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 31-Mar-2026 at 7:08 AM ET.

Recent Additions to the Intuitionistic Logic Explorer

Date	Label	Description
Theorem
26-Mar-2026	gsumgfsumlem 16633	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 gsumg F ) = ( G gsumg ( F o. S ) ) )
26-Mar-2026	gfsum0 16632	An empty finite group sum is the identity. (Contributed by Jim Kingdon, 26-Mar-2026.)
|- ( G e. CMnd -> ( G gfsumgf (/) ) = ( 0g ` G ) )
25-Mar-2026	gsumgfsum 16634	On an integer range, gsumg and gfsumgf 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 gsumg F ) = ( G gfsumgf F ) )
25-Mar-2026	gsumgfsum1 16631	On an integer range starting at one, gsumg and gfsumgf 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 gsumg F ) = ( G gfsumgf F ) )
24-Mar-2026	gfsumval 16630	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 gfsumgf F ) = ( W gsumg ( F o. G ) ) )
23-Mar-2026	df-gfsum 16629	Define the finite group sum (iterated sum) over an unordered finite set. As currently defined, df-igsum 13335 is indexed by consecutive integers, but in the case of a commutative monoid, the order of the sum doesn't matter and we can define a sum indexed by any finite set without needing to specify an order. (Contributed by Jim Kingdon, 23-Mar-2026.)
|- gfsumgf = ( 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 gsumg ( f o. g ) ) ) ) ) )
20-Mar-2026	exmidssfi 7125	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 15969	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 15968	A graph with one non-loop edge is a pseudograph. Variation of upgr1edc 15965 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 16196	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 16255	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 16118	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 11055	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 6398	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 16216	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 16207	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 16104	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 16101	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 16100	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 7090	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 16540	The powerset of 1o is not infinite. Since we cannot prove it is finite (see pw1fin 7097), this provides a concrete example of a set which we cannot show to be finite or infinite, as seen another way at inffiexmid 7093. (Contributed by Jim Kingdon, 14-Feb-2026.)
|- -. om ~<_ ~P 1o
14-Feb-2026	pw1ndom3 16539	The powerset of 1o does not dominate 3o. This is another way of saying that ~P 1o does not have three elements (like pwntru 4287). (Contributed by Steven Nguyen and Jim Kingdon, 14-Feb-2026.)
|- -. 3o ~<_ ~P 1o
14-Feb-2026	pw1ndom3lem 16538	Lemma for pw1ndom3 16539. (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 16555	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 16537	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 7089	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 16102	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 7083	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 16151	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 16193	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 16136	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 6998	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 16166	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 16160	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 16161	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 16130	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 5670	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 5669	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 4948	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	ifp2 986	Forward direction of dfifp2dc 987. 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 985	The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)
|- (if- ( ph , ps , ch ) -> DECID ph )
20-Jan-2026	cats1fvd 11340	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 11339	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 11343	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 11342	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 11341	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 2636	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 15999	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 7392	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 16547	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 7436	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 7435	A truth value which is inhabited is equal to true. This is a variation of pwntru 4287 and pwtrufal 16548. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- ( ( A e. ~P 1o /\ E. x x e. A ) -> A = 1o )
10-Jan-2026	1ndom2 7046	Two is not dominated by one. (Contributed by Jim Kingdon, 10-Jan-2026.)
|- -. 2o ~<_ 1o
9-Jan-2026	pw1map 16546	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 7434	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 11253	Closure of the prefix extractor. This extends pfxclg 11252 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 11251	The domain of the prefix extractor. (Contributed by Jim Kingdon, 8-Jan-2026.)
|- prefix Fn ( _V X. NN0 )
7-Jan-2026	pr1or2 7393	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 15964	Lemma for upgr1edc 15965. (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 15938	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 15937	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 15938). (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 6997	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 6994	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 6974	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 15927	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 15914	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 15913	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 15863	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 15862	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 3791	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 11158	Existence of the last symbol. The last symbol of a word is a set. See lsw0g 11155 or lswcl 11157 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 11221	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 11164	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 11153	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 13194	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 15876	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 15875	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 15874	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 15873	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 11100	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 16575	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 9794	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 14683	Simpler form of psrelbas 14682 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 14707	Lemma for mplsubgfi 14708. 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 14706	Lemma for mplsubgfi 14708. 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 7373	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 7326). (Contributed by Jim Kingdon, 25-Nov-2025.)
|- ( A. x e. ℕ∞ DECID x = ( i e. om |-> 1o ) <-> om e. WOmni )
23-Nov-2025	psrbagfi 14680	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 7378	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 14705	Lemma for mplsubgfi 14708. 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 9792	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 16545	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 16544	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 16552	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 13350	Lemma for prdsbas 13352 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 14700	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
4-Nov-2025	mplelbascoe 14699	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 14698	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 14697	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 7387	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 12501	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 11779	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 10499	Rational <_ is decidable. (Contributed by Jim Kingdon, 28-Oct-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A <_ B )
17-Oct-2025	plycoeid3 15474	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 7117	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 7115	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 3602	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 12804	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 14671	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 15415	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 1492	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 919), 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 1491	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 858), 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 1490	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 848), 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 15414	Real derivative of the identity function. (Contributed by Jim Kingdon, 3-Oct-2025.)
|- ( RR _D ( _I |` RR ) ) = ( RR X. { 1 } )
3-Oct-2025	dvconstre 15413	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 15410	Lemma for dvconstss 15415. Analogue of dvidlemap 15408 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 15409	Lemma for dvidre 15414 and dvconstre 15413. Analogue of dvidlemap 15408 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 14562	Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
|- ( A e. V -> (metUnif ` A ) e. _V )
28-Sep-2025	cndsex 14560	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 14561	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 14559	Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
|- ( D e. V -> ( MetOpen ` D ) e. _V )
24-Sep-2025	blfn 14558	The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
|- ball Fn _V
23-Sep-2025	elfzoext 10430	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 15477	Lemma for plycj 15478. (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 ) ) ) )
20-Sep-2025	plycolemc 15475	Lemma for plyco 15476. 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 10429	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 15798	Lemma for lgsquad 15802. 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 15797	Lemma for lgsquad 15802. 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 7126	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 6295	Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7415 (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	expghmap 14614	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 14596	The invertible complex numbers are exactly those apart from zero. This is recapb 8844 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
|- { z e. CC | z # 0 } = (Unit ` ℂfld )
9-Sep-2025	gsumfzfsumlemm 14594	Lemma for gsumfzfsum 14595. The case where the sum is inhabited. (Contributed by Jim Kingdon, 9-Sep-2025.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> B e. CC )   =>    |- ( ph -> (ℂfld gsumg ( k e. ( M ... N ) |-> B ) ) = sum_ k e. ( M ... N ) B )
9-Sep-2025	gsumfzfsumlem0 14593	Lemma for gsumfzfsum 14595. The case where the sum is empty. (Contributed by Jim Kingdon, 9-Sep-2025.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> N < M )   =>    |- ( ph -> (ℂfld gsumg ( k e. ( M ... N ) |-> B ) ) = sum_ k e. ( M ... N ) B )
9-Sep-2025	gsumfzmhm2 13924	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 gsumg ( k e. ( M ... N ) |-> X ) ) -> C = E )   =>    |- ( ph -> ( H gsumg ( k e. ( M ... N ) |-> D ) ) = E )
8-Sep-2025	gsumfzmhm 13923	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 gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
8-Sep-2025	5ndvds6 12489	5 does not divide 6. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 6
8-Sep-2025	5ndvds3 12488	5 does not divide 3. (Contributed by AV, 8-Sep-2025.)
|- -. 5 || 3
7-Sep-2025	5eluz3 9788	5 is an integer greater than or equal to 3. (Contributed by AV, 7-Sep-2025.)
|- 5 e. ( ZZ>= ` 3 )
6-Sep-2025	gsumfzconst 13921	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 gsumg ( k e. ( M ... N ) |-> X ) ) = ( ( ( N - M ) + 1 ) .x. X ) )
5-Sep-2025	uzuzle34 9791	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 ) )
31-Aug-2025	gsumfzmptfidmadd 13919	The sum of two group sums expressed as mappings with finite domain. (Contributed by AV, 23-Jul-2019.) (Revised by Jim Kingdon, 31-Aug-2025.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> C e. B )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> D e. B )   &    |- F = ( x e. ( M ... N ) |-> C )   &    |- H = ( x e. ( M ... N ) |-> D )   =>    |- ( ph -> ( G gsumg ( x e. ( M ... N ) |-> ( C .+ D ) ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
30-Aug-2025	gsumfzsubmcl 13918	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 gsumg F ) e. S )
30-Aug-2025	seqm1g 10729	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 10776	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 9875	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 9874. (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 10721	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 10718	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 11113	A zero-based sequence is a word. In iswrdinn0 11111 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	gsumfzcl 13575	Closure of a finite 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 gsumg F ) e. B )
16-Aug-2025	iswrdinn0 11111	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 )
15-Aug-2025	gsumfzz 13571	Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 15-Aug-2025.)
|- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ M e. ZZ /\ N e. ZZ ) -> ( G gsumg ( k e. ( M ... N ) |-> .0. ) ) = .0. )
14-Aug-2025	gsumfzval 13467	An expression for gsumg 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 gsumg F ) = if ( N < M , .0. , ( seq M ( .+ , F ) ` N ) ) )
13-Aug-2025	znidom 14664	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 14268	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 15781	Lemma for gausslemma2dlem1 15783. 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 15782	Lemma for gausslemma2dlem1 15783. (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 10498	Rational < is decidable. (Contributed by Jim Kingdon, 7-Aug-2025.)
|- ( ( A e. QQ /\ B e. QQ ) -> DECID A < B )
22-Jul-2025	ivthdich 15370	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 15360 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 15363, hovera 15364, and hoverb 15365, 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 8240, c < 1 or 0 < c, and that leads to z <_ 0 by hoverlt1 15366 or 0 <_ z by hovergt0 15367. (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 15369	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 15368	Lemma for ivthdich 15370. 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 15367	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 15366	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 15365	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 15364	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 2739	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 2738	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 2737	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 2736	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 15363	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 15333	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 15332	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 10692	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 7316	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 12604	Lemma for nninfct 12605. (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 16573	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 12605	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 10698	ℕ∞ is infinte. (Contributed by Jim Kingdon, 8-Jul-2025.)
|- om ~<_ ℕ∞
7-Jul-2025	ivthreinc 15362	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 15360). 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	fngsum 13464	Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)
|- gsumg Fn ( _V X. _V )
28-Jun-2025	iotaexel 5971	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-igsum 13335	Define a finite group sum (also called "iterated sum") of a structure. 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. 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. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 7-Dec-2014.) (Revised by Jim Kingdon, 27-Jun-2025.)
|- gsumg = ( 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 14192	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 14193. (Contributed by SN, 20-Jun-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. NzRing <-> O e. NzRing ) )
16-Jun-2025	fnpsr 14674	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 13137	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 5668	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 12878	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 10520	"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 2841	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 2874	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 12965	Exercise which may help in understanding the proof of 4sqlemsdc 12966. (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 5303	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 12966	Lemma for 4sq 12976. 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 12964 and 4sqexercise2 12965 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 12964	Exercise which may help in understanding the proof of 4sqlemsdc 12966. (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 12963	Lemma for 4sq 12976. 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 12962	Lemma for 4sq 12976. 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 12961	Lemma for 4sq 12976. 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 7127	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 14628	Set existence for ZRHom. (Contributed by Jim Kingdon, 19-May-2025.)
