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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.

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Last updated on 27-Mar-2025 at 6:31 AM ET.
Recent Additions to the Intuitionistic Logic Explorer
DateLabelDescription
Theorem
 
20-Mar-2025ccoslid 12686 Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- (comp = Slot (comp ` ndx ) /\ (comp ` ndx ) e. NN )
 
20-Mar-2025homslid 12684 Slot property of Hom. (Contributed by Jim Kingdon, 20-Mar-2025.)
|- ( Hom = Slot ( Hom ` ndx ) /\ ( Hom ` ndx ) e. NN )
 
19-Mar-2025ptex 12712 Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)
|- ( F e. V -> ( Xt_ ` F ) e. _V )
 
18-Mar-2025prdsex 12717 Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.)
|- ( ( S e. V /\ R e. W ) -> ( S X_s R ) e. _V )
 
13-Mar-2025imasex 12725 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-2025imasival 12726 Value of an image structure. The is a lemma for the theorems imasbas 12727, imasplusg 12728, and imasmulr 12729 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 >. } )
 
8-Mar-2025subgex 13034 The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)
|- ( G e. Grp -> (SubGrp ` G ) e. _V )
 
28-Feb-2025ringressid 13236 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 12529. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Ring -> ( G ↾s B ) e. Ring )
 
28-Feb-2025grpressid 12930 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 12529. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Grp -> ( G ↾s B ) e. Grp )
 
26-Feb-2025strext 12563 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 >. )
 
23-Feb-2025ltlenmkv 14753 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 14751). (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-2025neap0mkv 14752 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-2025lringuplu 13335 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-2025lringnz 13334 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-2025lringring 13333 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-2025lringnzr 13332 A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
|- ( R e. LRing -> R e. NzRing )
 
23-Feb-2025islring 13331 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-2025df-lring 13330 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-202501eq0ring 13328 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-2025nzrring 13325 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 )
 
21-Feb-2025dftap2 7249 Tight apartness with the apartness properties from df-pap 7246 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-2025aprap 13342 The relation given by df-apr 13337 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-2025setscomd 12502 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 >. ) )
 
17-Feb-2025aprcotr 13341 The apartness relation given by df-apr 13337 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-2025aprsym 13340 The apartness relation given by df-apr 13337 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-2025aprval 13338 Expand Definition df-apr 13337. (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 ) )
 
16-Feb-2025aprirr 13339 The apartness relation given by df-apr 13337 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-2025aptap 8606 Complex apartness (as defined at df-ap 8538) is a tight apartness (as defined at df-tap 7248). (Contributed by Jim Kingdon, 16-Feb-2025.)
|- # TAp CC
 
15-Feb-2025tapeq2 7251 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.)
|- ( A = B -> ( R TAp A <-> R TAp B ) )
 
14-Feb-2025exmidmotap 7259 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-2025exmidapne 7258 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-2025df-pap 7246 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-2025df-apr 13337 The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 13342. (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 ) ) } )
 
8-Feb-20252oneel 7254 (/) 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-2025tapeq1 7250 Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.)
|- ( R = S -> ( R TAp A <-> S TAp A ) )
 
6-Feb-20252omotap 7257 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-20252omotaplemst 7256 Lemma for 2omotap 7257. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( ( E* r r TAp 2o /\ -. -. ph ) -> ph )
 
6-Feb-20252omotaplemap 7255 Lemma for 2omotap 7257. (Contributed by Jim Kingdon, 6-Feb-2025.)
|- ( -. -. ph -> { <. u , v >. | ( ( u e. 2o /\ v e. 2o ) /\ ( ph /\ u =/= v ) ) } TAp 2o )
 
6-Feb-20252onetap 7253 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-2025netap 7252 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-2025df-tap 7248 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 ) ) )
 
31-Jan-20250subg 13057 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 ) )
 
28-Jan-2025dvdsrex 13265 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
|- ( R e. SRing -> ( ||r ` R ) e. _V )
 
24-Jan-2025reldvdsrsrg 13259 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-2025rerecapb 8799 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-2025recapb 8627 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-2025ressval3d 12530 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-2025strressid 12529 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 )
 
16-Jan-2025ressex 12524 Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.)
|- ( ( W e. X /\ A e. Y ) -> ( W ↾s A ) e. _V )
 
16-Jan-2025ressvalsets 12523 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 ) ) >. ) )
 
10-Jan-2025opprex 13243 Existence of the opposite ring. If you know that R is a ring, see opprring 13247. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> O e. _V )
 
10-Jan-2025mgpex 13133 Existence of the multiplication group. If R is known to be a semiring, see srgmgp 13149. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- M = (mulGrp ` R )   =>    |- ( R e. V -> M e. _V )
 
