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
13-Sep-2024	nninfisollemne 7471	Lemma for nninfisol 7473. A case where N is a successor and N and X are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
|- ( ph -> X e. ℕ∞ )   &    |- ( ph -> ( X ` N ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> N =/= (/) )   &    |- ( ph -> ( X ` U. N ) = (/) )   =>    |- ( ph -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
13-Sep-2024	nninfisollem0 7470	Lemma for nninfisol 7473. The case where N is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
|- ( ph -> X e. ℕ∞ )   &    |- ( ph -> ( X ` N ) = (/) )   &    |- ( ph -> N e. om )   &    |- ( ph -> N = (/) )   =>    |- ( ph -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
13-Sep-2024	fset0 6949	The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)
|- { f | f : (/) --> B } = { (/) }
13-Sep-2024	fsetsspwxp 6948	The class of all functions from A into B is a subclass of the power class of the cartesion product of A and B. (Contributed by AV, 13-Sep-2024.)
|- { f | f : A --> B } C_ ~P ( A X. B )
12-Sep-2024	nninfisol 7473	Finite elements of ℕ∞ are isolated. That is, given a natural number and any element of ℕ∞, it is decidable whether the natural number (when converted to an element of ℕ∞) is equal to the given element of ℕ∞. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence X to decide whether it is equal to N (in fact, you only need to look at two elements and N tells you where to look). By contrast, the point at infinity being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO) (nninfinfwlpo 7520). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
|- ( ( N e. om /\ X e. ℕ∞ ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
8-Sep-2024	relopabv 4904	A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4906. (Contributed by SN, 8-Sep-2024.)
|- Rel { <. x , y >. | ph }
7-Sep-2024	eulerthlemfi 13008	Lemma for eulerth 13013. The set S is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   =>    |- ( ph -> S e. Fin )
7-Sep-2024	modqexp 11106	Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. NN0 )   &    |- ( ph -> D e. QQ )   &    |- ( ph -> 0 < D )   &    |- ( ph -> ( A mod D ) = ( B mod D ) )   =>    |- ( ph -> ( ( A ^ C ) mod D ) = ( ( B ^ C ) mod D ) )
5-Sep-2024	eulerthlemh 13011	Lemma for eulerth 13013. A permutation of ( 1 ... ( phi ` N ) ). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   &    |- H = ( `' F o. ( y e. ( 1 ... ( phi ` N ) ) |-> ( ( A x. ( F ` y ) ) mod N ) ) )   =>    |- ( ph -> H : ( 1 ... ( phi ` N ) ) -1-1-onto-> ( 1 ... ( phi ` N ) ) )
2-Sep-2024	eulerthlemth 13012	Lemma for eulerth 13013. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( ( A ^ ( phi ` N ) ) mod N ) = ( 1 mod N ) )
2-Sep-2024	eulerthlema 13010	Lemma for eulerth 13013. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( ( ( A ^ ( phi ` N ) ) x. prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) ) mod N ) = ( prod_ x e. ( 1 ... ( phi ` N ) ) ( ( A x. ( F ` x ) ) mod N ) mod N ) )
2-Sep-2024	eulerthlemrprm 13009	Lemma for eulerth 13013. N and prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )   &    |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }   &    |- ( ph -> F : ( 1 ... ( phi ` N ) ) -1-1-onto-> S )   =>    |- ( ph -> ( N gcd prod_ x e. ( 1 ... ( phi ` N ) ) ( F ` x ) ) = 1 )
1-Sep-2024	qusmul2 14868	Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
|- Q = ( R /.s ( R ~QG I ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .X. = ( .r ` Q )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( [ X ] ( R ~QG I ) .X. [ Y ] ( R ~QG I ) ) = [ ( X .x. Y ) ] ( R ~QG I ) )
30-Aug-2024	fprodap0f 12405	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12390 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
|- F/ k ph   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A B # 0 )
28-Aug-2024	fprodrec 12398	The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
|- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. A ) -> B # 0 )   =>    |- ( ph -> prod_ k e. A ( 1 / B ) = ( 1 / prod_ k e. A B ) )
26-Aug-2024	exmidontri2or 7602	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
26-Aug-2024	exmidontri 7598	Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- (EXMID <-> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
26-Aug-2024	ontri2orexmidim 4719	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4718. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID ph )
26-Aug-2024	ontriexmidim 4669	Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4668. (Contributed by Jim Kingdon, 26-Aug-2024.)
|- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> DECID ph )
25-Aug-2024	onntri2or 7605	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
|- ( -. -. EXMID <-> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) )
25-Aug-2024	onntri3or 7604	Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.)
|- ( -. -. EXMID <-> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) )
25-Aug-2024	csbcow 3158	Composition law for chained substitutions into a class. Version of csbco 3157 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by GG, 25-Aug-2024.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
25-Aug-2024	cbvreuvw 2792	Version of cbvreuv 2788 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
25-Aug-2024	cbvrexvw 2791	Version of cbvrexv 2787 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
25-Aug-2024	cbvralvw 2790	Version of cbvralv 2786 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
25-Aug-2024	cbvabw 2363	Version of cbvab 2364 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x | ph } = { y | ps }
25-Aug-2024	nfsbv 2007	If z is not free in ph, it is not free in [ y / x ] ph when z is distinct from x and y. Version of nfsb 2006 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on x , y. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
|- F/ z ph   =>    |- F/ z [ y / x ] ph
25-Aug-2024	cbvexvw 1976	Change bound variable. See cbvexv 1974 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1501. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
25-Aug-2024	cbvalvw 1975	Change bound variable. See cbvalv 1973 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1501. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
25-Aug-2024	nfal 1629	If x is not free in ph, it is not free in A. y ph. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1563. (Revised by GG, 25-Aug-2024.)
|- F/ x ph   =>    |- F/ x A. y ph
24-Aug-2024	gcdcomd 12753	The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd N ) = ( N gcd M ) )
22-Aug-2024	fsetdmprc0 6950	The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
|- ( A e/ _V -> { f | f Fn A } = (/) )
22-Aug-2024	abanssr 3502	A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
|- { f | ( ph /\ ps ) } C_ { f | ps }
22-Aug-2024	abanssl 3501	A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
|- { f | ( ph /\ ps ) } C_ { f | ph }
21-Aug-2024	dvds2addd 12598	Deduction form of dvds2add 12594. (Contributed by SN, 21-Aug-2024.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K || M )   &    |- ( ph -> K || N )   =>    |- ( ph -> K || ( M + N ) )
20-Aug-2024	rspcedvdw 2936	Version of rspcedvd 2935 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on ph , x. (Contributed by SN, 20-Aug-2024.)