|- L = ( ZRHom ` R )   =>    |- ( R e. V -> L e. _V )
16-May-2025	rhmex 14164	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 13835	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 13538	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 14285	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 14275	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 13960	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 13147. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Rng -> ( G ↾s B ) e. Rng )
5-May-2025	ablressid 13915	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 13147. (Contributed by Jim Kingdon, 5-May-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Abel -> ( G ↾s B ) e. Abel )
30-Apr-2025	dvply2g 15483	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 14467	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 13146	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 14362	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 14571	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 14569	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 14480	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 14361	Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> ( LSubSp ` W ) e. _V )
25-Apr-2025	rspex 14481	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. V -> (RSpan ` W ) e. _V )
25-Apr-2025	lspex 14402	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 13801	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 13401	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 6988	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 13703	Existence of the group multiple operation. (Contributed by Jim Kingdon, 22-Apr-2025.)
|- ( G e. V -> (.g ` G ) e. _V )
21-Apr-2025	uspgruhgr 16031	An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
|- ( G e. USPGraph -> G e. UHGraph )
20-Apr-2025	uspgriedgedg 16023	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 16022	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 6215	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6216 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 5685	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 15708	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 8148. (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 15707	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 14516	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 14508	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 14495	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 14482	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 14436	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 14453	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 13602	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 14687	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 16613	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 13853	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 14540	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 14575	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 14564. (Revised by GG, 31-Mar-2025.)
|- ( abs o. - ) = ( dist ` ℂfld )
31-Mar-2025	cnfldle 14574	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 14564. (Revised by GG, 31-Mar-2025.)
|- <_ = ( le ` ℂfld )
31-Mar-2025	cnfldtset 14573	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 14570	The multiplication operation of the field of complex numbers. Version of cnfldmul 14571 using maps-to notation, which does not require ax-mulf 8148. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` ℂfld )
31-Mar-2025	mpocnfldadd 14568	The addition operation of the field of complex numbers. Version of cnfldadd 14569 using maps-to notation, which does not require ax-addf 8147. (Contributed by GG, 31-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x + y ) ) = ( +g ` ℂfld )
31-Mar-2025	df-cnfld 14564	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 14566, cnfldadd 14569, cnfldmul 14571, cnfldcj 14572, cnfldtset 14573, cnfldle 14574, cnfldds 14575, and cnfldbas 14567. 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 14531	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 14286	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 14284	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 14515	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 14489	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 14488	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 13314	Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- (comp = Slot (comp ` ndx ) /\ (comp ` ndx ) e. NN )
20-Mar-2025	homslid 13311	Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- ( Hom = Slot ( Hom ` ndx ) /\ ( Hom ` ndx ) e. NN )
19-Mar-2025	ptex 13340	Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
|- ( F e. V -> ( Xt_ ` F ) e. _V )
18-Mar-2025	prdsex 13345	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 15479	A polynomial is a continuous function. (Contributed by Mario Carneiro, 23-Jul-2014.) Avoid ax-mulf 8148. (Revised by GG, 16-Mar-2025.)
|- ( F e. (Poly ` S ) -> F e. ( CC -cn-> CC ) )
16-Mar-2025	expcn 15286	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 8148. (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 15283	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 8162	Multiplication is an operation on complex numbers. Version of ax-mulf 8148 using maps-to notation, proved from the axioms of set theory and ax-mulcl 8123. (Contributed by GG, 16-Mar-2025.)
|- ( x e. CC , y e. CC |-> ( x x. y ) ) : ( CC X. CC ) --> CC
13-Mar-2025	2idlss 14521	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 13381	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 14527	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 14524	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 14505	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 14504	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 14503	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 13382	Value of an image structure. The is a lemma for the theorems imasbas 13383, imasplusg 13384, and imasmulr 13385 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 14514	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 14513	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 13813	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 6122	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 13756	The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
|- ( G e. Grp -> (SubGrp ` G ) e. _V )
7-Mar-2025	ringrzd 14052	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 14051	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 13911	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 ) )
1-Mar-2025	quselbasg 13810	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 14539	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14540. Similar to qusmul2 14536. (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 14069	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 13147. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Ring -> ( G ↾s B ) e. Ring )
28-Feb-2025	grpressid 13637	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 13147. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
27-Feb-2025	imasringf1 14071	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 13181	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 14212	A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
|- ( A e. (SubRng ` R ) -> A e. (NrmSGrp ` R ) )
25-Feb-2025	rngansg 13956	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 13818	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 13819	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 13811	Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.) Generalization of quseccl 13813 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 16624	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 16622). (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 16623	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 14532	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 14534 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 14530	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 14203	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 14202	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 14201	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 14200	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
|- ( R e. LRing -> R e. NzRing )
23-Feb-2025	islring 14199	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 14198	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 14196	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 14190	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 13964	The quotient structure of a non-unital ring is a non-unital ring (qusring2 14072 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 13958	Ring multiplication distributes over subtraction. (subdir 8558 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 14063. (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 13957	Ring multiplication distributes over subtraction. (subdi 8557 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdi 14062. (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 ) ) )
22-Feb-2025	imasrngf1 13963	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 13962	The image structure of a non-unital ring is a non-unital ring (imasring 14070 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 13943	Restricted functionality of the multiplicative group on non-unital rings (mgpf 14017 analog). (Contributed by AV, 22-Feb-2025.)
|- (mulGrp |` Rng ) :Rng -->Smgrp
22-Feb-2025	imasabl 13916	The image structure of an abelian group is an abelian group (imasgrp 13691 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 13491	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 13490	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 7463	Tight apartness with the apartness properties from df-pap 7460 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 14529	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 14528	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 14526	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 14525	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 14523	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 14522	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 14512	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 14293	The relation given by df-apr 14288 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 13116	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 3649	The converse of ifbi 3624 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 3648	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 3647	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 14487	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 14292	The apartness relation given by df-apr 14288 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 14291	The apartness relation given by df-apr 14288 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 14289	Expand Definition df-apr 14288. (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 14223	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 13955	Double negation of a product in a non-unital ring (mul2neg 8570 analog). (Contributed by Mario Carneiro, 4-Dec-2014.) Generalization of ringm2neg 14061. (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 13954	Negation of a product in a non-unital ring (mulneg2 8568 analog). In contrast to ringmneg2 14060, 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 13953	Negation of a product in a non-unital ring (mulneg1 8567 analog). In contrast to ringmneg1 14059, 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 14290	The apartness relation given by df-apr 14288 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 13952	The zero of a non-unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringrz 14050. (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 13949	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 13948	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 13944	A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.)
|- ( R e. Rng -> R e. Grp )
16-Feb-2025	aptap 8823	Complex apartness (as defined at df-ap 8755) is a tight apartness (as defined at df-tap 7462). (Contributed by Jim Kingdon, 16-Feb-2025.)
|- # TAp CC
15-Feb-2025	subsubrng2 14222	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 14221	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 14220	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 14219	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 14218	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 14217	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 14084	A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14083. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Rng <-> O e. Rng ) )
15-Feb-2025	opprrng 14083	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 13961	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 13489	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 13485	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 7465	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 14216	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14240. (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 14215	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 14214	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 14213	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 14211	A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
|- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
14-Feb-2025	subrngrcl 14210	Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
|- ( A e. (SubRng ` R ) -> R e. Rng )
14-Feb-2025	subrngrng 14209	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 14208	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 14207	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 14206	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 14205	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 13959	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 13946	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 7473	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 7472	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 7460	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 14520	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 14519	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 14518	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 14517	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 14511	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 14288	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14293. (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 13945	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 13488	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 )
8-Feb-2025	2oneel 7468	(/) 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 7464	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 14692	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 14256	Restriction of the codomain of a (ring) homomorphism. resghm2b 13842 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 10963	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 7471	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 7470	Lemma for 2omotap 7471. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
6-Feb-2025	2omotaplemap 7469	Lemma for 2omotap 7471. (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 7467	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 7466	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 7462	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 13723	Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl 13718. (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 13779	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 13632	The right inverse of a group element. Deduction associated with grprinv 13627. (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 13631	The left inverse of a group element. Deduction associated with grplinv 13626. (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 13625	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 13610	The identity element of a group is a right identity. Deduction associated with grprid 13608. (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 13609	The identity element of a group is a left identity. Deduction associated with grplid 13607. (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 13588	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 14105	Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
|- ( R e. SRing -> ( ||r ` R ) e. _V )
24-Jan-2025	reldvdsrsrg 14099	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 9016	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 8844	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	ressval3d 13148	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 13147	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 4300	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 4213. (Revised by BJ, 17-Jan-2025.)
|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )
16-Jan-2025	ressex 13141	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 13140	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 ) ) >. ) )
12-Jan-2025	isrim 14176	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 14178	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 14168	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 14079	Existence of the opposite ring. If you know that R is a ring, see opprring 14085. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> O e. _V )
10-Jan-2025	mgpex 13931	Existence of the multiplication group. If R is known to be a semiring, see srgmgp 13974. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
6-Jan-2025	ord3 6589	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 3806	The singleton of an element of a class is a subset of the class (inference form of snssg 3805). 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 3805	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 3804	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 6991	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 6987	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 4375	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 2760	Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 2763 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 7372	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 7371	Lemma for nninfinfwlpo 7373. (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 7370	Lemma for nninfwlpoim 7372. (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 7369	Lemma for nninfwlpoim 7372. (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 7368	Lemma for nninfwlpoim 7372. (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 7367	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 7366	Lemma for nninfwlpor 7367. 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 6946	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 6921	The domain of a one-to-one set function is dominated by its codomain when the latter is a set. This variation of f1domg 6926 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 6920	The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng 6925 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 7365	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 7374	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 7364	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 6913	Dominance relation. This variation of brdomg 6914 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of brdomg 6914. (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 13136	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 )
22-Nov-2024	eliotaeu 5313	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 5312	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 13135	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 938	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 14450	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 14449	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 13251	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 16297	Contraposition when the antecedent is a negated stable proposition. See con1dc 861. (Contributed by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
11-Nov-2024	slotsdifdsndx 13301	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 13290	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 13289	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 13286	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 13270	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 6232	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 857	Contraposition when the antecedent is a negated stable proposition. See comment of condc 858. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
10-Nov-2024	slotsdifunifndx 13308	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 13143	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 14082	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 14081	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 14080	Lemma for opprbasg 14081 and oppraddg 14082. (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 15727	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 14649	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 14648	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 14647	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 14646	Lemma for znbas 14651. (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 14638	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 14637	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 14636	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 14635	Lemma for zlmbasg 14636 and zlmplusgg 14637. (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 14639	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 13285	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 13284	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 13283	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 10905	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 13298	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 13269	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 13267	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 13266	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 13265	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 15854	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 13281	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 13280	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 13279	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 14455	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 14452	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 14448	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 14447	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 14446	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 14445	Lemma for srabaseg 14446 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 13300	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 13299	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 13271	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 13250	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 13249	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 13237	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 13232	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 11087	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 11086	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 11085	Lemma for fiubz 11086 and fiubnn 11087. 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 15853	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 13307	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 13305	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 13304	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 13296	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 13295	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 13294	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 16289	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 16535 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 16284	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 16283 as its last step. (Contributed by BJ, 27-Oct-2024.)