5-Jan-2025imbibi 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-2025snss 3727 The singleton of an element of a class is a subset of the class (inference form of snssg 3726). 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-2025snssg 3726 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-2025snssb 3725 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 ) )
 
9-Dec-2024nninfwlpoim 7175 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-2024nninfwlpoimlemdc 7174 Lemma for nninfwlpoim 7175. (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-2024nninfwlpoimlemginf 7173 Lemma for nninfwlpoim 7175. (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-2024nninfwlpoimlemg 7172 Lemma for nninfwlpoim 7175. (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-2024nninfwlpor 7171 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-2024nninfwlporlem 7170 Lemma for nninfwlpor 7171. 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 )
 
6-Dec-2024nninfwlporlemd 7169 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-2024nninfwlpo 7176 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-2024nninfdcinf 7168 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 ) )
 
28-Nov-2024basmexd 12521 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-2024eliotaeu 5205 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-2024eliota 5204 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-2024basmex 12520 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 )
 
12-Nov-2024slotsdifipndx 12632 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-2024bj-con1st 14439 Contraposition when the antecedent is a negated stable proposition. See con1dc 856. (Contributed by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> ps ) -> ( -. ps -> ph ) ) )
 
11-Nov-2024slotsdifdsndx 12675 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-2024slotsdifplendx 12664 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-2024tsetndxnstarvndx 12648 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-2024const 852 Contraposition when the antecedent is a negated stable proposition. See comment of condc 853. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.)
|- (STAB ph -> ( ( -. ph -> -. ps ) -> ( ps -> ph ) ) )
 
10-Nov-2024slotsdifunifndx 12682 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-2024ressbasd 12526 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-2024oppraddg 13246 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-2024opprbasg 13245 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-2024opprsllem 13244 Lemma for opprbasg 13245 and oppraddg 13246. (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-2024lgsfvalg 14342 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 ) )
 
1-Nov-2024plendxnvscandx 12663 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-2024plendxnscandx 12662 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-2024plendxnmulrndx 12661 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-2024qsqeqor 10630 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-2024dsndxnmulrndx 12672 The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( dist ` ndx ) =/= ( .r ` ndx )
 
31-Oct-2024tsetndxnmulrndx 12647 The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) =/= ( .r ` ndx )
 
31-Oct-2024tsetndxnbasendx 12645 The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.)
|- (TopSet ` ndx ) =/= ( Base ` ndx )
 
31-Oct-2024basendxlttsetndx 12644 The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.)
|- ( Base ` ndx ) < (TopSet ` ndx )
 
31-Oct-2024tsetndxnn 12643 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-2024plendxnbasendx 12659 The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.)
|- ( le ` ndx ) =/= ( Base ` ndx )
 
30-Oct-2024basendxltplendx 12658 The index value of the Base slot is less than the index value of the le slot. (Contributed by AV, 30-Oct-2024.)
|- ( Base ` ndx ) < ( le ` ndx )
 
30-Oct-2024plendxnn 12657 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-2024dsndxntsetndx 12674 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-2024slotsdnscsi 12673 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-2024slotstnscsi 12649 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-2024ipndxnmulrndx 12631 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-2024ipndxnplusgndx 12630 The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.)
|- ( .i ` ndx ) =/= ( +g ` ndx )
 
29-Oct-2024vscandxnmulrndx 12618 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-2024scandxnmulrndx 12613 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-2024fiubnn 10809 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-2024fiubz 10808 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-2024fiubm 10807 Lemma for fiubz 10808 and fiubnn 10809. 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-2024unifndxntsetndx 12681 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-2024basendxltunifndx 12679 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-2024unifndxnn 12678 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-2024dsndxnbasendx 12670 The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.)
|- ( dist ` ndx ) =/= ( Base ` ndx )
 
28-Oct-2024basendxltdsndx 12669 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-2024dsndxnn 12668 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-2024bj-nnst 14431 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 14678 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-2024bj-imnimnn 14426 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 14425 as its last step. (Contributed by BJ, 27-Oct-2024.)
|- ( ph -> ps )   &    |- ( -. ph -> ps )   =>    |- -. -. ps
 
25-Oct-2024nnwosdc 12039 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-2024nnwodc 12036 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-2024uzwodc 12037 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-2024nnnotnotr 14678 Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 850, 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-2024unifndxnbasendx 12680 The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( UnifSet ` ndx ) =/= ( Base ` ndx )
 
21-Oct-2024ipndxnbasendx 12629 The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.)
|- ( .i ` ndx ) =/= ( Base ` ndx )
 