|- ( x = A -> ( ps <-> ch ) )   &    |- ( ph -> A e. B )   &    |- ( ph -> ch )   =>    |- ( ph -> E. x e. B ps )
18-Aug-2024	prdsmulr 14180	Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- B = ( Base ` P )   &    |- ( ph -> dom R = I )   &    |- .x. = ( .r ` P )   =>    |- ( ph -> .x. = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )
18-Aug-2024	prdsplusg 14179	Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- B = ( Base ` P )   &    |- ( ph -> dom R = I )   &    |- .+ = ( +g ` P )   =>    |- ( ph -> .+ = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )
18-Aug-2024	prdsbas 14178	Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   &    |- B = ( Base ` P )   &    |- ( ph -> dom R = I )   =>    |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )
18-Aug-2024	prdssca 14177	Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- ( ph -> S e. V )   &    |- ( ph -> R e. W )   =>    |- ( ph -> S = (Scalar ` P ) )
18-Aug-2024	prdsval 14175	Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- P = ( S X_s R )   &    |- K = ( Base ` S )   &    |- ( ph -> dom R = I )   &    |- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) )   &    |- ( ph -> .+ = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) )   &    |- ( ph -> .X. = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) )   &    |- ( ph -> .x. = ( f e. K , g e. B |-> ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) )   &    |- ( ph -> ., = ( f e. B , g e. B |-> ( S gsumg ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) )   &    |- ( ph -> O = ( Xt_ ` ( TopOpen o. R ) ) )   &    |- ( ph -> .<_ = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } )   &    |- ( ph -> D = ( f e. B , g e. B |-> sup ( ( ran ( x e. I |-> ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) )   &    |- ( ph -> H = ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) )   &    |- ( ph -> .xb = ( a e. ( B X. B ) , c e. B |-> ( d e. ( ( 2nd ` a ) H c ) , e e. ( H ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) )   &    |- ( ph -> S e. W )   &    |- ( ph -> R e. Z )   =>    |- ( ph -> P = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) u. ( { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } u. { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .xb >. } ) ) )
18-Aug-2024	df-prds 14172	Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
|- X_s = ( s e. _V , r e. _V |-> [_ X_ x e. dom r ( Base ` ( r ` x ) ) / v ]_ [_ ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) / h ]_ ( ( { <. ( Base ` ndx ) , v >. , <. ( +g ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( +g ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .r ` ndx ) , ( f e. v , g e. v |-> ( x e. dom r |-> ( ( f ` x ) ( .r ` ( r ` x ) ) ( g ` x ) ) ) ) >. } u. { <. (Scalar ` ndx ) , s >. , <. ( .s ` ndx ) , ( f e. ( Base ` s ) , g e. v |-> ( x e. dom r |-> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) ) ) >. , <. ( .i ` ndx ) , ( f e. v , g e. v |-> ( s gsumg ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) >. } ) u. ( { <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. r ) ) >. , <. ( le ` ndx ) , { <. f , g >. | ( { f , g } C_ v /\ A. x e. dom r ( f ` x ) ( le ` ( r ` x ) ) ( g ` x ) ) } >. , <. ( dist ` ndx ) , ( f e. v , g e. v |-> sup ( ( ran ( x e. dom r |-> ( ( f ` x ) ( dist ` ( r ` x ) ) ( g ` x ) ) ) u. { 0 } ) , RR* , < ) ) >. } u. { <. ( Hom ` ndx ) , h >. , <. (comp ` ndx ) , ( a e. ( v X. v ) , c e. v |-> ( d e. ( ( 2nd ` a ) h c ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. (comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) )
17-Aug-2024	fprodcl2lem 12374	Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
|- ( ph -> S C_ CC )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x x. y ) e. S )   &    |- ( ph -> A e. Fin )   &    |- ( ( ph /\ k e. A ) -> B e. S )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> prod_ k e. A B e. S )
16-Aug-2024	fprodunsn 12373	Multiply in an additional term in a finite product. See also fprodsplitsn 12402 which is the same but with a F/ k ph hypothesis in place of the distinct variable condition between ph and k. (Contributed by Jim Kingdon, 16-Aug-2024.)
|- F/_ k D   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. V )   &    |- ( ph -> -. B e. A )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( k = B -> C = D )   =>    |- ( ph -> prod_ k e. ( A u. { B } ) C = ( prod_ k e. A C x. D ) )
16-Aug-2024	if0ab 3641	Expression of a conditional class as a class abstraction when the False alternative is the empty class: in that case, the conditional class is the extension, in the True alternative, of the condition. (Contributed by BJ, 16-Aug-2024.)
|- if ( ph , A , (/) ) = { x e. A | ph }
15-Aug-2024	bj-charfundcALT 16847	Alternate proof of bj-charfundc 16846. It was expected to be much shorter since it uses bj-charfun 16845 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
15-Aug-2024	bj-charfun 16845	Properties of the characteristic function on the class X of the class A. (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   =>    |- ( ph -> ( ( F : X --> ~P 1o /\ ( F |` ( ( X i^i A ) u. ( X \ A ) ) ) : ( ( X i^i A ) u. ( X \ A ) ) --> 2o ) /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
15-Aug-2024	cnstab 8974	Equality of complex numbers is stable. Stability here means -. -. A = B -> A = B as defined at df-stab 843. This theorem for real numbers is Proposition 5.2 of [BauerHanson], p. 27. (Contributed by Jim Kingdon, 1-Aug-2023.) (Proof shortened by BJ, 15-Aug-2024.)
|- ( ( A e. CC /\ B e. CC ) -> STAB A = B )
15-Aug-2024	subap0d 8973	Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   =>    |- ( ph -> ( A - B ) # 0 )
15-Aug-2024	ifexd 4630	Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> if ( ps , A , B ) e. _V )
15-Aug-2024	ifelpwun 4629	Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.)
|- A e. _V   &    |- B e. _V   =>    |- if ( ph , A , B ) e. ~P ( A u. B )
15-Aug-2024	ifelpwund 4628	Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> if ( ps , A , B ) e. ~P ( A u. B ) )
15-Aug-2024	ifelpwung 4627	Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.)
|- ( ( A e. V /\ B e. W ) -> if ( ph , A , B ) e. ~P ( A u. B ) )
15-Aug-2024	ifidss 3656	A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , A , A ) C_ A
15-Aug-2024	ifssun 3655	A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , A , B ) C_ ( A u. B )
14-Aug-2024	hashf1lem1 11287	Lemma for hashf1 11289. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> -. z e. A )   &    |- ( ph -> ( (♯ ` A ) + 1 ) <_ (♯ ` B ) )   &    |- ( ph -> F : A -1-1-> B )   =>    |- ( ph -> { f | ( ( f |` A ) = F /\ f : ( A u. { z } ) -1-1-> B ) } ~~ ( B \ ran F ) )
14-Aug-2024	f1setexg 6951	The set of injections between two sets exists. (Contributed by AV, 14-Aug-2024.)
|- ( ( A e. V /\ B e. W ) -> { f | f : A -1-1-> B } e. _V )
12-Aug-2024	exmidontriimlem2 7578	Lemma for exmidontriim 7581. (Contributed by Jim Kingdon, 12-Aug-2024.)
|- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. y e. B ( A e. y \/ A = y \/ y e. A ) )   =>    |- ( ph -> ( A e. B \/ A. y e. B y e. A ) )
12-Aug-2024	exmidontriimlem1 7577	Lemma for exmidontriim 7581. A variation of r19.30dc 2698. (Contributed by Jim Kingdon, 12-Aug-2024.)