|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps
25-Oct-2024	nnwosdc 12603	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 12600	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 12601	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 16535	Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 855, 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 13306	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 13248	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 13230	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 12706	Lemma for isprm5 12707. 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 13149	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 14358	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 14296 of a left module, see also islmod 14298. (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 13937	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 13297	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 13282	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 13268	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 13238	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 13236	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 13235	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 13231	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 13220	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 13219	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 13218	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 13189	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 13187	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 9839	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 11797	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 11796	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 15852	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 15851	Index value of the df-edgf 15849 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
|- (.ef ` ndx ) = ; 1 8
13-Oct-2024	prdsvallem 13348	Lemma for prdsval 13349. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 13349, dependency on df-hom 13177 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 12876	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 10031	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 10030	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 10965	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 12854	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 9838	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 12853	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 12852	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 10964	Special case of ltexp2 15658 which we use here because we haven't yet defined df-rpcxp 15576 which is used in the current proof of ltexp2 15658. (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 10492	The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 8146.) (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 10491	An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 8146.) (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 10489	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 , < ) )
1-Oct-2024	infex2g 7227	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
|- ( A e. C -> inf ( B , A , R ) e. _V )
30-Sep-2024	unbendc 13068	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 12695	Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( N e. NN -> DECID N e. Prime )
30-Sep-2024	dcfi 7174	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 5977	Change bound variable in a restricted description binder. Version of cbvriotav 5979 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 5290	Change bound variables in a description binder. Version of cbviotav 5291 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 13061	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 13060	Lemma for ssnnct 13061. 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 10490	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 9825	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 13064	Lemma for nninfdc 13067. 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 12599	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 12598. (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 13062	Lemma for nninfdc 13067. (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 13065	Lemma for nninfdc 13067. The function from nninfdclemf 13063 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 13067	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 13066	Lemma for nninfdc 13067. The function from nninfdclemf 13063 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 13063	Lemma for nninfdc 13067. 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 12598	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 6911	Equinumerosity relation. This variation of bren 6912 does not require the Axiom of Union. (Contributed by NM, 15-Jun-1998.) Extract from a subproof of bren 6912. (Revised by BTernaryTau, 23-Sep-2024.)
|- ( ( A e. V /\ B e. W ) -> ( A ~~ B <-> E. f f : A -1-1-onto-> B ) )
19-Sep-2024	ssomct 13059	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 6594	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 1385	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 5828	Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 5497. (Contributed by AV, 18-Sep-2024.)
|- ( ( F : A --> B /\ Fun G ) -> ( F o. G ) : ( `' G " A ) --> B )
17-Sep-2024	fncofn 5827	Composition of a function with domain and a function as a function with domain. Generalization of fnco 5437. (Contributed by AV, 17-Sep-2024.)
|- ( ( F Fn A /\ Fun G ) -> ( F o. G ) Fn ( `' G " A ) )
14-Sep-2024	nnpredlt 4720	The predecessor (see nnpredcl 4719) of a nonzero natural number is less than (see df-iord 4461) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
13-Sep-2024	nninfisollemeq 7325	Lemma for nninfisol 7326. 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 7324	Lemma for nninfisol 7326. 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 7323	Lemma for nninfisol 7326. 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 )
12-Sep-2024	nninfisol 7326	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 7373). (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 4852	A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4854. (Contributed by SN, 8-Sep-2024.)
|- Rel { <. x , y >. | ph }
7-Sep-2024	eulerthlemfi 12793	Lemma for eulerth 12798. 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 10921	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 12796	Lemma for eulerth 12798. 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 12797	Lemma for eulerth 12798. 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 12795	Lemma for eulerth 12798. (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 12794	Lemma for eulerth 12798. 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 14536	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 12190	A finite product of terms apart from zero is apart from zero. A version of fprodap0 12175 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 12183	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 7454	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 7450	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 4668	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4667. (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 4618	Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4617. (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 7457	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 7456	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 3136	Composition law for chained substitutions into a class. Version of csbco 3135 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 2771	Version of cbvreuv 2767 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 2770	Version of cbvrexv 2766 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 2769	Version of cbvralv 2765 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 2352	Version of cbvab 2353 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 1998	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 1997 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 1967	Change bound variable. See cbvexv 1965 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1494. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
25-Aug-2024	cbvalvw 1966	Change bound variable. See cbvalv 1964 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1494. (Revised by GG, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
25-Aug-2024	nfal 1622	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 1556. (Revised by GG, 25-Aug-2024.)
|- F/ x ph   =>    |- F/ x A. y ph
24-Aug-2024	gcdcomd 12538	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 ) )
21-Aug-2024	dvds2addd 12383	Deduction form of dvds2add 12379. (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 ) )
18-Aug-2024	prdsmulr 13354	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 13353	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 13352	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 13351	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 13349	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 13343	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 12159	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	if0ab 16351	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. Remark: a consequence which could be formalized is the inclusion |- if ( ph , A , (/) ) C_ A and therefore, using elpwg 3658, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 7458 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)
|- if ( ph , A , (/) ) = { x e. A | ph }
16-Aug-2024	fprodunsn 12158	Multiply in an additional term in a finite product. See also fprodsplitsn 12187 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 ) )
15-Aug-2024	bj-charfundcALT 16354	Alternate proof of bj-charfundc 16353. It was expected to be much shorter since it uses bj-charfun 16352 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 16352	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 8818	Equality of complex numbers is stable. Stability here means -. -. A = B -> A = B as defined at df-stab 836. 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 8817	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	fmelpw1o 7458	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 856, which translate to 1o and (/) respectively by iftrue 3608 and iffalse 3611, giving pwtrufal 16548). As proved in if0ab 16351, the associated element of ~P 1o is the extension, in ~P 1o, of the formula ph. (Contributed by BJ, 15-Aug-2024.)
|- if ( ph , 1o , (/) ) e. ~P 1o
15-Aug-2024	ifexd 4579	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 4578	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 4577	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 4576	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 3619	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 3618	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 )
12-Aug-2024	exmidontriimlem2 7430	Lemma for exmidontriim 7433. (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 7429	Lemma for exmidontriim 7433. A variation of r19.30dc 2678. (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 856	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 855 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 1695 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 4283): 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 2520 does not hold. Actually, -. -. EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying -. EXMID and noncontradiction holds (pm3.24 698). (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 7433	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 7432	Lemma for exmidontriim 7433. 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 7431	Lemma for exmidontriim 7433. 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 7320	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 7317	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 4771 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
|- ( i e. om |-> 1o ) e. ℕ∞
9-Aug-2024	ss1o0el1o 7100	Reformulation of ss1o0el1 4285 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
9-Aug-2024	pw1dc0el 7098	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 4285	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 7101	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 )
7-Aug-2024	pw1fin 7097	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	elomssom 4701	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 4702. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
6-Aug-2024	bj-charfunbi 16356	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 16355	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 16353	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	prodssdc 12143	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 16350	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 16349	The maps-to notation defines a function (deduction form). Note: one should similarly prove a deduction form of funopab4 5361, then prove funmptd 16349 from it, and then prove funmpt 5362 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 16301	The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID F.
5-Aug-2024	bj-dctru 16299	The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID T.
5-Aug-2024	bj-stfal 16288	The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB F.
5-Aug-2024	bj-sttru 16286	The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB T.
5-Aug-2024	prod1dc 12140	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	2ssom 6687	The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
2-Aug-2024	onntri52 7455	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 7453	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 7452	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 7451	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 7449	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 7448	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 7449), (3) implies (5) (onntri35 7448), (5) implies (1) (onntri51 7451), (2) implies (4) (onntri24 7453), (4) implies (5) (onntri45 7452), and (5) implies (2) (onntri52 7455). 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 7450 and exmidontri2or 7454, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7456 and (4) by onntri2or 7457. (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 2520	The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1695. (Contributed by Jim Kingdon, 1-Aug-2024.)
|- ( -. -. A. x e. A ph -> A. x e. A -. -. ph )
31-Jul-2024	3nsssucpw1 7447	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 7446	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 14677	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 7445	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 7444	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 7443	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 7442	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 7441	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 7440	The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 1o
30-Jul-2024	pw1ne0 7439	The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)
29-Jul-2024	grpcld 13590	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 7437	The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
28-Jul-2024	exmidpweq 7096	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 16616	Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 16615 by means of dceqnconst 16614. (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 16612	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 16618	Lemma for nconstwlpo 16620. 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 16617	Lemma for nconstwlpo 16620. 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 )
24-Jul-2024	tridceq 16610	Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 16597 and redcwlpo 16609). Thus, this is an analytic analogue to lpowlpo 7361. (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 16605	Weak omniscience stated in terms of equality with 0. Like iswomninn 16604 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	lpowlpo 7361	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7360. There is an analogue in terms of analytic omniscience principles at tridceq 16610. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
23-Jul-2024	nconstwlpolem 16619	Lemma for nconstwlpo 16620. (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 16614	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 16609 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 16611	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 5964	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 1546 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 16620	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 )
15-Jul-2024	fprodseq 12137	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 2535	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 2534	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 15695	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 12745, 2logb9irrap 15694 and sqrt2cxp2logb9e3 15692. 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 15694	Example for logbgcd1irrap 15687. 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 13408	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 13407	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 13406	Lemma for ercpbl 13407. (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 13405	Value of the function in qusval 13399. (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 13404	Value of the function in qusval 13399. (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 15685	Lemma for logbgcd1irrap 15687. 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 15495	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 15656	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 15686	Lemma for logbgcd1irrap 15687. 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 10973	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 15594	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 15662	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 15595	Deduction form of logrpap0 15594. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( log ` A ) # 0 )
3-Jul-2024	logrpap0b 15593	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 16543	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 16542	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 16589	An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
|- ( 0 (,) 1 ) ~~ RR
27-Jun-2024	iooref1o 16588	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 16621	Lemma for neapmkv 16622. 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 16607	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 16606	Lemma for ismkvnn 16607. 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 7354	Lemma for enmkv 7355. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
24-Jun-2024	neapmkv 16622	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 16615	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 16597 for more discussion of decidability of real number apartness. This is a weaker form of dceqnconst 16614 and in fact this theorem can be proved using dceqnconst 16614 as shown at dcapnconstALT 16616. (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 7355	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 6592 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 16608	Lemma for redcwlpo 16609. A biconditionalized version of trilpolemeq1 16594. (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 16609	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 16608). 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 10497 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 16604	Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7359 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 16603	Lemma for iswomnimap 7359. 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 7363	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 6592 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 7362	Lemma for enwomni 7363. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
19-Jun-2024	rpabscxpbnd 15657	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 15639	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 616	Deduction associated with biadani 614. 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 15622	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 15621	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 15619	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 15615	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 15613	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 15612	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 15611	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 15576	Define the power function on complex numbers. Because df-relog 15575 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 16599	Version of trirec0 16598 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 16598	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 16597). (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 7360	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 7359	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 7358	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 7357	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 14041	A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. CMnd )
1-Jun-2024	ringabld 14040	A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Abel )
1-Jun-2024	cmnmndd 13888	A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
1-Jun-2024	ablcmnd 13872	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 13589	A group is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mnd )
1-Jun-2024	fndmi 5427	The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
|- F Fn A   =>    |- dom F = A
29-May-2024	pw1nct 16554	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 16553	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 )
26-May-2024	elpwi2 4246	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 14712	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 15441	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 2754	Rule used to change bound variables, using implicit substitution. Version of cbvralf 2756 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 1553 and ax-bndl 1555 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 2714	Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 2764 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 2118	Rule used to change bound variables, using implicit substitution. Version of cbvmo 2117 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 15491	Lemma for eflt 15492. The converse of efltim 12252 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 15492	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 653	A negated syllogism inference. (Contributed by Wolf Lammen, 20-May-2024.)