21-Oct-2024scandxnbasendx 12611 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-2024isprm5lem 12140 Lemma for isprm5 12141. 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-2024resseqnbasd 12531 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-2024mgpress 13139 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-2024dsndxnplusgndx 12671 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-2024plendxnplusgndx 12660 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-2024tsetndxnplusgndx 12646 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-2024vscandxnscandx 12619 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-2024vscandxnplusgndx 12617 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-2024vscandxnbasendx 12616 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-2024scandxnplusgndx 12612 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-2024starvndxnmulrndx 12601 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-2024starvndxnplusgndx 12600 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-2024starvndxnbasendx 12599 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-2024basendxltplusgndx 12571 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-2024plusgndxnn 12569 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-2024elnndc 9611 Membership of an integer in NN is decidable. (Contributed by Jim Kingdon, 17-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN )
 
14-Oct-20242zinfmin 11250 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-2024mingeb 11249 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-2024pcxnn0cl 12309 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-2024xnn0letri 9802 Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> ( A <_ B \/ B <_ A ) )
 
13-Oct-2024xnn0dcle 9801 Decidability of <_ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
|- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
 
9-Oct-2024nn0leexp2 10689 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-2024pclemdc 12287 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-2024elnn0dc 9610 Membership of an integer in NN0 is decidable. (Contributed by Jim Kingdon, 8-Oct-2024.)
|- ( N e. ZZ -> DECID N e. NN0 )
 
7-Oct-2024pclemub 12286 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-2024pclem0 12285 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-2024nn0ltexp2 10688 Special case of ltexp2 14296 which we use here because we haven't yet defined df-rpcxp 14216 which is used in the current proof of ltexp2 14296. (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-2024suprzcl2dc 11955 The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7931.) (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-2024zsupssdc 11954 An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7931.) (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-2024suprzubdc 11952 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-2024infex2g 7032 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
|- ( A e. C -> inf ( B , A , R ) e. _V )
 
30-Sep-2024unbendc 12454 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-2024prmdc 12129 Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( N e. NN -> DECID N e. Prime )
 
30-Sep-2024dcfi 6979 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 )
 
29-Sep-2024ssnnct 12447 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-2024ssnnctlemct 12446 Lemma for ssnnct 12447. 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-2024nninfdcex 11953 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-2024infregelbex 9597 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-2024nninfdclemp1 12450 Lemma for nninfdc 12453. 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-2024nnminle 12035 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 12034. (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-2024nninfdclemcl 12448 Lemma for nninfdc 12453. (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-2024nninfdclemlt 12451 Lemma for nninfdc 12453. The function from nninfdclemf 12449 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-2024nninfdc 12453 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-2024nninfdclemf1 12452 Lemma for nninfdc 12453. The function from nninfdclemf 12449 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-2024nninfdclemf 12449 Lemma for nninfdc 12453. 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-2024nnmindc 12034 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 )
 
19-Sep-2024ssomct 12445 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 ) )
 
14-Sep-2024nnpredlt 4623 The predecessor (see nnpredcl 4622) of a nonzero natural number is less than (see df-iord 4366) that number. (Contributed by Jim Kingdon, 14-Sep-2024.)
|- ( ( A e. om /\ A =/= (/) ) -> U. A e. A )
 
13-Sep-2024nninfisollemeq 7129 Lemma for nninfisol 7130. 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-2024nninfisollemne 7128 Lemma for nninfisol 7130. 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-2024nninfisollem0 7127 Lemma for nninfisol 7130. 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-2024nninfisol 7130 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). (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 )
 
7-Sep-2024eulerthlemfi 12227 Lemma for eulerth 12232. 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-2024modqexp 10646 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-2024eulerthlemh 12230 Lemma for eulerth 12232. 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-2024eulerthlemth 12231 Lemma for eulerth 12232. 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-2024eulerthlema 12229 Lemma for eulerth 12232. (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-2024eulerthlemrprm 12228 Lemma for eulerth 12232. 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 )
 
30-Aug-2024fprodap0f 11643 A finite product of terms apart from zero is apart from zero. A version of fprodap0 11628 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-2024fprodrec 11636 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-2024exmidontri2or 7241 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-2024exmidontri 7237 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-2024ontri2orexmidim 4571 Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4570. (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-2024ontriexmidim 4521 Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4520. (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-2024onntri2or 7244 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-2024onntri3or 7243 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-2024csbcow 3068 Composition law for chained substitutions into a class. Version of csbco 3067 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.)
|- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B
 
25-Aug-2024cbvreuvw 2709 Version of cbvreuv 2705 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E! x e. A ph <-> E! y e. A ps )
 
25-Aug-2024cbvrexvw 2708 Version of cbvrexv 2704 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x e. A ph <-> E. y e. A ps )
 
25-Aug-2024cbvralvw 2707 Version of cbvralv 2703 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x e. A ph <-> A. y e. A ps )
 