|- ( ( A. x e. A ( ph \/ ps \/ ch ) /\ EXMID ) -> ( E. x e. A ph \/ E. x e. A ps \/ A. x e. A ch ) )
11-Aug-2024	nndc 863	Double negation of decidability of a formula. Intuitionistic logic refutes the negation of decidability (but does not prove decidability) of any formula. This should not trick the reader into thinking that -. -. EXMID is provable in intuitionistic logic. Indeed, if we could quantify over formula metavariables, then generalizing nnexmid 862 over ph would give " |- A. ph -. -. DECID ph", but EXMID is " A. phDECID ph", so proving -. -. EXMID would amount to proving " -. -. A. phDECID ph", which is not implied by the above theorem. Indeed, the converse of nnal 1702 does not hold. Since our system does not allow quantification over formula metavariables, we can reproduce this argument by representing formulas as subsets of ~P 1o, like we do in our definition of EXMID (df-exmid 4332): then, we can prove A. x e. ~P 1o -. -. DECID x = 1o but we cannot prove -. -. A. x e. ~P 1oDECID x = 1o because the converse of nnral 2540 does not hold. Actually, -. -. EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying -. EXMID and noncontradiction holds (pm3.24 705). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of -. -. EXMID. (Revised by BJ, 11-Aug-2024.)
|- -. -. DECID ph
10-Aug-2024	exmidontriim 7581	Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
10-Aug-2024	exmidontriimlem4 7580	Lemma for exmidontriim 7581. The induction step for the induction on A. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. z e. A A. y e. On ( z e. y \/ z = y \/ y e. z ) )   =>    |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
10-Aug-2024	exmidontriimlem3 7579	Lemma for exmidontriim 7581. What we get to do based on induction on both A and B. (Contributed by Jim Kingdon, 10-Aug-2024.)
|- ( ph -> A e. On )   &    |- ( ph -> B e. On )   &    |- ( ph -> EXMID )   &    |- ( ph -> A. z e. A A. y e. On ( z e. y \/ z = y \/ y e. z ) )   &    |- ( ph -> A. y e. B ( A e. y \/ A = y \/ y e. A ) )   =>    |- ( ph -> ( A e. B \/ A = B \/ B e. A ) )
10-Aug-2024	nnnninf2 7467	Canonical embedding of suc om into ℕ∞. (Contributed by BJ, 10-Aug-2024.)
|- ( N e. suc om -> ( i e. om |-> if ( i e. N , 1o , (/) ) ) e. ℕ∞ )
10-Aug-2024	infnninf 7464	The point at infinity in ℕ∞ is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as ( om X. { 1o } ), as fconstmpt 4822 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
|- ( i e. om |-> 1o ) e. ℕ∞
10-Aug-2024	mapssfsetg 6946	The value of the set exponentiation ( B ^m A ) is a subset of the class of functions from A to B. (Contributed by AV, 10-Aug-2024.)
|- ( B e. V -> ( B ^m A ) C_ { f | f : A --> B } )
9-Aug-2024	ss1o0el1o 7220	Reformulation of ss1o0el1 4334 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
9-Aug-2024	pw1dc0el 7218	Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
|- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
9-Aug-2024	ss1o0el1 4334	A subclass of { (/) } contains the empty set if and only if it equals { (/) }. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)
|- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )
8-Aug-2024	pw1dc1 7221	If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
|- (EXMID <-> A. x e. ~P 1oDECID x = 1o )
8-Aug-2024	mapfset 6945	If B is a set, the value of the set exponentiation ( B ^m A ) is the class of all functions from A to B. Generalisation of mapvalg 6932 to arbitrary domains. Note that the class { f | f : A --> B } can only contain set-functions, as opposed to arbitrary class-functions. When A is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 6950), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)
|- ( B e. V -> { f | f : A --> B } = ( B ^m A ) )
8-Aug-2024	fdmexb 5939	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
|- ( F : A --> B -> ( A e. _V <-> F e. _V ) )
8-Aug-2024	fndmexb 5938	The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
|- ( F Fn A -> ( A e. _V <-> F e. _V ) )
7-Aug-2024	psrbagaddclfi 15063	The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G e. D /\ I e. Fin ) -> ( F oF + G ) e. D )
7-Aug-2024	psrbagfsupp 15057	Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( F e. D -> F finSupp 0 )
7-Aug-2024	pw1fin 7217	Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
|- (EXMID <-> ~P 1o e. Fin )
7-Aug-2024	mapfoss 6947	The value of the set exponentiation ( B ^m A ) is a superset of the set of all functions from A onto B. (Contributed by AV, 7-Aug-2024.)
|- { f | f : A -onto-> B } C_ ( B ^m A )
7-Aug-2024	elomssom 4752	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4753. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
6-Aug-2024	bj-charfunbi 16849	In an ambient set X, if membership in A is stable, then it is decidable if and only if A has a characteristic function. This characterization can be applied to singletons when the set X has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> X e. V )   &    |- ( ph -> A. x e. X STAB x e. A )   =>    |- ( ph -> ( A. x e. X DECID x e. A <-> E. f e. ( 2o ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) = 1o /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) ) )
6-Aug-2024	bj-charfunr 16848	If a class A has a "weak" characteristic function on a class X, then negated membership in A is decidable (in other words, membership in A is testable) in X. The hypothesis imposes that X be a set. As usual, it could be formulated as |- ( ph -> ( F : X --> om /\ ... ) ) to deal with general classes, but that extra generality would not make the theorem much more useful. The theorem would still hold if the codomain of f were any class with testable equality to the point where ( X \ A ) is sent. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> E. f e. ( om ^m X ) ( A. x e. ( X i^i A ) ( f ` x ) =/= (/) /\ A. x e. ( X \ A ) ( f ` x ) = (/) ) )   =>    |- ( ph -> A. x e. X DECID -. x e. A )
6-Aug-2024	bj-charfundc 16846	Properties of the characteristic function on the class X of the class A, provided membership in A is decidable in X. (Contributed by BJ, 6-Aug-2024.)
|- ( ph -> F = ( x e. X |-> if ( x e. A , 1o , (/) ) ) )   &    |- ( ph -> A. x e. X DECID x e. A )   =>    |- ( ph -> ( F : X --> 2o /\ ( A. x e. ( X i^i A ) ( F ` x ) = 1o /\ A. x e. ( X \ A ) ( F ` x ) = (/) ) ) )
6-Aug-2024	psrbagconf1o 15066	Bag complementation is a bijection on the set of bags dominated by a given bag F. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- S = { y e. D | y oR <_ F }   =>    |- ( F e. D -> ( x e. S |-> ( F oF - x ) ) : S -1-1-onto-> S )
6-Aug-2024	psrbagconcl 15065	The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- S = { y e. D | y oR <_ F }   =>    |- ( ( F e. D /\ X e. S ) -> ( F oF - X ) e. S )
6-Aug-2024	prodssdc 12358	Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.)