|- ( ph -> ps )   &    |- ( -. ph -> ch )   =>    |- ( -. ps -> ch )
19-May-2024	apdifflemr 16601	Lemma for apdiff 16602. (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 16600	Lemma for apdiff 16602. 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 16602	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 14304	A left module is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> W e. LMod )   =>    |- ( ph -> W e. Grp )
16-May-2024	crnggrpd 14016	A commutative ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Grp )
16-May-2024	crngringd 14015	A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Ring )
16-May-2024	ringgrpd 14011	A ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Grp )
15-May-2024	reeff1oleme 15489	Lemma for reeff1o 15490. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
14-May-2024	df-relog 15575	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 6153	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 )
12-May-2024	dvdstrd 12384	The divides relation is transitive, a deduction version of dvdstr 12382. (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 15571	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 15569	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 15570	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 13058	The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 13054 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 13055	Variation of ctiunct 13054 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	ifpnst 994	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 7483	Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7482, 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 7481	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 7482	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 991	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 7480	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 999	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 7479	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 7478	Lemma for cc2 7479. (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 7477	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 14439	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 13057	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 998	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 12131	Lemma for prodmodc 12132. (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 ) )
11-Apr-2024	prodmodclem2a 12130	Lemma for prodmodc 12132. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-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 ) )   &    |- H = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( K ` j ) / k ]_ B , 1 ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> f : ( 1 ... N ) -1-1-onto-> A )   &    |- ( ph -> K Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   =>    |- ( ph -> seq M ( x. , F ) ~~> ( seq 1 ( x. , G ) ` N ) )
11-Apr-2024	prodmodclem3 12129	Lemma for prodmodc 12132. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-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 ) )   &    |- H = ( j e. NN |-> if ( j <_ (♯ ` A ) , [_ ( K ` j ) / k ]_ B , 1 ) )   &    |- ( ph -> ( M e. NN /\ N e. NN ) )   &    |- ( ph -> f : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ph -> K : ( 1 ... N ) -1-1-onto-> A )   =>    |- ( ph -> ( seq 1 ( x. , G ) ` M ) = ( seq 1 ( x. , H ) ` N ) )
10-Apr-2024	jcnd 656	Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)
|- ( ph -> ps )   &    |- ( ph -> -. ch )   =>    |- ( ph -> -. ( ps -> ch ) )
4-Apr-2024	prodrbdclem 12125	Lemma for prodrbdc 12128. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) )
24-Mar-2024	prodfdivap 12101	The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) # 0 )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) / ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )
24-Mar-2024	prodfrecap 12100	The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( 1 / ( F ` k ) ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   =>    |- ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )
23-Mar-2024	prodfap0 12099	The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )   =>    |- ( ph -> ( seq M ( x. , F ) ` N ) # 0 )
22-Mar-2024	prod3fmul 12095	The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , G ) ` N ) ) )
21-Mar-2024	df-proddc 12105	Define the product of a series with an index set of integers A. This definition takes most of the aspects of df-sumdc 11908 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 21-Mar-2024.)
|- prod_ k e. A B = ( iota x ( 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. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> y ) /\ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( x. , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 1 ) ) ) ` m ) ) ) )
19-Mar-2024	cos02pilt1 15568	Cosine is less than one between zero and 2 x. pi. (Contributed by Jim Kingdon, 19-Mar-2024.)
|- ( A e. ( 0 (,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )
19-Mar-2024	cosq34lt1 15567	Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.)
|- ( A e. ( pi [,) ( 2 x. pi ) ) -> ( cos ` A ) < 1 )
14-Mar-2024	coseq0q4123 15551	Location of the zeroes of cosine in ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ). (Contributed by Jim Kingdon, 14-Mar-2024.)
|- ( A e. ( -u ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( ( cos ` A ) = 0 <-> A = ( pi / 2 ) ) )
14-Mar-2024	cosq23lt0 15550	The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)
|- ( A e. ( ( pi / 2 ) (,) ( 3 x. ( pi / 2 ) ) ) -> ( cos ` A ) < 0 )
9-Mar-2024	pilem3 15500	Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
9-Mar-2024	exmidonfin 7398	If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 7054 and nnon 4706. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
|- ( om = ( On i^i Fin ) -> EXMID )
9-Mar-2024	exmidonfinlem 7397	Lemma for exmidonfin 7398. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
|- A = { { x e. { (/) } | ph } , { x e. { (/) } | -. ph } }   =>    |- ( om = ( On i^i Fin ) -> DECID ph )
8-Mar-2024	sin0pilem2 15499	Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
|- E. q e. ( 2 (,) 4 ) ( ( sin ` q ) = 0 /\ A. x e. ( 0 (,) q ) 0 < ( sin ` x ) )
8-Mar-2024	sin0pilem1 15498	Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.)
|- E. p e. ( 1 (,) 2 ) ( ( cos ` p ) = 0 /\ A. x e. ( p (,) ( 2 x. p ) ) 0 < ( sin ` x ) )
7-Mar-2024	cosz12 15497	Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.)
|- E. p e. ( 1 (,) 2 ) ( cos ` p ) = 0
6-Mar-2024	cos12dec 12322	Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.)
|- ( ( A e. ( 1 [,] 2 ) /\ B e. ( 1 [,] 2 ) /\ A < B ) -> ( cos ` B ) < ( cos ` A ) )
2-Mar-2024	clwwlknonmpo 16237	(ClWWalksNOn ` G ) is an operator mapping a vertex v and a nonnegative integer n to the set of closed walks on v of length n as words over the set of vertices in a graph G. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)
|- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
2-Mar-2024	scaffvalg 14313	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
|- B = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .xb = ( .sf ` W )   &    |- .x. = ( .s ` W )   =>    |- ( W e. V -> .xb = ( x e. K , y e. B |-> ( x .x. y ) ) )
2-Mar-2024	dvrfvald 14140	Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> I = ( invr ` R ) )   &    |- ( ph -> ./ = (/r ` R ) )   &    |- ( ph -> R e. SRing )   =>    |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
2-Mar-2024	plusffvalg 13438	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .+^ = ( +f ` G )   =>    |- ( G e. V -> .+^ = ( x e. B , y e. B |-> ( x .+ y ) ) )
25-Feb-2024	insubm 13561	The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
|- ( ( A e. (SubMnd ` M ) /\ B e. (SubMnd ` M ) ) -> ( A i^i B ) e. (SubMnd ` M ) )
25-Feb-2024	mul2lt0pn 9992	The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 0 )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( B x. A ) < 0 )
25-Feb-2024	mul2lt0np 9991	The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 0 )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> ( A x. B ) < 0 )
25-Feb-2024	lt0ap0 8821	A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ( A e. RR /\ A < 0 ) -> A # 0 )
25-Feb-2024	negap0d 8804	The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> -u A # 0 )
24-Feb-2024	lt0ap0d 8822	A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 0 )   =>    |- ( ph -> A # 0 )
20-Feb-2024	ivthdec 15361	The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` B ) < U /\ U < ( F ` A ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` y ) < ( F ` x ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
20-Feb-2024	ivthinclemex 15359	Lemma for ivthinc 15360. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E! z e. ( A (,) B ) ( A. q e. L q < z /\ A. r e. R z < r ) )
19-Feb-2024	ivthinclemuopn 15355	Lemma for ivthinc 15360. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   &    |- ( ph -> S e. R )   =>    |- ( ph -> E. q e. R q < S )
19-Feb-2024	dedekindicc 15350	A Dedekind cut identifies a unique real number. Similar to df-inp 7679 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E! x e. ( A (,) B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
19-Feb-2024	grpsubfvalg 13621	Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- I = ( invg ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. V -> .- = ( x e. B , y e. B |-> ( x .+ ( I ` y ) ) ) )
18-Feb-2024	ivthinclemloc 15358	Lemma for ivthinc 15360. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. R ) ) )
18-Feb-2024	ivthinclemdisj 15357	Lemma for ivthinc 15360. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> ( L i^i R ) = (/) )
18-Feb-2024	ivthinclemur 15356	Lemma for ivthinc 15360. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. r e. ( A [,] B ) ( r e. R <-> E. q e. R q < r ) )
18-Feb-2024	ivthinclemlr 15354	Lemma for ivthinc 15360. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )
18-Feb-2024	ivthinclemum 15352	Lemma for ivthinc 15360. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E. r e. ( A [,] B ) r e. R )
18-Feb-2024	ivthinclemlm 15351	Lemma for ivthinc 15360. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   =>    |- ( ph -> E. q e. ( A [,] B ) q e. L )
17-Feb-2024	0subm 13560	The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
|- .0. = ( 0g ` G )   =>    |- ( G e. Mnd -> { .0. } e. (SubMnd ` G ) )
17-Feb-2024	mndissubm 13551	If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Mnd /\ H e. Mnd ) -> ( ( S C_ B /\ .0. e. S /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. (SubMnd ` G ) ) )
17-Feb-2024	mgmsscl 13437	If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.)
|- B = ( Base ` G )   &    |- S = ( Base ` H )   =>    |- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( X e. S /\ Y e. S ) ) -> ( X ( +g ` G ) Y ) e. S )
15-Feb-2024	dedekindicclemeu 15348	Lemma for dedekindicc 15350. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   &    |- ( ph -> C e. ( A [,] B ) )   &    |- ( ph -> ( A. q e. L q < C /\ A. r e. U C < r ) )   &    |- ( ph -> D e. ( A [,] B ) )   &    |- ( ph -> ( A. q e. L q < D /\ A. r e. U D < r ) )   &    |- ( ph -> C < D )   =>    |- ( ph -> F. )
15-Feb-2024	dedekindicclemlu 15347	Lemma for dedekindicc 15350. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E. x e. ( A [,] B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
15-Feb-2024	dedekindicclemlub 15346	Lemma for dedekindicc 15350. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E. x e. ( A [,] B ) ( A. y e. L -. x < y /\ A. y e. ( A [,] B ) ( y < x -> E. z e. L y < z ) ) )
15-Feb-2024	dedekindicclemloc 15345	Lemma for dedekindicc 15350. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> A. x e. ( A [,] B ) A. y e. ( A [,] B ) ( x < y -> ( E. z e. L x < z \/ A. z e. L z < y ) ) )
15-Feb-2024	dedekindicclemub 15344	Lemma for dedekindicc 15350. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. ( A [,] B ) A. y e. L y < x )
15-Feb-2024	dedekindicclemuub 15343	Lemma for dedekindicc 15350. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> C e. U )   =>    |- ( ph -> A. z e. L z < C )
14-Feb-2024	suplociccex 15342	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8245 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
|- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B < C )   &    |- ( ph -> A C_ ( B [,] C ) )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ( B [,] C ) A. y e. ( B [,] C ) ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) )   =>    |- ( ph -> E. x e. ( B [,] C ) ( A. y e. A -. x < y /\ A. y e. ( B [,] C ) ( y < x -> E. z e. A y < z ) ) )
14-Feb-2024	suplociccreex 15341	An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8245 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.)
|- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> B < C )   &    |- ( ph -> A C_ ( B [,] C ) )   &    |- ( ph -> E. x x e. A )   &    |- ( ph -> A. x e. ( B [,] C ) A. y e. ( B [,] C ) ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) )   =>    |- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
10-Feb-2024	cbvexdvaw 1978	Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 1976 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( E. x ps <-> E. y ch ) )
10-Feb-2024	cbvaldvaw 1977	Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1975 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
6-Feb-2024	ivthinclemlopn 15353	Lemma for ivthinc 15360. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   &    |- L = { w e. ( A [,] B ) | ( F ` w ) < U }   &    |- R = { w e. ( A [,] B ) | U < ( F ` w ) }   &    |- ( ph -> Q e. L )   =>    |- ( ph -> E. r e. L Q < r )
5-Feb-2024	ivthinc 15360	The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> U e. RR )   &    |- ( ph -> A < B )   &    |- ( ph -> ( A [,] B ) C_ D )   &    |- ( ph -> F e. ( D -cn-> CC ) )   &    |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. RR )   &    |- ( ph -> ( ( F ` A ) < U /\ U < ( F ` B ) ) )   &    |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ ( y e. ( A [,] B ) /\ x < y ) ) -> ( F ` x ) < ( F ` y ) )   =>    |- ( ph -> E. c e. ( A (,) B ) ( F ` c ) = U )
2-Feb-2024	dedekindeulemuub 15334	Lemma for dedekindeu 15340. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A e. U )   =>    |- ( ph -> A. z e. L z < A )
31-Jan-2024	dedekindeulemeu 15339	Lemma for dedekindeu 15340. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A e. RR )   &    |- ( ph -> ( A. q e. L q < A /\ A. r e. U A < r ) )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( A. q e. L q < B /\ A. r e. U B < r ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> F. )
31-Jan-2024	dedekindeulemlu 15338	Lemma for dedekindeu 15340. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR ( A. q e. L q < x /\ A. r e. U x < r ) )
31-Jan-2024	dedekindeulemlub 15337	Lemma for dedekindeu 15340. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR ( A. y e. L -. x < y /\ A. y e. RR ( y < x -> E. z e. L y < z ) ) )
31-Jan-2024	dedekindeulemloc 15336	Lemma for dedekindeu 15340. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> A. x e. RR A. y e. RR ( x < y -> ( E. z e. L x < z \/ A. z e. L z < y ) ) )
31-Jan-2024	dedekindeulemub 15335	Lemma for dedekindeu 15340. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E. x e. RR A. y e. L y < x )
30-Jan-2024	axsuploc 8245	An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 8146 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y < x /\ A. x e. RR A. y e. RR ( x < y -> ( E. z e. A x < z \/ A. z e. A z < y ) ) ) ) -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )
30-Jan-2024	iotam 5316	Representation of "the unique element such that ph " with a class expression A which is inhabited (that means that "the unique element such that ph " exists). (Contributed by AV, 30-Jan-2024.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( ( A e. V /\ E. w w e. A /\ A = ( iota x ph ) ) -> ps )
29-Jan-2024	sgrpidmndm 13496	A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ( G e. Smgrp /\ E. e e. B ( E. w w e. e /\ e = .0. ) ) -> G e. Mnd )
29-Jan-2024	ccatw2s1p1g 11215	Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.)
|- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = X )
28-Jan-2024	ccat2s1fstg 11218	The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 28-Jan-2024.)
|- ( ( ( W e. Word V /\ 0 < (♯ ` W ) ) /\ ( X e. A /\ Y e. B ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
28-Jan-2024	ccat2s1fvwd 11217	Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.)
|- ( ph -> W e. Word V )   &    |- ( ph -> I e. NN0 )   &    |- ( ph -> I < (♯ ` W ) )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )
26-Jan-2024	elovmporab1w 6218	Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by GG, 26-Jan-2024.)
|- O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } )   &    |- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V )   =>    |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) )
26-Jan-2024	opabidw 4349	The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Version of opabid 4348 with a disjoint variable condition. (Contributed by NM, 14-Apr-1995.) (Revised by GG, 26-Jan-2024.)
|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
26-Jan-2024	invdisjrab 4080	The restricted class abstractions { x e. B | C = y } for distinct y e. A are disjoint. (Contributed by AV, 6-May-2020.) (Proof shortened by GG, 26-Jan-2024.)
|- Disj y e. A { x e. B | C = y }
24-Jan-2024	axpre-suploclemres 8114	Lemma for axpre-suploc 8115. The result. The proof just needs to define B as basically the same set as A (but expressed as a subset of R. rather than a subset of RR), and apply suplocsr 8022. (Contributed by Jim Kingdon, 24-Jan-2024.)
|- ( ph -> A C_ RR )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. RR A. y e. A y <RR x )   &    |- ( ph -> A. x e. RR A. y e. RR ( x <RR y -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) )   &    |- B = { w e. R. | <. w , 0R >. e. A }   =>    |- ( ph -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )
23-Jan-2024	ax-pre-suploc 8146	An inhabited, bounded-above, located set of reals has a supremum. Locatedness here means that given x < y, either there is an element of the set greater than x, or y is an upper bound. Although this and ax-caucvg 8145 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 8145. (Contributed by Jim Kingdon, 23-Jan-2024.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y <RR x /\ A. x e. RR A. y e. RR ( x <RR y -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )
23-Jan-2024	axpre-suploc 8115	An inhabited, bounded-above, located set of reals has a supremum. Locatedness here means that given x < y, either there is an element of the set greater than x, or y is an upper bound. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 8146. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)
|- ( ( ( A C_ RR /\ E. x x e. A ) /\ ( E. x e. RR A. y e. A y <RR x /\ A. x e. RR A. y e. RR ( x <RR y -> ( E. z e. A x <RR z \/ A. z e. A z <RR y ) ) ) ) -> E. x e. RR ( A. y e. A -. x <RR y /\ A. y e. RR ( y <RR x -> E. z e. A y <RR z ) ) )
22-Jan-2024	suplocsr 8022	An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> E. x e. R. ( A. y e. A -. x <R y /\ A. y e. R. ( y <R x -> E. z e. A y <R z ) ) )
21-Jan-2024	bj-el2oss1o 16320	Shorter proof of el2oss1o 6606 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( A e. 2o -> A C_ 1o )
21-Jan-2024	ltm1sr 7990	Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.)
|- ( A e. R. -> ( A +R -1R ) <R A )
20-Jan-2024	mndinvmod 13521	Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) )
20-Jan-2024	ccats1val1g 11209	Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by JJ, 20-Jan-2024.)
|- ( ( W e. Word V /\ S e. Y /\ I e. ( 0..^ (♯ ` W ) ) ) -> ( ( W ++ <" S "> ) ` I ) = ( W ` I ) )
19-Jan-2024	suplocsrlempr 8020	Lemma for suplocsr 8022. The set B has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.)
|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }   &    |- ( ph -> A C_ R. )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> E. v e. P. ( A. w e. B -. v <P w /\ A. w e. P. ( w <P v -> E. u e. B w <P u ) ) )
18-Jan-2024	ccatval1 11167	Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.)
|- ( ( S e. Word A /\ T e. Word B /\ I e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` I ) = ( S ` I ) )
18-Jan-2024	ccat0 11166	The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.)
|- ( ( S e. Word A /\ T e. Word B ) -> ( ( S ++ T ) = (/) <-> ( S = (/) /\ T = (/) ) ) )
18-Jan-2024	suplocsrlemb 8019	Lemma for suplocsr 8022. The set B is located. (Contributed by Jim Kingdon, 18-Jan-2024.)
|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }   &    |- ( ph -> A C_ R. )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> A. u e. P. A. v e. P. ( u <P v -> ( E. q e. B u <P q \/ A. q e. B q <P v ) ) )
16-Jan-2024	suplocsrlem 8021	Lemma for suplocsr 8022. The set A has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.)
|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }   &    |- ( ph -> A C_ R. )   &    |- ( ph -> C e. A )   &    |- ( ph -> E. x e. R. A. y e. A y <R x )   &    |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )   =>    |- ( ph -> E. x e. R. ( A. y e. A -. x <R y /\ A. y e. R. ( y <R x -> E. z e. A y <R z ) ) )
15-Jan-2024	eqg0el 13809	Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
|- .~ = ( G ~QG H )   =>    |- ( ( G e. Grp /\ H e. (SubGrp ` G ) ) -> ( [ X ] .~ = H <-> X e. H ) )
14-Jan-2024	wlklenvclwlk 16184	The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)
|- ( W e. Word (Vtx ` G ) -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. (Walks ` G ) -> (♯ ` F ) = (♯ ` W ) ) )
14-Jan-2024	suplocexprlemlub 7937	Lemma for suplocexpr 7938. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> ( y <P B -> E. z e. A y <P z ) )
14-Jan-2024	suplocexprlemub 7936	Lemma for suplocexpr 7938. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> A. y e. A -. B <P y )
10-Jan-2024	nfcsbw 3162	Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3163 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x [_ A / y ]_ B
10-Jan-2024	nfsbcw 3160	Bound-variable hypothesis builder for class substitution. Version of nfsbc 3050 with a disjoint variable condition, which in the future may make it possible to reduce axiom usage. (Contributed by NM, 7-Sep-2014.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x [. A / y ]. ph
10-Jan-2024	nfsbcdw 3159	Version of nfsbcd 3049 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x [. A / y ]. ps )
10-Jan-2024	cbvcsbw 3129	Version of cbvcsb 3130 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
|- F/_ y C   &    |- F/_ x D   &    |- ( x = y -> C = D )   =>    |- [_ A / x ]_ C = [_ A / y ]_ D
10-Jan-2024	cbvsbcw 3057	Version of cbvsbc 3058 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
10-Jan-2024	cbvrex2vw 2777	Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2779 with a disjoint variable condition, which does not require ax-13 2202. (Contributed by FL, 2-Jul-2012.) (Revised by GG, 10-Jan-2024.)
|- ( x = z -> ( ph <-> ch ) )   &    |- ( y = w -> ( ch <-> ps ) )   =>    |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps )
10-Jan-2024	cbvral2vw 2776	Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2778 with a disjoint variable condition, which does not require ax-13 2202. (Contributed by NM, 10-Aug-2004.) (Revised by GG, 10-Jan-2024.)
|- ( x = z -> ( ph <-> ch ) )   &    |- ( y = w -> ( ch <-> ps ) )   =>    |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps )
10-Jan-2024	cbvrexw 2759	Rule used to change bound variables, using implicit substitution. Version of cbvrexfw 2755 with more disjoint variable conditions. Although we don't do so yet, we expect the disjoint variable conditions will allow us to remove reliance on ax-i12 1553 and ax-bndl 1555 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
10-Jan-2024	cbvralw 2758	Rule used to change bound variables, using implicit substitution. Version of cbvral 2761 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 1553 and ax-bndl 1555 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
10-Jan-2024	cbvrexfw 2755	Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2757 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 1553 and ax-bndl 1555 in the proof. (Contributed by FL, 27-Apr-2008.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/_ y A   &    |- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
10-Jan-2024	nfralw 2567	Bound-variable hypothesis builder for restricted quantification. See nfralya 2570 for a version with y and A distinct instead of x and y. (Contributed by NM, 1-Sep-1999.) (Revised by GG, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x A. y e. A ph
10-Jan-2024	nfraldw 2562	Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2565 for a version with y and A distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y e. A ps )
10-Jan-2024	nfabdw 2391	Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2392 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by GG, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x { y | ps } )
10-Jan-2024	cbvex2vw 1980	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) (Revised by GG, 10-Jan-2024.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( E. x E. y ph <-> E. z E. w ps )
10-Jan-2024	cbval2vw 1979	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) (Revised by GG, 10-Jan-2024.)