25-Aug-2024cbvabw 2300 Version of cbvab 2301 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- { x | ph } = { y | ps }
 
25-Aug-2024nfsbv 1947 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 1946 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 Gino Giotto, 25-Aug-2024.)
|- F/ z ph   =>    |- F/ z [ y / x ] ph
 
25-Aug-2024cbvexvw 1920 Change bound variable. See cbvexv 1918 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( E. x ph <-> E. y ps )
 
25-Aug-2024cbvalvw 1919 Change bound variable. See cbvalv 1917 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.)
|- ( x = y -> ( ph <-> ps ) )   =>    |- ( A. x ph <-> A. y ps )
 
25-Aug-2024nfal 1576 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 1510. (Revised by Gino Giotto, 25-Aug-2024.)
|- F/ x ph   =>    |- F/ x A. y ph
 
24-Aug-2024gcdcomd 11974 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-2024dvds2addd 11835 Deduction form of dvds2add 11831. (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 ) )
 
17-Aug-2024fprodcl2lem 11612 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-2024if0ab 14493 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 3583, |- ( A e. V -> if ( ph , A , (/) ) e. ~P A ), from which fmelpw1o 14494 could be derived, yielding an alternative proof. (Contributed by BJ, 16-Aug-2024.)

|- if ( ph , A , (/) ) = { x e. A | ph }
 
16-Aug-2024fprodunsn 11611 Multiply in an additional term in a finite product. See also fprodsplitsn 11640 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-2024bj-charfundcALT 14497 Alternate proof of bj-charfundc 14496. It was expected to be much shorter since it uses bj-charfun 14495 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-2024bj-charfun 14495 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-2024fmelpw1o 14494 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 851, which translate to 1o and (/) respectively by iftrue 3539 and iffalse 3542, giving pwtrufal 14683).

As proved in if0ab 14493, 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-2024cnstab 8601 Equality of complex numbers is stable. Stability here means -. -. A = B -> A = B as defined at df-stab 831. 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-2024subap0d 8600 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-2024ifexd 4484 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-2024ifelpwun 4483 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-2024ifelpwund 4482 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-2024ifelpwung 4481 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-2024ifidss 3549 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-2024ifssun 3548 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-2024exmidontriimlem2 7220 Lemma for exmidontriim 7223. (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-2024exmidontriimlem1 7219 Lemma for exmidontriim 7223. A variation of r19.30dc 2624. (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-2024nndc 851 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 850 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 1649 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 4195): 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 2467 does not hold.

Actually, -. -. EXMID is not provable in intuitionistic logic since intuitionistic logic has models satisfying -. EXMID and noncontradiction holds (pm3.24 693). (Contributed by BJ, 9-Oct-2019.) Add explanation on non-provability of -. -. EXMID. (Revised by BJ, 11-Aug-2024.)

|- -. -. DECID ph
 
10-Aug-2024exmidontriim 7223 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-2024exmidontriimlem4 7222 Lemma for exmidontriim 7223. 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-2024exmidontriimlem3 7221 Lemma for exmidontriim 7223. 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-2024nnnninf2 7124 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-2024infnninf 7121 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 4673 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-2024ss1o0el1o 6911 Reformulation of ss1o0el1 4197 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
 
9-Aug-2024pw1dc0el 6910 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-2024ss1o0el1 4197 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-2024pw1dc1 6912 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-2024pw1fin 6909 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-2024elomssom 4604 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 4605. (Revised by BJ, 7-Aug-2024.)
|- ( A e. om -> A C_ om )
 
6-Aug-2024bj-charfunbi 14499 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-2024bj-charfunr 14498 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-2024bj-charfundc 14496 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-2024prodssdc 11596 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-2024fnmptd 14492 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-2024funmptd 14491 The maps-to notation defines a function (deduction form).

Note: one should similarly prove a deduction form of funopab4 5253, then prove funmptd 14491 from it, and then prove funmpt 5254 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-2024bj-dcfal 14443 The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID F.
 
5-Aug-2024bj-dctru 14441 The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)
|- DECID T.
 
5-Aug-2024bj-stfal 14430 The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB F.
 
5-Aug-2024bj-sttru 14428 The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)
|- STAB T.
 
5-Aug-2024prod1dc 11593 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-20242ssom 6524 The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.)
|- 2o C_ om
 
2-Aug-2024onntri52 7242 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-2024onntri24 7240 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-2024onntri45 7239 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-2024onntri51 7238 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-2024onntri13 7236 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-2024onntri35 7235 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 7236), (3) implies (5) (onntri35 7235), (5) implies (1) (onntri51 7238), (2) implies (4) (onntri24 7240), (4) implies (5) (onntri45 7239), and (5) implies (2) (onntri52 7242).