|- ( ph -> A C_ B )   &    |- ( ( ph /\ k e. A ) -> C e. CC )   &    |- ( ph -> E. n e. ( ZZ>= ` M ) E. y ( y # 0 /\ seq n ( x. , ( k e. ( ZZ>= ` M ) |-> if ( k e. B , C , 1 ) ) ) ~~> y ) )   &    |- ( ph -> A. j e. ( ZZ>= ` M )DECID j e. A )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. ( B \ A ) ) -> C = 1 )   &    |- ( ph -> B C_ ( ZZ>= ` M ) )   &    |- ( ph -> A. j e. ( ZZ>= ` M )DECID j e. B )   =>    |- ( ph -> prod_ k e. A C = prod_ k e. B C )
5-Aug-2024	fnmptd 16844	The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.)
|- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ( ph /\ x e. A ) -> B e. V )   =>    |- ( ph -> F Fn A )
5-Aug-2024	funmptd 16843	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5414, then prove funmptd 16843 from it, and then prove funmpt 5415 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.)
|- ( ph -> F = ( x e. A |-> B ) )   =>    |- ( ph -> Fun F )
5-Aug-2024	bj-dcfal 16795	The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID F.
5-Aug-2024	bj-dctru 16793	The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID T.
5-Aug-2024	bj-stfal 16782	The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB F.
5-Aug-2024	bj-sttru 16780	The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB T.
5-Aug-2024	psrbagcon 15064	The analogue of the statement " 0 <_ G <_ F implies 0 <_ F - G <_ F " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( ( F oF - G ) e. D /\ ( F oF - G ) oR <_ F ) )
5-Aug-2024	psrbaglecl 15062	The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G e. D )
5-Aug-2024	psrbaglesupp 15060	The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) C_ ( `' F " NN ) )
5-Aug-2024	prod1dc 12355	Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.)
|- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) \/ A e. Fin ) -> prod_ k e. A 1 = 1 )
5-Aug-2024	fcdmnn0fsuppg 9620	Version of fcdmnn0fsupp 9618 avoiding ax-coll 4246 by assuming F is a set rather than its domain I. (Contributed by SN, 5-Aug-2024.)
|- ( ( F e. V /\ F : I --> NN0 ) -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) )
5-Aug-2024	fcdmnn0suppg 9619	Version of fcdmnn0supp 9617 avoiding ax-coll 4246 by assuming F is a set rather than its domain I. (Contributed by SN, 5-Aug-2024.)
|- ( ( F e. V /\ F : I --> NN0 ) -> ( F supp 0 ) = ( `' F " NN ) )
5-Aug-2024	2ssom 6797	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
5-Aug-2024	suppssdc 6500	Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)
|- ( ph -> F : A --> B )   &    |- ( ( ph /\ k e. ( A \ W ) ) -> ( F ` k ) = Z )   &    |- ( ph -> A. x e. A DECID x e. W )   =>    |- ( ph -> ( F supp Z ) C_ W )
5-Aug-2024	elsuppfng 6482	An element of the support of a function with a given domain. This version of elsuppfn 6483 assumes F is a set rather than its domain X, avoiding ax-coll 4246. (Contributed by SN, 5-Aug-2024.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( S e. ( F supp Z ) <-> ( S e. X /\ ( F ` S ) =/= Z ) ) )
5-Aug-2024	suppvalfng 6480	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 6481 assumes F is a set rather than its domain X, avoiding ax-coll 4246. (Contributed by SN, 5-Aug-2024.)
|- ( ( F Fn X /\ F e. V /\ Z e. W ) -> ( F supp Z ) = { i e. X | ( F ` i ) =/= Z } )
4-Aug-2024	en3d 7055	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> ( x e. A -> C e. B ) )   &    |- ( ph -> ( y e. B -> D e. A ) )   &    |- ( ph -> ( ( x e. A /\ y e. B ) -> ( x = D <-> y = C ) ) )   =>    |- ( ph -> A ~~ B )
4-Aug-2024	en2d 7054	Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> ( x e. A -> C e. X ) )   &    |- ( ph -> ( y e. B -> D e. Y ) )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> A ~~ B )
2-Aug-2024	onntri52 7603	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. EXMID -> -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
2-Aug-2024	onntri24 7601	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) )
2-Aug-2024	onntri45 7600	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ) -> -. -. EXMID )
2-Aug-2024	onntri51 7599	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. EXMID -> -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2-Aug-2024	onntri13 7597	Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) )
2-Aug-2024	onntri35 7596	Double negated ordinal trichotomy. There are five equivalent statements: (1) -. -. A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ), (2) -. -. A. x e. On A. y e. On ( x C_ y \/ y C_ x ), (3) A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ), (4) A. x e. On A. y e. On -. -. ( x C_ y \/ y C_ x ), and (5) -. -. EXMID. That these are all equivalent is expressed by (1) implies (3) (onntri13 7597), (3) implies (5) (onntri35 7596), (5) implies (1) (onntri51 7599), (2) implies (4) (onntri24 7601), (4) implies (5) (onntri45 7600), and (5) implies (2) (onntri52 7603). Another way of stating this is that EXMID is equivalent to trichotomy, either the x e. y \/ x = y \/ y e. x or the x C_ y \/ y C_ x form, as shown in exmidontri 7598 and exmidontri2or 7602, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7604 and (4) by onntri2or 7605. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
|- ( A. x e. On A. y e. On -. -. ( x e. y \/ x = y \/ y e. x ) -> -. -. EXMID )
1-Aug-2024	nnral 2540	The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1702. (Contributed by Jim Kingdon, 1-Aug-2024.)
|- ( -. -. A. x e. A ph -> A. x e. A -. -. ph )
31-Jul-2024	3nsssucpw1 7595	Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
|- ( -. EXMID -> -. 3o C_ suc ~P 1o )
31-Jul-2024	sucpw1nss3 7594	Negated excluded middle implies that the successor of the power set of 1o is not a subset of 3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
|- ( -. EXMID -> -. suc ~P 1o C_ 3o )
30-Jul-2024	psrbagf 15056	A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( F e. D -> F : I --> NN0 )
30-Jul-2024	3nelsucpw1 7593	Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- -. 3o e. suc ~P 1o
30-Jul-2024	sucpw1nel3 7592	The successor of the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- -. suc ~P 1o e. 3o
30-Jul-2024	sucpw1ne3 7591	Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ( -. EXMID -> suc ~P 1o =/= 3o )
30-Jul-2024	pw1nel3 7590	Negated excluded middle implies that the power set of 1o is not an element of 3o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ( -. EXMID -> -. ~P 1o e. 3o )
30-Jul-2024	pw1ne3 7589	The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 3o
30-Jul-2024	pw1ne1 7588	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 1o
30-Jul-2024	pw1ne0 7587	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)
30-Jul-2024	fsuppeqg 6488	Version of fsuppeq 6487 avoiding ax-coll 4246 by assuming F is a set rather than its domain I. (Contributed by SN, 30-Jul-2024.)