|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )   =>    |- ( A. x A. y ph <-> A. z A. w ps )
10-Jan-2024	cbv2w 1796	Rule used to change bound variables, using implicit substitution. Version of cbv2 1795 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by GG, 10-Jan-2024.)
|- F/ x ph   &    |- F/ y ph   &    |- ( ph -> F/ y ps )   &    |- ( ph -> F/ x ch )   &    |- ( ph -> ( x = y -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( A. x ps <-> A. y ch ) )
9-Jan-2024	suplocexprlemloc 7934	Lemma for suplocexpr 7938. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) )
9-Jan-2024	suplocexprlemdisj 7933	Lemma for suplocexpr 7938. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
9-Jan-2024	suplocexprlemru 7932	Lemma for suplocexpr 7938. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
9-Jan-2024	suplocexprlemrl 7930	Lemma for suplocexpr 7938. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   =>    |- ( ph -> A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )
9-Jan-2024	suplocexprlem2b 7927	Lemma for suplocexpr 7938. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
9-Jan-2024	suplocexprlemell 7926	Lemma for suplocexpr 7938. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.)
|- ( B e. U. ( 1st " A ) <-> E. x e. A B e. ( 1st ` x ) )
7-Jan-2024	suplocexpr 7938	An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   =>    |- ( ph -> E. x e. P. ( A. y e. A -. x <P y /\ A. y e. P. ( y <P x -> E. z e. A y <P z ) ) )
7-Jan-2024	suplocexprlemex 7935	Lemma for suplocexpr 7938. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> B e. P. )
7-Jan-2024	suplocexprlemmu 7931	Lemma for suplocexpr 7938. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   &    |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.   =>    |- ( ph -> E. s e. Q. s e. ( 2nd ` B ) )
7-Jan-2024	suplocexprlemml 7929	Lemma for suplocexpr 7938. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   =>    |- ( ph -> E. s e. Q. s e. U. ( 1st " A ) )
7-Jan-2024	suplocexprlemss 7928	Lemma for suplocexpr 7938. A is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
|- ( ph -> E. x x e. A )   &    |- ( ph -> E. x e. P. A. y e. A y <P x )   &    |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )   =>    |- ( ph -> A C_ P. )
5-Jan-2024	dedekindicclemicc 15349	Lemma for dedekindicc 15350. Same as dedekindicc 15350, except that we merely show x to be an element of ( A [,] B ). Later we will strengthen that to ( A (,) B ). (Contributed by Jim Kingdon, 5-Jan-2024.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> L C_ ( A [,] B ) )   &    |- ( ph -> U C_ ( A [,] B ) )   &    |- ( ph -> E. q e. ( A [,] B ) q e. L )   &    |- ( ph -> E. r e. ( A [,] B ) r e. U )   &    |- ( ph -> A. q e. ( A [,] B ) ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. ( A [,] B ) ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. ( A [,] B ) A. r e. ( A [,] B ) ( q < r -> ( q e. L \/ r e. U ) ) )   &    |- ( ph -> A < B )   =>    |- ( ph -> E! x e. ( A [,] B ) ( A. q e. L q < x /\ A. r e. U x < r ) )
5-Jan-2024	dedekindeu 15340	A Dedekind cut identifies a unique real number. Similar to df-inp 7679 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.)
|- ( ph -> L C_ RR )   &    |- ( ph -> U C_ RR )   &    |- ( ph -> E. q e. RR q e. L )   &    |- ( ph -> E. r e. RR r e. U )   &    |- ( ph -> A. q e. RR ( q e. L <-> E. r e. L q < r ) )   &    |- ( ph -> A. r e. RR ( r e. U <-> E. q e. U q < r ) )   &    |- ( ph -> ( L i^i U ) = (/) )   &    |- ( ph -> A. q e. RR A. r e. RR ( q < r -> ( q e. L \/ r e. U ) ) )   =>    |- ( ph -> E! x e. RR ( A. q e. L q < x /\ A. r e. U x < r ) )
1-Jan-2024	ccatlen 11165	The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.)
|- ( ( S e. Word A /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
31-Dec-2023	dvmptsubcn 15440	Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
|- ( ( ph /\ x e. CC ) -> A e. CC )   &    |- ( ( ph /\ x e. CC ) -> B e. V )   &    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> B ) )   &    |- ( ( ph /\ x e. CC ) -> C e. CC )   &    |- ( ( ph /\ x e. CC ) -> D e. W )   &    |- ( ph -> ( CC _D ( x e. CC |-> C ) ) = ( x e. CC |-> D ) )   =>    |- ( ph -> ( CC _D ( x e. CC |-> ( A - C ) ) ) = ( x e. CC |-> ( B - D ) ) )
31-Dec-2023	dvmptnegcn 15439	Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
|- ( ( ph /\ x e. CC ) -> A e. CC )   &    |- ( ( ph /\ x e. CC ) -> B e. V )   &    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> B ) )   =>    |- ( ph -> ( CC _D ( x e. CC |-> -u A ) ) = ( x e. CC |-> -u B ) )
31-Dec-2023	dvmptcmulcn 15438	Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.)
|- ( ( ph /\ x e. CC ) -> A e. CC )   &    |- ( ( ph /\ x e. CC ) -> B e. V )   &    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> B ) )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( CC _D ( x e. CC |-> ( C x. A ) ) ) = ( x e. CC |-> ( C x. B ) ) )
31-Dec-2023	rinvmod 13889	Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6211. (Contributed by AV, 31-Dec-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E* w e. B ( A .+ w ) = .0. )
31-Dec-2023	brm 4137	If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.)
|- ( A R B -> E. x x e. R )
30-Dec-2023	dvmptccn 15432	Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( CC _D ( x e. CC |-> A ) ) = ( x e. CC |-> 0 ) )
30-Dec-2023	dvmptidcn 15431	Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.)
|- ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 )
30-Dec-2023	eqwrd 11147	Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)
|- ( ( U e. Word S /\ W e. Word T ) -> ( U = W <-> ( (♯ ` U ) = (♯ ` W ) /\ A. i e. ( 0..^ (♯ ` U ) ) ( U ` i ) = ( W ` i ) ) ) )
29-Dec-2023	mndbn0 13507	The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 13506). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
|- B = ( Base ` G )   =>    |- ( G e. Mnd -> B =/= (/) )
28-Dec-2023	mulgnn0gsum 13708	Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- F = ( x e. ( 1 ... N ) |-> X )   =>    |- ( ( N e. NN0 /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )
28-Dec-2023	mulgnngsum 13707	Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- F = ( x e. ( 1 ... N ) |-> X )   =>    |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( G gsumg F ) )
26-Dec-2023	gsumfzreidx 13917	Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with M = 1. (Contributed by AV, 26-Dec-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> F : ( M ... N ) --> B )   &    |- ( ph -> H : ( M ... N ) -1-1-onto-> ( M ... N ) )   =>    |- ( ph -> ( G gsumg F ) = ( G gsumg ( F o. H ) ) )
26-Dec-2023	gsumsplit1r 13474	Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. V )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> F : ( M ... ( N + 1 ) ) --> B )   =>    |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` ( M ... N ) ) ) .+ ( F ` ( N + 1 ) ) ) )
26-Dec-2023	lidrididd 13458	If there is a left and right identity element for any binary operation (group operation) .+, the left identity element (and therefore also the right identity element according to lidrideqd 13457) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
|- ( ph -> L e. B )   &    |- ( ph -> R e. B )   &    |- ( ph -> A. x e. B ( L .+ x ) = x )   &    |- ( ph -> A. x e. B ( x .+ R ) = x )   &    |- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   =>    |- ( ph -> L = .0. )
26-Dec-2023	lidrideqd 13457	If there is a left and right identity element for any binary operation (group operation) .+, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
|- ( ph -> L e. B )   &    |- ( ph -> R e. B )   &    |- ( ph -> A. x e. B ( L .+ x ) = x )   &    |- ( ph -> A. x e. B ( x .+ R ) = x )   =>    |- ( ph -> L = R )
25-Dec-2023	ctfoex 7311	A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
|- ( E. f f : om -onto-> ( A ⊔ 1o ) -> A e. _V )
23-Dec-2023	enct 13047	Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- ( A ~~ B -> ( E. f f : om -onto-> ( A ⊔ 1o ) <-> E. g g : om -onto-> ( B ⊔ 1o ) ) )
23-Dec-2023	enctlem 13046	Lemma for enct 13047. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- ( A ~~ B -> ( E. f f : om -onto-> ( A ⊔ 1o ) -> E. g g : om -onto-> ( B ⊔ 1o ) ) )
23-Dec-2023	omct 7310	om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- E. f f : om -onto-> ( om ⊔ 1o )
21-Dec-2023	dvcoapbr 15424	The chain rule for derivatives at a point. The u # C -> ( G ` u ) # ( G ` C ) hypothesis constrains what functions work for G. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : Y --> X )   &    |- ( ph -> Y C_ T )   &    |- ( ph -> A. u e. Y ( u # C -> ( G ` u ) # ( G ` C ) ) )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> T C_ CC )   &    |- ( ph -> ( G ` C ) ( S _D F ) K )   &    |- ( ph -> C ( T _D G ) L )   &    |- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> C ( T _D ( F o. G ) ) ( K x. L ) )
19-Dec-2023	apsscn 8820	The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.)
|- { x e. A | x # B } C_ CC
19-Dec-2023	aprcl 8819	Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
|- ( A # B -> ( A e. CC /\ B e. CC ) )
18-Dec-2023	limccoap 15395	Composition of two limits. This theorem is only usable in the case where x # X implies R(x) # C so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
|- ( ( ph /\ x e. { w e. A | w # X } ) -> R e. { w e. B | w # C } )   &    |- ( ( ph /\ y e. { w e. B | w # C } ) -> S e. CC )   &    |- ( ph -> C e. ( ( x e. { w e. A | w # X } |-> R ) lim CC X ) )   &    |- ( ph -> D e. ( ( y e. { w e. B | w # C } |-> S ) lim CC C ) )   &    |- ( y = R -> S = T )   =>    |- ( ph -> D e. ( ( x e. { w e. A | w # X } |-> T ) lim CC X ) )
16-Dec-2023	cnreim 11532	Complex apartness in terms of real and imaginary parts. See also apreim 8776 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.)
|- ( ( A e. CC /\ B e. CC ) -> ( A # B <-> ( ( Re ` A ) # ( Re ` B ) \/ ( Im ` A ) # ( Im ` B ) ) ) )
14-Dec-2023	cnopnap 15328	The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
|- ( A e. CC -> { w e. CC | w # A } e. ( MetOpen ` ( abs o. - ) ) )
14-Dec-2023	cnovex 14913	The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.)
|- ( ( J e. Top /\ K e. Top ) -> ( J Cn K ) e. _V )
13-Dec-2023	reopnap 15263	The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
|- ( A e. RR -> { w e. RR | w # A } e. ( topGen ` ran (,) ) )
12-Dec-2023	cnopncntop 15261	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
|- CC e. ( MetOpen ` ( abs o. - ) )
12-Dec-2023	unicntopcntop 15259	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
|- CC = U. ( MetOpen ` ( abs o. - ) )
4-Dec-2023	bj-pm2.18st 16296	Clavius law for stable formulas. See pm2.18dc 860. (Contributed by BJ, 4-Dec-2023.)