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 7237 and exmidontri2or 7241, respectively. Thus -. -. EXMID is equivalent to (1) or (2). In addition, -. -. EXMID is equivalent to (3) by onntri3or 7243 and (4) by onntri2or 7244.

(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-2024nnral 2467 The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1649. (Contributed by Jim Kingdon, 1-Aug-2024.)
|- ( -. -. A. x e. A ph -> A. x e. A -. -. ph )
 
31-Jul-20243nsssucpw1 7234 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-2024sucpw1nss3 7233 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-20243nelsucpw1 7232 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-2024sucpw1nel3 7231 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-2024sucpw1ne3 7230 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-2024pw1nel3 7229 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-2024pw1ne3 7228 The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 3o
 
30-Jul-2024pw1ne1 7227 The power set of 1o is not one. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= 1o
 
30-Jul-2024pw1ne0 7226 The power set of 1o is not zero. (Contributed by Jim Kingdon, 30-Jul-2024.)
|- ~P 1o =/= (/)
 
29-Jul-2024grpcld 12889 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-2024pw1on 7224 The power set of 1o is an ordinal. (Contributed by Jim Kingdon, 29-Jul-2024.)
|- ~P 1o e. On
 
28-Jul-2024exmidpweq 6908 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-2024dcapnconstALT 14745 Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 14744 by means of dceqnconst 14743. (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-2024reap0 14742 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-2024nconstwlpolemgt0 14747 Lemma for nconstwlpo 14749. 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-2024nconstwlpolem0 14746 Lemma for nconstwlpo 14749. 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-2024tridceq 14740 Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14727 and redcwlpo 14739). Thus, this is an analytic analogue to lpowlpo 7165. (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-2024iswomni0 14735 Weak omniscience stated in terms of equality with 0. Like iswomninn 14734 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-2024lpowlpo 7165 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7164. There is an analogue in terms of analytic omniscience principles at tridceq 14740. (Contributed by Jim Kingdon, 24-Jul-2024.)
|- ( om e. Omni -> om e. WOmni )
 
23-Jul-2024nconstwlpolem 14748 Lemma for nconstwlpo 14749. (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-2024dceqnconst 14743 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 14739 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-2024redc0 14741 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-2024canth 5828 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 1500 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-2024nconstwlpo 14749 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-2024fprodseq 11590 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-2024rexbid2 2482 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-2024ralbid2 2481 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-20242irrexpqap 14332 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 12179, 2logb9irrap 14331 and sqrt2cxp2logb9e3 14329. 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-20242logb9irrap 14331 Example for logbgcd1irrap 14324. 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-2024erlecpbl 12750 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-2024ercpbl 12749 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-2024ercpbllemg 12748 Lemma for ercpbl 12749. (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-2024divsfvalg 12747 Value of the function in qusval 12743. (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 ] .~ )
 
11-Jul-2024logbgcd1irraplemexp 14322 Lemma for logbgcd1irrap 14324. 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-2024reapef 14135 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-2024apcxp2 14294 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-2024logbgcd1irraplemap 14323 Lemma for logbgcd1irrap 14324. 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-2024apexp1 10697 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-2024logrpap0 14234 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-2024rplogbval 14299 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-2024logrpap0d 14235 Deduction form of logrpap0 14234. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( log ` A ) # 0 )
 
3-Jul-2024logrpap0b 14233 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-20242o01f 14682 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-2024012of 14681 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-2024iooreen 14719 An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.)
|- ( 0 (,) 1 ) ~~ RR
 
27-Jun-2024iooref1o 14718 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-2024neapmkvlem 14750 Lemma for neapmkv 14751. 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-2024ismkvnn 14737 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-2024ismkvnnlem 14736 Lemma for ismkvnn 14737. 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-2024enmkvlem 7158 Lemma for enmkv 7159. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
|- ( A ~~ B -> ( A e. Markov -> B e. Markov ) )
 
24-Jun-2024neapmkv 14751 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-2024dcapnconst 14744 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 14727 for more discussion of decidability of real number apartness.