|- ( ( F e. V /\ Z e. W ) -> ( F : I --> S -> ( F supp Z ) = ( `' F " ( S \ { Z } ) ) ) )
29-Jul-2024	ringcld 14324	Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .x. Y ) e. B )
29-Jul-2024	grpcld 13821	Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
29-Jul-2024	pw1on 7585	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
29-Jul-2024	isfsuppd 7290	Deduction form of isfsupp 7289. (Contributed by SN, 29-Jul-2024.)
|- ( ph -> R e. V )   &    |- ( ph -> Z e. W )   &    |- ( ph -> Fun R )   &    |- ( ph -> ( R supp Z ) e. Fin )   =>    |- ( ph -> R finSupp Z )
28-Jul-2024	exmidpweq 7216	Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
|- (EXMID <-> ~P 1o = 2o )
27-Jul-2024	dcapnconstALT 17124	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 17123 by means of dceqnconst 17122. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( A. x e. RR DECID x # 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )
27-Jul-2024	reap0 17120	Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. z e. RR DECID z # 0 )
26-Jul-2024	nconstwlpolemgt0 17126	Lemma for nconstwlpo 17128. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
|- ( ph -> G : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )   &    |- ( ph -> E. x e. NN ( G ` x ) = 1 )   =>    |- ( ph -> 0 < A )
26-Jul-2024	nconstwlpolem0 17125	Lemma for nconstwlpo 17128. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.)
|- ( ph -> G : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )   &    |- ( ph -> A. x e. NN ( G ` x ) = 0 )   =>    |- ( ph -> A = 0 )
26-Jul-2024	fvmptd4 5800	Deduction version of fvmpt 5782 (where the substitution hypothesis does not have the antecedent ph). (Contributed by SN, 26-Jul-2024.)
|- ( x = A -> B = C )   &    |- ( ph -> F = ( x e. D |-> B ) )   &    |- ( ph -> A e. D )   &    |- ( ph -> C e. V )   =>    |- ( ph -> ( F ` A ) = C )
24-Jul-2024	tridceq 17118	Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 17104 and redcwlpo 17117). Thus, this is an analytic analogue to lpowlpo 7508. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) -> A. x e. RR A. y e. RR DECID x = y )
24-Jul-2024	iswomni0 17113	Weak omniscience stated in terms of equality with 0. Like iswomninn 17112 but with zero in place of one. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( A e. V -> ( A e. WOmni <-> A. f e. ( { 0 , 1 } ^m A )DECID A. x e. A ( f ` x ) = 0 ) )
24-Jul-2024	asclmulg 15046	Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .^ = (.g ` W )   &    |- .* = (.g ` F )   =>    |- ( ( W e. AssAlg /\ N e. NN0 /\ X e. K ) -> ( A ` ( N .* X ) ) = ( N .^ ( A ` X ) ) )
24-Jul-2024	lpowlpo 7508	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7507. There is an analogue in terms of analytic omniscience principles at tridceq 17118. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
23-Jul-2024	nconstwlpolem 17127	Lemma for nconstwlpo 17128. (Contributed by Jim Kingdon, 23-Jul-2024.)
|- ( ph -> F : RR --> ZZ )   &    |- ( ph -> ( F ` 0 ) = 0 )   &    |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )   &    |- ( ph -> G : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( G ` i ) )   =>    |- ( ph -> ( A. y e. NN ( G ` y ) = 0 \/ -. A. y e. NN ( G ` y ) = 0 ) )
23-Jul-2024	dceqnconst 17122	Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 17117 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
|- ( A. x e. RR DECID x = 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )
23-Jul-2024	redc0 17119	Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.)
|- ( A. x e. RR A. y e. RR DECID x = y <-> A. z e. RR DECID z = 0 )
23-Jul-2024	canth 6036	No set A is equinumerous to its power set (Cantor's theorem), i.e., no function can map A onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. (Use nex 1553 if you want the form -. E. f f : A -onto-> ~P A.) (Contributed by NM, 7-Aug-1994.) (Revised by Noah R Kingdon, 23-Jul-2024.)
|- A e. _V   =>    |- -. F : A -onto-> ~P A
22-Jul-2024	nconstwlpo 17128	Existence of a certain non-constant function from reals to integers implies om e. WOmni (the Weak Limited Principle of Omniscience or WLPO). Based on Exercise 11.6(ii) of [HoTT], p. (varies). (Contributed by BJ and Jim Kingdon, 22-Jul-2024.)
|- ( ph -> F : RR --> ZZ )   &    |- ( ph -> ( F ` 0 ) = 0 )   &    |- ( ( ph /\ x e. RR+ ) -> ( F ` x ) =/= 0 )   =>    |- ( ph -> om e. WOmni )
16-Jul-2024	unexd 4892	The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( A u. B ) e. _V )
16-Jul-2024	difexd 4277	Existence of a difference. (Contributed by SN, 16-Jul-2024.)
|- ( ph -> A e. V )   =>    |- ( ph -> ( A \ B ) e. _V )
15-Jul-2024	fprodseq 12352	The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
|- ( k = ( F ` n ) -> B = C )   &    |- ( ph -> M e. NN )   &    |- ( ph -> F : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ n e. ( 1 ... M ) ) -> ( G ` n ) = C )   =>    |- ( ph -> prod_ k e. A B = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ M , ( G ` n ) , 1 ) ) ) ` M ) )
14-Jul-2024	rexbid2 2555	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
|- F/ x ph   &    |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )   =>    |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )
14-Jul-2024	ralbid2 2554	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
|- F/ x ph   &    |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )   =>    |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )
12-Jul-2024	2irrexpqap 16086	There exist real numbers a and b which are irrational (in the sense of being apart from any rational number) such that ( a ^ b ) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers ( sqr ` 2 ) and ( 2 logb 9 ), see sqrt2irrap 12960, 2logb9irrap 16085 and sqrt2cxp2logb9e3 16083. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
|- E. a e. RR E. b e. RR ( A. p e. QQ a # p /\ A. q e. QQ b # q /\ ( a ^c b ) e. QQ )
12-Jul-2024	2logb9irrap 16085	Example for logbgcd1irrap 16078. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.)
|- ( Q e. QQ -> ( 2 logb 9 ) # Q )
12-Jul-2024	erlecpbl 13655	Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A N B <-> C N D ) ) )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( A N B <-> C N D ) ) )
12-Jul-2024	ercpbl 13654	Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ( ph /\ ( a e. V /\ b e. V ) ) -> ( a .+ b ) e. V )   &    |- ( ph -> ( ( A .~ C /\ B .~ D ) -> ( A .+ B ) .~ ( C .+ D ) ) )   =>    |- ( ( ph /\ ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( F ` A ) = ( F ` C ) /\ ( F ` B ) = ( F ` D ) ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) ) )
12-Jul-2024	ercpbllemg 13653	Lemma for ercpbl 13654. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. V )   =>    |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> A .~ B ) )
12-Jul-2024	divsfvalg 13652	Value of the function in qusval 13646. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   &    |- ( ph -> A e. V )   =>    |- ( ph -> ( F ` A ) = [ A ] .~ )
12-Jul-2024	divsfval 13651	Value of the function in qusval 13646. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.)