|- (STAB ph -> ( ( -. ph -> ph ) -> ph ) )
4-Dec-2023	bj-nnclavius 16283	Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
|- ( ( -. ph -> ph ) -> -. -. ph )
2-Dec-2023	dvmulxx 15421	The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 15419. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> C e. dom ( S _D F ) )   &    |- ( ph -> C e. dom ( S _D G ) )   =>    |- ( ph -> ( ( S _D ( F oF x. G ) ) ` C ) = ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) )
1-Dec-2023	dvmulxxbr 15419	The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 15421. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> C ( S _D F ) K )   &    |- ( ph -> C ( S _D G ) L )   &    |- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> C ( S _D ( F oF x. G ) ) ( ( K x. ( G ` C ) ) + ( L x. ( F ` C ) ) ) )
29-Nov-2023	subctctexmid 16551	If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.)
|- ( ph -> A. x ( E. s ( s C_ om /\ E. f f : s -onto-> x ) -> E. g g : om -onto-> ( x ⊔ 1o ) ) )   &    |- ( ph -> om e. Markov )   =>    |- ( ph -> EXMID )
29-Nov-2023	ismkvnex 7348	The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
|- ( A e. V -> ( A e. Markov <-> A. f e. ( 2o ^m A ) ( -. -. E. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = 1o ) ) )
28-Nov-2023	ccfunen 7476	Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.)
|- ( ph -> CCHOICE )   &    |- ( ph -> A ~~ om )   &    |- ( ph -> A. x e. A E. w w e. x )   =>    |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
28-Nov-2023	exmid1stab 4296	If every proposition is stable, excluded middle follows. We are thinking of x as a proposition and x = { (/) } as " x is true". (Contributed by Jim Kingdon, 28-Nov-2023.)
|- ( ( ph /\ x C_ { (/) } ) -> STAB x = { (/) } )   =>    |- ( ph -> EXMID )
27-Nov-2023	df-cc 7475	The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7414 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.)
|- (CCHOICE <-> A. x ( dom x ~~ om -> E. f ( f C_ x /\ f Fn dom x ) ) )
26-Nov-2023	offeq 6244	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : B --> T )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   &    |- ( ph -> H : C --> U )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = D )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = E )   &    |- ( ( ph /\ x e. C ) -> ( D R E ) = ( H ` x ) )   =>    |- ( ph -> ( F oF R G ) = H )
25-Nov-2023	dvaddxx 15420	The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 15418. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> C e. dom ( S _D F ) )   &    |- ( ph -> C e. dom ( S _D G ) )   =>    |- ( ph -> ( ( S _D ( F oF + G ) ) ` C ) = ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) )
25-Nov-2023	dvaddxxbr 15418	The sum rule for derivatives at a point. That is, if the derivative of F at C is K and the derivative of G at C is L, then the derivative of the pointwise sum of those two functions at C is K + L. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
|- ( ph -> F : X --> CC )   &    |- ( ph -> X C_ S )   &    |- ( ph -> G : X --> CC )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> C ( S _D F ) K )   &    |- ( ph -> C ( S _D G ) L )   &    |- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> C ( S _D ( F oF + G ) ) ( K + L ) )
25-Nov-2023	dcnn 853	Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 847. The relation between dcn 847 and dcnn 853 is analogous to that between notnot 632 and notnotnot 637 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 853 means that a proposition is testable if and only if its negation is testable, and dcn 847 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
|- (DECID -. ph <-> DECID -. -. ph )
24-Nov-2023	bj-dcst 16307	Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)
|- (DECID STAB ph <-> STAB ph )
24-Nov-2023	bj-nnbidc 16303	If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16290. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (DECID ph <-> ph ) )
24-Nov-2023	bj-dcstab 16302	A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
|- (DECID ph -> STAB ph )
24-Nov-2023	bj-fadc 16300	A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> DECID ph )
24-Nov-2023	bj-trdc 16298	A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( ph -> DECID ph )
24-Nov-2023	bj-stal 16295	The universal quantification of a stable formula is stable. See bj-stim 16292 for implication, stabnot 838 for negation, and bj-stan 16293 for conjunction. (Contributed by BJ, 24-Nov-2023.)
|- ( A. xSTAB ph -> STAB A. x ph )
24-Nov-2023	bj-stand 16294	The conjunction of two stable formulas is stable. Deduction form of bj-stan 16293. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16293 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
|- ( ph -> STAB ps )   &    |- ( ph -> STAB ch )   =>    |- ( ph -> STAB ( ps /\ ch ) )
24-Nov-2023	bj-stan 16293	The conjunction of two stable formulas is stable. See bj-stim 16292 for implication, stabnot 838 for negation, and bj-stal 16295 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
|- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )
24-Nov-2023	bj-stim 16292	A conjunction with a stable consequent is stable. See stabnot 838 for negation , bj-stan 16293 for conjunction , and bj-stal 16295 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
|- (STAB ps -> STAB ( ph -> ps ) )
24-Nov-2023	bj-nnbist 16290	If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if ph is a classical tautology, then -. -. ph is an intuitionistic tautology. Therefore, if ph is a classical tautology, then ph is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16303). (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (STAB ph <-> ph ) )
24-Nov-2023	bj-fast 16287	A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> STAB ph )
24-Nov-2023	bj-trst 16285	A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
|- ( ph -> STAB ph )
24-Nov-2023	bj-nnan 16282	The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ( ph /\ ps ) -> ( -. -. ph /\ -. -. ps ) )
24-Nov-2023	bj-nnim 16281	The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ( ph -> ps ) -> ( ph -> -. -. ps ) )
24-Nov-2023	bj-nnsn 16279	As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)
|- ( ( ph -> -. ps ) <-> ( -. -. ph -> -. ps ) )
24-Nov-2023	nnal 1695	The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. A. x ph -> A. x -. -. ph )
22-Nov-2023	ofvalg 6240	Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )   &    |- ( ( ph /\ X e. S ) -> ( C R D ) e. U )   =>    |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )
21-Nov-2023	exmidac 7417	The axiom of choice implies excluded middle. See acexmid 6012 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.)
|- (CHOICE -> EXMID )
21-Nov-2023	exmidaclem 7416	Lemma for exmidac 7417. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.)
|- A = { x e. { (/) , { (/) } } | ( x = (/) \/ y = { (/) } ) }   &    |- B = { x e. { (/) , { (/) } } | ( x = { (/) } \/ y = { (/) } ) }   &    |- C = { A , B }   =>    |- (CHOICE -> EXMID )
21-Nov-2023	exmid1dc 4288	A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4281 or ordtriexmid 4617. In this context x = { (/) } can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
|- ( ( ph /\ x C_ { (/) } ) -> DECID x = { (/) } )   =>    |- ( ph -> EXMID )
20-Nov-2023	acfun 7415	A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.)
|- ( ph -> CHOICE )   &    |- ( ph -> A e. V )   &    |- ( ph -> A. x e. A E. w w e. x )   =>    |- ( ph -> E. f ( f Fn A /\ A. x e. A ( f ` x ) e. x ) )
18-Nov-2023	rnrhmsubrg 14259	The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
|- ( F e. ( M RingHom N ) -> ran F e. (SubRing ` N ) )
18-Nov-2023	condc 858	Contraposition of a decidable proposition. This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)
|- (DECID ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
18-Nov-2023	stdcn 852	A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 847. (Contributed by BJ, 18-Nov-2023.)
|- (STAB ph <-> (DECID -. ph -> DECID ph ) )
17-Nov-2023	cnplimclemr 15386	Lemma for cnplimccntop 15387. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> B e. A )   &    |- ( ph -> ( F ` B ) e. ( F lim CC B ) )   =>    |- ( ph -> F e. ( ( J CnP K ) ` B ) )
17-Nov-2023	cnplimclemle 15385	Lemma for cnplimccntop 15387. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> B e. A )   &    |- ( ph -> ( F ` B ) e. ( F lim CC B ) )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> D e. RR+ )   &    |- ( ph -> Z e. A )   &    |- ( ( ph /\ Z # B /\ ( abs ` ( Z - B ) ) < D ) -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < ( E / 2 ) )   &    |- ( ph -> ( abs ` ( Z - B ) ) < D )   =>    |- ( ph -> ( abs ` ( ( F ` Z ) - ( F ` B ) ) ) < E )
14-Nov-2023	limccnp2cntop 15394	The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.)
|- ( ( ph /\ x e. A ) -> R e. X )   &    |- ( ( ph /\ x e. A ) -> S e. Y )   &    |- ( ph -> X C_ CC )   &    |- ( ph -> Y C_ CC )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( ( K tX K ) ↾t ( X X. Y ) )   &    |- ( ph -> C e. ( ( x e. A |-> R ) lim CC B ) )   &    |- ( ph -> D e. ( ( x e. A |-> S ) lim CC B ) )   &    |- ( ph -> H e. ( ( J CnP K ) ` <. C , D >. ) )   =>    |- ( ph -> ( C H D ) e. ( ( x e. A |-> ( R H S ) ) lim CC B ) )
10-Nov-2023	rpmaxcl 11777	The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> sup ( { A , B } , RR , < ) e. RR+ )
9-Nov-2023	limccnp2lem 15393	Lemma for limccnp2cntop 15394. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.)
|- ( ( ph /\ x e. A ) -> R e. X )   &    |- ( ( ph /\ x e. A ) -> S e. Y )   &    |- ( ph -> X C_ CC )   &    |- ( ph -> Y C_ CC )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( ( K tX K ) ↾t ( X X. Y ) )   &    |- ( ph -> C e. ( ( x e. A |-> R ) lim CC B ) )   &    |- ( ph -> D e. ( ( x e. A |-> S ) lim CC B ) )   &    |- ( ph -> H e. ( ( J CnP K ) ` <. C , D >. ) )   &    |- F/ x ph   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> L e. RR+ )   &    |- ( ph -> A. r e. X A. s e. Y ( ( ( C ( ( abs o. - ) |` ( X X. X ) ) r ) < L /\ ( D ( ( abs o. - ) |` ( Y X. Y ) ) s ) < L ) -> ( ( C H D ) ( abs o. - ) ( r H s ) ) < E ) )   &    |- ( ph -> F e. RR+ )   &    |- ( ph -> A. x e. A ( ( x # B /\ ( abs ` ( x - B ) ) < F ) -> ( abs ` ( R - C ) ) < L ) )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> A. x e. A ( ( x # B /\ ( abs ` ( x - B ) ) < G ) -> ( abs ` ( S - D ) ) < L ) )   =>    |- ( ph -> E. d e. RR+ A. x e. A ( ( x # B /\ ( abs ` ( x - B ) ) < d ) -> ( abs ` ( ( R H S ) - ( C H D ) ) ) < E ) )
4-Nov-2023	ellimc3apf 15377	Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- F/_ z F   =>    |- ( ph -> ( C e. ( F lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( ( F ` z ) - C ) ) < x ) ) ) )
3-Nov-2023	limcmpted 15380	Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.)
|- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- ( ( ph /\ z e. A ) -> D e. CC )   =>    |- ( ph -> ( C e. ( ( z e. A |-> D ) lim CC B ) <-> ( C e. CC /\ A. x e. RR+ E. y e. RR+ A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < y ) -> ( abs ` ( D - C ) ) < x ) ) ) )
1-Nov-2023	unct 13056	The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.)