This is a weaker form of dceqnconst 14743 and in fact this theorem can be proved using dceqnconst 14743 as shown at dcapnconstALT 14745. (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-2024enmkv 7159 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 6430 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-2024redcwlpolemeq1 14738 Lemma for redcwlpo 14739. A biconditionalized version of trilpolemeq1 14724. (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-2024redcwlpo 14739 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 14738). 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 10246 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-2024iswomninn 14734 Weak omniscience stated in terms of natural numbers. Similar to iswomnimap 7163 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-2024iswomninnlem 14733 Lemma for iswomnimap 7163. 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-2024enwomni 7167 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 6430 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-2024enwomnilem 7166 Lemma for enwomni 7167. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
|- ( A ~~ B -> ( A e. WOmni -> B e. WOmni ) )
 
19-Jun-2024rpabscxpbnd 14295 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-2024rpcxpsqrt 14278 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 ) )
 
13-Jun-2024rpcxpadd 14262 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-2024cxpap0 14261 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-2024rpcncxpcl 14259 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-2024rpcxp0 14255 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-2024cxpexpnn 14253 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-2024cxpexprp 14252 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-2024rpcxpef 14251 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-2024df-rpcxp 14216 Define the power function on complex numbers. Because df-relog 14215 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-2024trirec0xor 14729 Version of trirec0 14728 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-2024trirec0 14728 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 14727). (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-2024omniwomnimkv 7164 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-2024iswomnimap 7163 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-2024iswomni 7162 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-2024df-womni 7161 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-2024cmnmndd 13109 A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. CMnd )   =>    |- ( ph -> G e. Mnd )
 
1-Jun-2024grpmndd 12888 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
|- ( ph -> G e. Grp )   =>    |- ( ph -> G e. Mnd )
 
29-May-2024pw1nct 14688 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-2024sssneq 14687 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-2024elpwi2 4158 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
 
24-May-2024dvmptcjx 14122 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-2024cbvralfw 2694 Rule used to change bound variables, using implicit substitution. Version of cbvralf 2696 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 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 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 )
 
22-May-2024efltlemlt 14131 Lemma for eflt 14132. The converse of efltim 11705 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-2024eflt 14132 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 ) ) )
 
19-May-2024apdifflemr 14731 Lemma for apdiff 14732. (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-2024apdifflemf 14730 Lemma for apdiff 14732. 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-2024apdiff 14732 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-2024crnggrpd 13191 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Grp )
 
16-May-2024crngringd 13190 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. CRing )   =>    |- ( ph -> R e. Ring )
 
16-May-2024ringgrpd 13186 A ring is a group. (Contributed by SN, 16-May-2024.)
|- ( ph -> R e. Ring )   =>    |- ( ph -> R e. Grp )
 
15-May-2024reeff1oleme 14129 Lemma for reeff1o 14130. (Contributed by Jim Kingdon, 15-May-2024.)
|- ( U e. ( 0 (,) _e ) -> E. x e. RR ( exp ` x ) = U )
 
14-May-2024df-relog 14215 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 )
 
12-May-2024dvdstrd 11836 The divides relation is transitive, a deduction version of dvdstr 11834. (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-2024ioocosf1o 14211 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-2024cos0pilt1 14209 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-2024cos11 14210 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-2024omiunct 12444 The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12440 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-2024ctiunctal 12441 Variation of ctiunct 12440 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 ) )
 
3-May-2024cc4n 7269 Countable choice with a simpler restriction on how every set in the countable collection needs to be inhabited. That is, compared with cc4 7268, 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-2024cc4f 7267 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-2024cc4 7268 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 ) )
 
29-Apr-2024cc3 7266 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 ) )
 
27-Apr-2024cc2 7265 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-2024cc2lem 7264 Lemma for cc2 7265. (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-2024cc1 7263 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 ) )
 
19-Apr-2024omctfn 12443 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 ) ) )
 
13-Apr-2024prodmodclem2 11584 Lemma for prodmodc 11585. (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-2024prodmodclem2a 11583 Lemma for prodmodc 11585. (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-2024prodmodclem3 11582 Lemma for prodmodc 11585. (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-2024jcnd 652 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-2024prodrbdclem 11578 Lemma for prodrbdc 11581. (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-2024prodfdivap 11554 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-2024prodfrecap 11553 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-2024prodfap0 11552 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-2024prod3fmul 11548 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-2024df-proddc 11558 Define the product of a series with an index set of integers A. This definition takes most of the aspects of df-sumdc 11361 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-2024cos02pilt1 14208 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-2024cosq34lt1 14207 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-2024coseq0q4123 14191 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-2024cosq23lt0 14190 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-2024pilem3 14140 Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.)
|- ( pi e. ( 2 (,) 4 ) /\ ( sin ` pi ) = 0 )
 
9-Mar-2024exmidonfin 7192 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 6871 and nnon 4609. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
|- ( om = ( On i^i Fin ) -> EXMID )
 