|- ( ph -> .~ Er V )   &    |- ( ph -> V e. W )   &    |- F = ( x e. V |-> [ x ] .~ )   =>    |- ( ph -> ( F ` A ) = [ A ] .~ )
11-Jul-2024	logbgcd1irraplemexp 16076	Lemma for logbgcd1irrap 16078. Apartness of X ^ N and B ^ M. (Contributed by Jim Kingdon, 11-Jul-2024.)
|- ( ph -> X e. ( ZZ>= ` 2 ) )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> ( X gcd B ) = 1 )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( X ^ N ) # ( B ^ M ) )
11-Jul-2024	reapef 15881	Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A # B <-> ( exp ` A ) # ( exp ` B ) ) )
10-Jul-2024	apcxp2 16047	Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( A ^c B ) # ( A ^c C ) ) )
9-Jul-2024	logbgcd1irraplemap 16077	Lemma for logbgcd1irrap 16078. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
|- ( ph -> X e. ( ZZ>= ` 2 ) )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> ( X gcd B ) = 1 )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( B logb X ) # ( M / N ) )
9-Jul-2024	apexp1 11158	Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
|- ( ( A e. CC /\ B e. CC /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
5-Jul-2024	logrpap0 15982	The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.)
|- ( ( A e. RR+ /\ A # 1 ) -> ( log ` A ) # 0 )
3-Jul-2024	rplogbval 16053	Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
3-Jul-2024	logrpap0d 15983	Deduction form of logrpap0 15982. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( log ` A ) # 0 )
3-Jul-2024	logrpap0b 15981	The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( A e. RR+ -> ( A # 1 <-> ( log ` A ) # 0 ) )
28-Jun-2024	2o01f 17036	Mapping zero and one between om and NN0 style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( G |` 2o ) : 2o --> { 0 , 1 }
28-Jun-2024	012of 17035	Mapping zero and one between NN0 and om style integers. (Contributed by Jim Kingdon, 28-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( `' G |` { 0 , 1 } ) : { 0 , 1 } --> 2o
27-Jun-2024	iooreen 17096	An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
|- ( 0 (,) 1 ) ~~ RR
27-Jun-2024	iooref1o 17095	A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
|- F = ( x e. RR |-> ( 1 / ( 1 + ( exp ` x ) ) ) )   =>    |- F : RR -1-1-onto-> ( 0 (,) 1 )
25-Jun-2024	neapmkvlem 17129	Lemma for neapmkv 17130. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ( ph /\ A =/= 1 ) -> A # 1 )   =>    |- ( ph -> ( -. A. x e. NN ( F ` x ) = 1 -> E. x e. NN ( F ` x ) = 0 ) )
25-Jun-2024	ismkvnn 17115	The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A e. V -> ( A e. Markov <-> A. f e. ( { 0 , 1 } ^m A ) ( -. A. x e. A ( f ` x ) = 1 -> E. x e. A ( f ` x ) = 0 ) ) )
25-Jun-2024	ismkvnnlem 17114	Lemma for ismkvnn 17115. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( A e. V -> ( A e. Markov <-> A. f e. ( { 0 , 1 } ^m A ) ( -. A. x e. A ( f ` x ) = 1 -> E. x e. A ( f ` x ) = 0 ) ) )
25-Jun-2024	enmkvlem 7501	Lemma for enmkv 7502. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
24-Jun-2024	neapmkv 17130	If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.)
|- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> om e. Markov )
24-Jun-2024	dcapnconst 17123	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See trilpo 17104 for more discussion of decidability of real number apartness. This is a weaker form of dceqnconst 17122 and in fact this theorem can be proved using dceqnconst 17122 as shown at dcapnconstALT 17124. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.)
|- ( A. x e. RR DECID x # 0 -> E. f ( f : RR --> ZZ /\ ( f ` 0 ) = 0 /\ A. x e. RR+ ( f ` x ) =/= 0 ) )
24-Jun-2024	enmkv 7502	Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either om e. Markov or NN0 e. Markov. The former is a better match to conventional notation in the sense that df2o3 6702 says that 2o = { (/) , 1o } whereas the corresponding relationship does not exist between 2 and { 0 , 1 }. (Contributed by Jim Kingdon, 24-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov <-> B e. Markov ) )
21-Jun-2024	redcwlpolemeq1 17116	Lemma for redcwlpo 17117. A biconditionalized version of trilpolemeq1 17101. (Contributed by Jim Kingdon, 21-Jun-2024.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   =>    |- ( ph -> ( A = 1 <-> A. x e. NN ( F ` x ) = 1 ) )
20-Jun-2024	redcwlpo 17117	Decidability of real number equality implies the Weak Limited Principle of Omniscience (WLPO). We expect that we'd need some form of countable choice to prove the converse. Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 17116). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10681 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A. x e. RR A. y e. RR DECID x = y -> om e. WOmni )
20-Jun-2024	iswomninn 17112	Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7506 but it will sometimes be more convenient to use 0 and 1 rather than (/) and 1o. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A e. V -> ( A e. WOmni <-> A. f e. ( { 0 , 1 } ^m A )DECID A. x e. A ( f ` x ) = 1 ) )
20-Jun-2024	iswomninnlem 17111	Lemma for iswomnimap 7506. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( A e. V -> ( A e. WOmni <-> A. f e. ( { 0 , 1 } ^m A )DECID A. x e. A ( f ` x ) = 1 ) )
20-Jun-2024	enwomni 7510	Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either om e. WOmni or NN0 e. WOmni. The former is a better match to conventional notation in the sense that df2o3 6702 says that 2o = { (/) , 1o } whereas the corresponding relationship does not exist between 2 and { 0 , 1 }. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni <-> B e. WOmni ) )
20-Jun-2024	enwomnilem 7509	Lemma for enwomni 7510. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
19-Jun-2024	rpabscxpbnd 16048	Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   &    |- ( ph -> 0 < ( Re ` B ) )   &    |- ( ph -> M e. RR )   &    |- ( ph -> ( abs ` A ) <_ M )   =>    |- ( ph -> ( abs ` ( A ^c B ) ) <_ ( ( M ^c ( Re ` B ) ) x. ( exp ` ( ( abs ` B ) x. pi ) ) ) )
16-Jun-2024	rpcxpsqrt 16030	The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other n-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.)
|- ( A e. RR+ -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
16-Jun-2024	biadanid 622	Deduction associated with biadani 620. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ch ) -> ( ps <-> th ) )   =>    |- ( ph -> ( ps <-> ( ch /\ th ) ) )
13-Jun-2024	rpcxpadd 16013	Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )
12-Jun-2024	cxpap0 16012	Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) # 0 )
12-Jun-2024	rpcncxpcl 16010	Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
12-Jun-2024	rpcxp0 16006	Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( A e. RR+ -> ( A ^c 0 ) = 1 )
12-Jun-2024	cxpexpnn 16004	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
12-Jun-2024	cxpexprp 16003	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
12-Jun-2024	rpcxpef 16002	Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
12-Jun-2024	df-rpcxp 15963	Define the power function on complex numbers. Because df-relog 15962 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.)