|- ( ( E. f f : om -onto-> ( A ⊔ 1o ) /\ E. g g : om -onto-> ( B ⊔ 1o ) ) -> E. h h : om -onto-> ( ( A u. B ) ⊔ 1o ) )
31-Oct-2023	ctiunct 13054	A sequence of enumerations gives an enumeration of the union. We refer to "sequence of enumerations" rather than "countably many countable sets" because the hypothesis provides more than countability for each B ( x ): it refers to B ( x ) together with the G ( x ) which enumerates it. Theorem 8.1.19 of [AczelRathjen], p. 74. For "countably many countable sets" the key hypothesis would be ( ph /\ x e. A ) -> E. g g : om -onto-> ( B ⊔ 1o ). This is almost omiunct 13058 (which uses countable choice) although that is for a countably infinite collection not any countable collection. Compare with the case of two sets instead of countably many, as seen at unct 13056, which says that the union of two countable sets is countable . The proof proceeds by mapping a natural number to a pair of natural numbers (by xpomen 13009) and using the first number to map to an element x of A and the second number to map to an element of B(x) . In this way we are able to map to every element of U_ x e. A B. Although it would be possible to work directly with countability expressed as F : om -onto-> ( A ⊔ 1o ), we instead use functions from subsets of the natural numbers via ctssdccl 7304 and ctssdc 7306. (Contributed by Jim Kingdon, 31-Oct-2023.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- ( ( ph /\ x e. A ) -> G : om -onto-> ( B ⊔ 1o ) )   =>    |- ( ph -> E. h h : om -onto-> ( U_ x e. A B ⊔ 1o ) )
30-Oct-2023	ctssdccl 7304	A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7306 but expressed in terms of classes rather than E.. (Contributed by Jim Kingdon, 30-Oct-2023.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- S = { x e. om | ( F ` x ) e. (inl " A ) }   &    |- G = ( `'inl o. F )   =>    |- ( ph -> ( S C_ om /\ G : S -onto-> A /\ A. n e. om DECID n e. S ) )
28-Oct-2023	ctiunctlemfo 13053	Lemma for ctiunct 13054. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   &    |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )   &    |- F/_ x H   &    |- F/_ x U   =>    |- ( ph -> H : U -onto-> U_ x e. A B )
28-Oct-2023	ctiunctlemf 13052	Lemma for ctiunct 13054. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   &    |- H = ( n e. U |-> ( [_ ( F ` ( 1st ` ( J ` n ) ) ) / x ]_ G ` ( 2nd ` ( J ` n ) ) ) )   =>    |- ( ph -> H : U --> U_ x e. A B )
28-Oct-2023	ctiunctlemudc 13051	Lemma for ctiunct 13054. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   =>    |- ( ph -> A. n e. om DECID n e. U )
28-Oct-2023	ctiunctlemuom 13050	Lemma for ctiunct 13054. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   =>    |- ( ph -> U C_ om )
28-Oct-2023	ctiunctlemu2nd 13049	Lemma for ctiunct 13054. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   &    |- ( ph -> N e. U )   =>    |- ( ph -> ( 2nd ` ( J ` N ) ) e. [_ ( F ` ( 1st ` ( J ` N ) ) ) / x ]_ T )
28-Oct-2023	ctiunctlemu1st 13048	Lemma for ctiunct 13054. (Contributed by Jim Kingdon, 28-Oct-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ( ph /\ x e. A ) -> T C_ om )   &    |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )   &    |- ( ( ph /\ x e. A ) -> G : T -onto-> B )   &    |- ( ph -> J : om -1-1-onto-> ( om X. om ) )   &    |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }   &    |- ( ph -> N e. U )   =>    |- ( ph -> ( 1st ` ( J ` N ) ) e. S )
28-Oct-2023	pm2.521gdc 873	A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
|- (DECID ph -> ( -. ( ph -> ps ) -> ( ch -> ph ) ) )
28-Oct-2023	stdcndc 850	A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
|- ( (STAB ph /\ DECID -. ph ) <-> DECID ph )
28-Oct-2023	conax1k 658	Weakening of conax1 657. General instance of pm2.51 659 and of pm2.52 660. (Contributed by BJ, 28-Oct-2023.)
|- ( -. ( ph -> ps ) -> ( ch -> -. ps ) )
28-Oct-2023	conax1 657	Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
|- ( -. ( ph -> ps ) -> -. ps )
25-Oct-2023	divcnap 15282	Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- K = ( J ↾t { x e. CC | x # 0 } )   =>    |- ( y e. CC , z e. { x e. CC | x # 0 } |-> ( y / z ) ) e. ( ( J tX K ) Cn J )
23-Oct-2023	cnm 8045	A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
|- ( A e. CC -> E. x x e. A )
23-Oct-2023	oprssdmm 6329	Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)
|- ( ( ph /\ u e. S ) -> E. v v e. u )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   &    |- ( ph -> Rel F )   =>    |- ( ph -> ( S X. S ) C_ dom F )
22-Oct-2023	addcncntoplem 15278	Lemma for addcncntop 15279, subcncntop 15280, and mulcncntop 15281. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- .+ : ( CC X. CC ) --> CC   &    |- ( ( a e. RR+ /\ b e. CC /\ c e. CC ) -> E. y e. RR+ E. z e. RR+ A. u e. CC A. v e. CC ( ( ( abs ` ( u - b ) ) < y /\ ( abs ` ( v - c ) ) < z ) -> ( abs ` ( ( u .+ v ) - ( b .+ c ) ) ) < a ) )   =>    |- .+ e. ( ( J tX J ) Cn J )
22-Oct-2023	txmetcnp 15235	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
|- J = ( MetOpen ` C )   &    |- K = ( MetOpen ` D )   &    |- L = ( MetOpen ` E )   =>    |- ( ( ( C e. ( *Met ` X ) /\ D e. ( *Met ` Y ) /\ E e. ( *Met ` Z ) ) /\ ( A e. X /\ B e. Y ) ) -> ( F e. ( ( ( J tX K ) CnP L ) ` <. A , B >. ) <-> ( F : ( X X. Y ) --> Z /\ A. z e. RR+ E. w e. RR+ A. u e. X A. v e. Y ( ( ( A C u ) < w /\ ( B D v ) < w ) -> ( ( A F B ) E ( u F v ) ) < z ) ) ) )
22-Oct-2023	xmetxpbl 15225	The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point C with radius R. (Contributed by Jim Kingdon, 22-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- ( ph -> R e. RR* )   &    |- ( ph -> C e. ( X X. Y ) )   =>    |- ( ph -> ( C ( ball ` P ) R ) = ( ( ( 1st ` C ) ( ball ` M ) R ) X. ( ( 2nd ` C ) ( ball ` N ) R ) ) )
21-Oct-2023	pr2cv2 7395	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
|- ( { A , B } ~~ 2o -> B e. _V )
21-Oct-2023	pr2cv1 7394	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
|- ( { A , B } ~~ 2o -> A e. _V )
15-Oct-2023	xmettxlem 15226	Lemma for xmettx 15227. (Contributed by Jim Kingdon, 15-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- J = ( MetOpen ` M )   &    |- K = ( MetOpen ` N )   &    |- L = ( MetOpen ` P )   =>    |- ( ph -> L C_ ( J tX K ) )
11-Oct-2023	xmettx 15227	The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   &    |- J = ( MetOpen ` M )   &    |- K = ( MetOpen ` N )   &    |- L = ( MetOpen ` P )   =>    |- ( ph -> L = ( J tX K ) )
11-Oct-2023	xmetxp 15224	The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
|- P = ( u e. ( X X. Y ) , v e. ( X X. Y ) |-> sup ( { ( ( 1st ` u ) M ( 1st ` v ) ) , ( ( 2nd ` u ) N ( 2nd ` v ) ) } , RR* , < ) )   &    |- ( ph -> M e. ( *Met ` X ) )   &    |- ( ph -> N e. ( *Met ` Y ) )   =>    |- ( ph -> P e. ( *Met ` ( X X. Y ) ) )
8-Oct-2023	pr2cv 7396	If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
|- ( { A , B } ~~ 2o -> ( A e. _V /\ B e. _V ) )
7-Oct-2023	df-iress 13083	Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the Base set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range, defining a function using the base set and applying that, or explicitly truncating the slot before use. (Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.) (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.)
|- ↾s = ( w e. _V , x e. _V |-> ( w sSet <. ( Base ` ndx ) , ( x i^i ( Base ` w ) ) >. ) )
29-Sep-2023	syl2anc2 412	Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
|- ( ph -> ps )   &    |- ( ps -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
27-Sep-2023	fnpr2ob 13416	Biconditional version of fnpr2o 13415. (Contributed by Jim Kingdon, 27-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
25-Sep-2023	xpsval 13428	Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
|- T = ( R X.s S )   &    |- X = ( Base ` R )   &    |- Y = ( Base ` S )   &    |- ( ph -> R e. V )   &    |- ( ph -> S e. W )   &    |- F = ( x e. X , y e. Y |-> { <. (/) , x >. , <. 1o , y >. } )   &    |- G = (Scalar ` R )   &    |- U = ( G X_s { <. (/) , R >. , <. 1o , S >. } )   =>    |- ( ph -> T = ( `' F "s U ) )
25-Sep-2023	fvpr1o 13418	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
|- ( B e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
25-Sep-2023	fvpr0o 13417	The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
|- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
25-Sep-2023	fnpr2o 13415	Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
|- ( ( A e. V /\ B e. W ) -> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
25-Sep-2023	df-xps 13380	Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
|- X.s = ( r e. _V , s e. _V |-> ( `' ( x e. ( Base ` r ) , y e. ( Base ` s ) |-> { <. (/) , x >. , <. 1o , y >. } ) "s ( (Scalar ` r ) X_s { <. (/) , r >. , <. 1o , s >. } ) ) )
12-Sep-2023	pwntru 4287	A slight strengthening of pwtrufal 16548. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
|- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )
11-Sep-2023	pwtrufal 16548	A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4286. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4284), then this theorem states there are no truth values other than true and false, as described in section 1.1 of [Bauer], p. 481. (Contributed by Mario Carneiro and Jim Kingdon, 11-Sep-2023.)
|- ( A C_ { (/) } -> -. -. ( A = (/) \/ A = { (/) } ) )
9-Sep-2023	mathbox 16278	(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.) A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm. Guidelines: Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details. (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)
|- ph   =>    |- ph
6-Sep-2023	djuexb 7237	The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> ( A ⊔ B ) e. _V )
3-Sep-2023	pwf1oexmid 16550	An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
|- T = U_ x e. N ( { x } X. 1o )   =>    |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> ( ran G = ~P 1o <-> ( N = 2o /\ EXMID ) ) )
3-Sep-2023	pwle2 16549	An exercise related to N copies of a singleton and the power set of a singleton (where the latter can also be thought of as representing truth values). Posed as an exercise by Martin Escardo online. (Contributed by Jim Kingdon, 3-Sep-2023.)
|- T = U_ x e. N ( { x } X. 1o )   =>    |- ( ( N e. om /\ G : T -1-1-> ~P 1o ) -> N C_ 2o )
30-Aug-2023	isomninn 16585	Omniscience stated in terms of natural numbers. Similar to isomnimap 7330 but it will sometimes be more convenient to use 0 and 1 rather than (/) and 1o. (Contributed by Jim Kingdon, 30-Aug-2023.)
|- ( A e. V -> ( A e. Omni <-> A. f e. ( { 0 , 1 } ^m A ) ( E. x e. A ( f ` x ) = 0 \/ A. x e. A ( f ` x ) = 1 ) ) )
30-Aug-2023	isomninnlem 16584	Lemma for isomninn 16585. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.)
|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( A e. V -> ( A e. Omni <-> A. f e. ( { 0 , 1 } ^m A ) ( E. x e. A ( f ` x ) = 0 \/ A. x e. A ( f ` x ) = 1 ) ) )