9-Mar-2024exmidonfinlem 7191 Lemma for exmidonfin 7192. (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-2024sin0pilem2 14139 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-2024sin0pilem1 14138 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-2024cosz12 14137 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-2024cos12dec 11774 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-2024dvrfvald 13300 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-2024plusffvalg 12780 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-2024insubm 12871 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-2024mul2lt0pn 9763 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-2024mul2lt0np 9762 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-2024lt0ap0 8604 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-2024negap0d 8587 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-2024lt0ap0d 8605 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-2024ivthdec 14058 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-2024ivthinclemex 14056 Lemma for ivthinc 14057. 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-2024ivthinclemuopn 14052 Lemma for ivthinc 14057. 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-2024dedekindicc 14047 A Dedekind cut identifies a unique real number. Similar to df-inp 7464 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-2024grpsubfvalg 12917 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-2024ivthinclemloc 14055 Lemma for ivthinc 14057. 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-2024ivthinclemdisj 14054 Lemma for ivthinc 14057. 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-2024ivthinclemur 14053 Lemma for ivthinc 14057. 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-2024ivthinclemlr 14051 Lemma for ivthinc 14057. 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-2024ivthinclemum 14049 Lemma for ivthinc 14057. 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-2024ivthinclemlm 14048 Lemma for ivthinc 14057. 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-20240subm 12870 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-2024mndissubm 12865 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-2024mgmsscl 12779 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-2024dedekindicclemeu 14045 Lemma for dedekindicc 14047. 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-2024dedekindicclemlu 14044 Lemma for dedekindicc 14047. 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-2024dedekindicclemlub 14043 Lemma for dedekindicc 14047. 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-2024dedekindicclemloc 14042 Lemma for dedekindicc 14047. 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-2024dedekindicclemub 14041 Lemma for dedekindicc 14047. 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-2024dedekindicclemuub 14040 Lemma for dedekindicc 14047. 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-2024suplociccex 14039 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8029 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-2024suplociccreex 14038 An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8029 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 ) ) )
 
6-Feb-2024ivthinclemlopn 14050 Lemma for ivthinc 14057. 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-2024ivthinc 14057 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-2024dedekindeulemuub 14031 Lemma for dedekindeu 14037. 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-2024dedekindeulemeu 14036 Lemma for dedekindeu 14037. 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-2024dedekindeulemlu 14035 Lemma for dedekindeu 14037. 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-2024dedekindeulemlub 14034 Lemma for dedekindeu 14037. 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-2024dedekindeulemloc 14033 Lemma for dedekindeu 14037. 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-2024dedekindeulemub 14032 Lemma for dedekindeu 14037. 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-2024axsuploc 8029 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 7931 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-2024iotam 5208 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-2024sgrpidmndm 12820 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 )
 
24-Jan-2024axpre-suploclemres 7899 Lemma for axpre-suploc 7900. 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 7807. (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-2024ax-pre-suploc 7931 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 7930 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7930.

(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-2024axpre-suploc 7900 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 7931. (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-2024suplocsr 7807 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-2024bj-el2oss1o 14462 Shorter proof of el2oss1o 6443 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-2024ltm1sr 7775 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-2024mndinvmod 12845 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. ) )
 
19-Jan-2024suplocsrlempr 7805 Lemma for suplocsr 7807. 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-2024suplocsrlemb 7804 Lemma for suplocsr 7807. 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-2024suplocsrlem 7806 Lemma for suplocsr 7807. 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 ) ) )
 
14-Jan-2024suplocexprlemlub 7722 Lemma for suplocexpr 7723. 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-2024suplocexprlemub 7721 Lemma for suplocexpr 7723. 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-2024nfcsbw 3093 Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3094 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x [_ A / y ]_ B
 
10-Jan-2024nfsbcdw 3091 Version of nfsbcd 2982 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x [. A / y ]. ps )
 
10-Jan-2024cbvcsbw 3061 Version of cbvcsb 3062 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
|- F/_ y C   &    |- F/_ x D   &    |- ( x = y -> C = D )   =>    |- [_ A / x ]_ C = [_ A / y ]_ D
 
10-Jan-2024cbvsbcw 2990 Version of cbvsbc 2991 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [. A / x ]. ph <-> [. A / y ]. ps )
 
10-Jan-2024cbvrex2vw 2715 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2717 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 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-2024cbvral2vw 2714 Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2716 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 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-2024cbvralw 2698 Rule used to change bound variables, using implicit substitution. Version of cbvral 2699 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 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 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-2024cbvrexfw 2695 Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2697 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 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-2024nfralw 2514 Bound-variable hypothesis builder for restricted quantification. See nfralya 2517 for a version with y and A distinct instead of x and y. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/_ x A   &    |- F/ x ph   =>    |- F/ x A. y e. A ph
 
10-Jan-2024nfraldw 2509 Not-free for restricted universal quantification where x and y are distinct. See nfraldya 2512 for a version with y and A distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/ x A. y e. A ps )
 