|- ^c = ( x e. RR+ , y e. CC |-> ( exp ` ( y x. ( log ` x ) ) ) )
10-Jun-2024	trirec0xor 17106	Version of trirec0 17105 with exclusive-or. The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/_ x = 0 ) )
10-Jun-2024	trirec0 17105	Every real number having a reciprocal or equaling zero is equivalent to real number trichotomy. This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 17104). (Contributed by Jim Kingdon, 10-Jun-2024.)
|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) <-> A. x e. RR ( E. z e. RR ( x x. z ) = 1 \/ x = 0 ) )
9-Jun-2024	omniwomnimkv 7507	A set is omniscient if and only if it is weakly omniscient and Markov. The case A = om says that LPO <-> WLPO /\ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
|- ( A e. Omni <-> ( A e. WOmni /\ A e. Markov ) )
9-Jun-2024	iswomnimap 7506	The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
|- ( A e. V -> ( A e. WOmni <-> A. f e. ( 2o ^m A )DECID A. x e. A ( f ` x ) = 1o ) )
9-Jun-2024	iswomni 7505	The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
|- ( A e. V -> ( A e. WOmni <-> A. f ( f : A --> 2o -> DECID A. x e. A ( f ` x ) = 1o ) ) )
9-Jun-2024	df-womni 7504	A weakly omniscient set is one where we can decide whether a predicate (here represented by a function f) holds (is equal to 1o) for all elements or not. Generalization of definition 2.4 of [Pierik], p. 9. In particular, om e. WOmni is known as the Weak Limited Principle of Omniscience (WLPO). The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.)
|- WOmni = { y | A. f ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) }
1-Jun-2024	ringcmnd 14342	A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. CMnd )
1-Jun-2024	ringabld 14341	A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Abel )
1-Jun-2024	cmnmndd 14113	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
1-Jun-2024	ablcmnd 14097	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. CMnd )
1-Jun-2024	grpmndd 13820	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mnd )
1-Jun-2024	fndmi 5481	The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
|- F Fn A   =>    |- dom F = A
29-May-2024	pw1nct 17045	A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.)
|- ( A. r ( r C_ ( ~P 1o X. om ) -> ( A. p e. ~P 1o E. n e. om p r n -> E. m e. om A. q e. ~P 1o q r m ) ) -> -. E. f f : om -onto-> ( ~P 1o ⊔ 1o ) )
28-May-2024	sssneq 17044	Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)
|- ( A C_ { B } -> A. y e. A A. z e. A y = z )
28-May-2024	fvun2d 5772	The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( F u. G ) ` X ) = ( G ` X ) )
28-May-2024	fvun1d 5771	The value of a union when the argument is in the first domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> ( A i^i B ) = (/) )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( ( F u. G ) ` X ) = ( F ` X ) )
26-May-2024	elpwi2 4294	Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)
|- B e. V   &    |- A C_ B   =>    |- A e. ~P B
25-May-2024	mplnegfi 15098	The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- N = ( invg ` R )   &    |- M = ( invg ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
24-May-2024	dvmptcjx 15827	Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.)
|- ( ( ph /\ x e. X ) -> A e. CC )   &    |- ( ( ph /\ x e. X ) -> B e. V )   &    |- ( ph -> ( RR _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )   &    |- ( ph -> X C_ RR )   =>    |- ( ph -> ( RR _D ( x e. X |-> ( * ` A ) ) ) = ( x e. X |-> ( * ` B ) ) )
23-May-2024	cbvralfw 2775	Rule used to change bound variables, using implicit substitution. Version of cbvralf 2777 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1560 and ax-bndl 1562 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by GG, 23-May-2024.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
23-May-2024	cbvrmow 2735	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 2785 with a disjoint variable condition. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 23-May-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x e. A ph <-> E* y e. A ps )
23-May-2024	cbvmow 2127	Rule used to change bound variables, using implicit substitution. Version of cbvmo 2126 with a disjoint variable condition. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 23-May-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E* x ph <-> E* y ps )
22-May-2024	efltlemlt 15877	Lemma for eflt 15878. The converse of efltim 12467 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( exp ` A ) < ( exp ` B ) )   &    |- ( ph -> D e. RR+ )   &    |- ( ph -> ( ( abs ` ( A - B ) ) < D -> ( abs ` ( ( exp ` A ) - ( exp ` B ) ) ) < ( ( exp ` B ) - ( exp ` A ) ) ) )   =>    |- ( ph -> A < B )
21-May-2024	eflt 15878	The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( exp ` A ) < ( exp ` B ) ) )
20-May-2024	nsyl5 659	A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.)
|- ( ph -> ps )   &    |- ( -. ph -> ch )   =>    |- ( -. ps -> ch )
19-May-2024	apdifflemr 17108	Lemma for apdiff 17109. (Contributed by Jim Kingdon, 19-May-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> S e. QQ )   &    |- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )   &    |- ( ( ph /\ S =/= 0 ) -> ( abs ` ( A - 0 ) ) # ( abs ` ( A - ( 2 x. S ) ) ) )   =>    |- ( ph -> A # S )
18-May-2024	apdifflemf 17107	Lemma for apdiff 17109. Being apart from the point halfway between Q and R suffices for A to be a different distance from Q and from R. (Contributed by Jim Kingdon, 18-May-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> Q e. QQ )   &    |- ( ph -> R e. QQ )   &    |- ( ph -> Q < R )   &    |- ( ph -> ( ( Q + R ) / 2 ) # A )   =>    |- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )
17-May-2024	apdiff 17109	The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)
|- ( A e. RR -> ( A. q e. QQ A # q <-> A. q e. QQ A. r e. QQ ( q =/= r -> ( abs ` ( A - q ) ) # ( abs ` ( A - r ) ) ) ) )
16-May-2024	lmodgrpd 14635	A left module is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> W e. LMod )   =>    |- ( ph -> W e. Grp )
16-May-2024	drnggrpd 14614	A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. DivRing )   =>    |- ( ph -> R e. Grp )
16-May-2024	drngringd 14613	A division ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. DivRing )   =>    |- ( ph -> R e. Ring )
16-May-2024	crnggrpd 14316	A commutative ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Grp )
16-May-2024	crngringd 14315	A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Ring )
16-May-2024	ringgrpd 14311	A ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Grp )
15-May-2024	reeff1oleme 15875	Lemma for reeff1o 15876. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
14-May-2024	df-relog 15962	Define the natural logarithm function. Defining the logarithm on complex numbers is similar to square root - there are ways to define it but they tend to make use of excluded middle. Therefore, we merely define logarithms on positive reals. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Jim Kingdon, 14-May-2024.)
|- log = `' ( exp |` RR )
14-May-2024	fvmpopr2d 6225	Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)
|- ( ph -> F = ( a e. A , b e. B |-> C ) )   &    |- ( ph -> P = <. a , b >. )   &    |- ( ( ph /\ a e. A /\ b e. B ) -> C e. V )   =>    |- ( ( ph /\ a e. A /\ b e. B ) -> ( F ` P ) = C )
13-May-2024	fndmexd 5581	If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.)
|- ( ph -> F e. V )   &    |- ( ph -> F Fn D )   =>    |- ( ph -> D e. _V )
12-May-2024	dvdstrd 12599	The divides relation is transitive, a deduction version of dvdstr 12597. (Contributed by metakunt, 12-May-2024.)
|- ( ph -> K e. ZZ )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> K || M )   &    |- ( ph -> M || N )   =>    |- ( ph -> K || N )
7-May-2024	ioocosf1o 15958	The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.)
|- ( cos |` ( 0 (,) pi ) ) : ( 0 (,) pi ) -1-1-onto-> ( -u 1 (,) 1 )
7-May-2024	cos0pilt1 15956	Cosine is between minus one and one on the open interval between zero and pi. (Contributed by Jim Kingdon, 7-May-2024.)
|- ( A e. ( 0 (,) pi ) -> ( cos ` A ) e. ( -u 1 (,) 1 ) )
6-May-2024	cos11 15957	Cosine is one-to-one over the closed interval from 0 to pi. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Jim Kingdon, 6-May-2024.)
|- ( ( A e. ( 0 [,] pi ) /\ B e. ( 0 [,] pi ) ) -> ( A = B <-> ( cos ` A ) = ( cos ` B ) ) )
5-May-2024	omiunct 13337	The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 13333 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ( ph /\ x e. om ) -> E. g g : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. om B ⊔ 1o ) )
5-May-2024	ctiunctal 13334	Variation of ctiunct 13333 which allows x to be present in ph. (Contributed by Jim Kingdon, 5-May-2024.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- ( ph -> A. x e. A G : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
5-May-2024	suppssrgst 6502	A function is zero outside its support. Version of suppssrst 6501 avoiding ax-coll 4246 by assuming F is a set rather than its domain A. (Contributed by SN, 5-May-2024.)
|- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp Z ) C_ W )   &    |- ( ph -> F e. V )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. u e. B A. v e. B STAB u = v )   =>    |- ( ( ph /\ X e. ( A \ W ) ) -> ( F ` X ) = Z )
5-May-2024	ifpnst 1001	Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.) (Proof shortened by Wolf Lammen, 5-May-2024.)
|- (STAB ph -> (if- ( ph , ps , ch ) <-> if- ( -. ph , ch , ps ) ) )
3-May-2024	cc4n 7637	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7636, the hypotheses only require an A(n) for each value of n, not a single set A which suffices for every n e. om. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A. n e. N { x e. A | ps } e. V )   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f Fn N /\ A. n e. N ch ) )
3-May-2024	cc4f 7635	Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 3-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A e. V )   &    |- F/_ n A   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )
1-May-2024	cc4 7636	Countable choice by showing the existence of a function f which can choose a value at each index n such that ch holds. (Contributed by Mario Carneiro, 7-Apr-2013.) (Revised by Jim Kingdon, 1-May-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A e. V )   &    |- ( ph -> N ~~ om )   &    |- ( x = ( f ` n ) -> ( ps <-> ch ) )   &    |- ( ph -> A. n e. N E. x e. A ps )   =>    |- ( ph -> E. f ( f : N --> A /\ A. n e. N ch ) )
30-Apr-2024	ifpdfbidc 998	Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)
|- (DECID ph -> ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) ) )
29-Apr-2024	cc3 7634	Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A. n e. N F e. _V )   &    |- ( ph -> A. n e. N E. w w e. F )   &    |- ( ph -> N ~~ om )   =>    |- ( ph -> E. f ( f Fn N /\ A. n e. N ( f ` n ) e. F ) )
28-Apr-2024	ifpbi23d 1006	Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)
|- ( ph -> ( ch <-> et ) )   &    |- ( ph -> ( th <-> ze ) )   =>    |- ( ph -> (if- ( ps , ch , th ) <-> if- ( ps , et , ze ) ) )
27-Apr-2024	cc2 7633	Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> F Fn om )   &    |- ( ph -> A. x e. om E. w w e. ( F ` x ) )   =>    |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )
27-Apr-2024	cc2lem 7632	Lemma for cc2 7633. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ph -> F Fn om )   &    |- ( ph -> A. x e. om E. w w e. ( F ` x ) )   &    |- A = ( n e. om |-> ( { n } X. ( F ` n ) ) )   &    |- G = ( n e. om |-> ( 2nd ` ( f ` ( A ` n ) ) ) )   =>    |- ( ph -> E. g ( g Fn om /\ A. n e. om ( g ` n ) e. ( F ` n ) ) )
27-Apr-2024	cc1 7631	Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.)
|- (CCHOICE -> A. x ( ( x ~~ om /\ A. z e. x E. w w e. z ) -> E. f A. z e. x ( f ` z ) e. z ) )
24-Apr-2024	lsppropd 14771	If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> B C_ W )   &    |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) e. W )   &    |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   &    |- ( ph -> P = ( Base ` (Scalar ` K ) ) )   &    |- ( ph -> P = ( Base ` (Scalar ` L ) ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> L e. Y )   =>    |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )
19-Apr-2024	omctfn 13336	Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.)
|- ( ph -> CCHOICE )   &    |- ( ( ph /\ x e. om ) -> E. g g : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. f ( f Fn om /\ A. x e. om ( f ` x ) : om -onto-> ( B ⊔ 1o ) ) )
17-Apr-2024	ifpbi123d 1005	Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
|- ( ph -> ( ps <-> ta ) )   &    |- ( ph -> ( ch <-> et ) )   &    |- ( ph -> ( th <-> ze ) )   =>    |- ( ph -> (if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )
13-Apr-2024	prodmodclem2 12346	Lemma for prodmodc 12347. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( f ` j ) / k ]_ B , 1 ) )   =>    |- ( ( ph /\ E. m e. ZZ ( ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A ) /\ ( E. n e. ( ZZ>= ` m ) E. y ( y # 0 /\ seq n ( x. , F ) ~~> y ) /\ seq m ( x. , F ) ~~> x ) ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ z = ( seq 1 ( x. , G ) ` m ) ) -> x = z ) )
13-Apr-2024	sspwd 3704	The powerclass preserves inclusion (deduction form). (Contributed by BJ, 13-Apr-2024.)
|- ( ph -> A C_ B )   =>    |- ( ph -> ~P A C_ ~P B )
13-Apr-2024	sspwi 3703	The powerclass preserves inclusion (inference form). (Contributed by BJ, 13-Apr-2024.)
|- A C_ B   =>    |- ~P A C_ ~P B
13-Apr-2024	sspw 3702	The powerclass preserves inclusion. See sspwb 4356 for the biconditional version. (Contributed by NM, 13-Oct-1996.) Extract forward implication of sspwb 4356 since it requires fewer axioms. (Revised by BJ, 13-Apr-2024.)
|- ( A C_ B -> ~P A C_ ~P B )