10-Jan-2024nfabdw 2338 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2339 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
|- F/ y ph   &    |- ( ph -> F/ x ps )   =>    |- ( ph -> F/_ x { y | ps } )
 
10-Jan-2024cbv2w 1750 Rule used to change bound variables, using implicit substitution. Version of cbv2 1749 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 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-2024suplocexprlemloc 7719 Lemma for suplocexpr 7723. 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-2024suplocexprlemdisj 7718 Lemma for suplocexpr 7723. 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-2024suplocexprlemru 7717 Lemma for suplocexpr 7723. 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-2024suplocexprlemrl 7715 Lemma for suplocexpr 7723. 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-2024suplocexprlem2b 7712 Lemma for suplocexpr 7723. 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-2024suplocexprlemell 7711 Lemma for suplocexpr 7723. 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-2024suplocexpr 7723 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-2024suplocexprlemex 7720 Lemma for suplocexpr 7723. 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-2024suplocexprlemmu 7716 Lemma for suplocexpr 7723. 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-2024suplocexprlemml 7714 Lemma for suplocexpr 7723. 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-2024suplocexprlemss 7713 Lemma for suplocexpr 7723. 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-2024dedekindicclemicc 14046 Lemma for dedekindicc 14047. Same as dedekindicc 14047, 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-2024dedekindeu 14037 A Dedekind cut identifies a unique real number. Similar to df-inp 7464 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 ) )
 
31-Dec-2023dvmptsubcn 14121 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-2023dvmptnegcn 14120 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-2023dvmptcmulcn 14119 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-2023rinvmod 13110 Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6067. (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-2023brm 4053 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-2023dvmptccn 14115 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-2023dvmptidcn 14114 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 )
 
29-Dec-2023mndbn0 12831 The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 12830). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
|- B = ( Base ` G )   =>    |- ( G e. Mnd -> B =/= (/) )
 
26-Dec-2023lidrididd 12800 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 12799) 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-2023lidrideqd 12799 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-2023ctfoex 7116 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-2023enct 12433 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-2023enctlem 12432 Lemma for enct 12433. 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-2023omct 7115 om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- E. f f : om -onto-> ( om ⊔ 1o )
 
21-Dec-2023dvcoapbr 14107 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-2023apsscn 8603 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-2023aprcl 8602 Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.)
|- ( A # B -> ( A e. CC /\ B e. CC ) )
 
18-Dec-2023limccoap 14083 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-2023cnreim 10986 Complex apartness in terms of real and imaginary parts. See also apreim 8559 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-2023cnopnap 14030 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-2023cnovex 13632 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-2023reopnap 13974 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-2023cnopncntop 13973 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-2023unicntopcntop 13972 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-2023bj-pm2.18st 14438 Clavius law for stable formulas. See pm2.18dc 855. (Contributed by BJ, 4-Dec-2023.)
|- (STAB ph -> ( ( -. ph -> ph ) -> ph ) )
 
4-Dec-2023bj-nnclavius 14425 Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
|- ( ( -. ph -> ph ) -> -. -. ph )
 
2-Dec-2023dvmulxx 14104 The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14102. (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-2023dvmulxxbr 14102 The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 14104. (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-2023subctctexmid 14686 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-2023ismkvnex 7152 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-2023ccfunen 7262 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-2023exmid1stab 4208 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-2023df-cc 7261 The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7204 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-2023offeq 6095 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-2023dvaddxx 14103 The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 14101. (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-2023dvaddxxbr 14101 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-2023dcnn 848 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 842. The relation between dcn 842 and dcnn 848 is analogous to that between notnot 629 and notnotnot 634 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 848 means that a proposition is testable if and only if its negation is testable, and dcn 842 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-2023bj-dcst 14449 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-2023bj-nnbidc 14445 If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 14432. (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (DECID ph <-> ph ) )
 
24-Nov-2023bj-dcstab 14444 A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)
|- (DECID ph -> STAB ph )
 
24-Nov-2023bj-fadc 14442 A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> DECID ph )
 
24-Nov-2023bj-trdc 14440 A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)
|- ( ph -> DECID ph )
 
24-Nov-2023bj-stal 14437 The universal quantification of a stable formula is stable. See bj-stim 14434 for implication, stabnot 833 for negation, and bj-stan 14435 for conjunction. (Contributed by BJ, 24-Nov-2023.)
|- ( A. xSTAB ph -> STAB A. x ph )
 
24-Nov-2023bj-stand 14436 The conjunction of two stable formulas is stable. Deduction form of bj-stan 14435. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 14435 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-2023bj-stan 14435 The conjunction of two stable formulas is stable. See bj-stim 14434 for implication, stabnot 833 for negation, and bj-stal 14437 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
|- ( (STAB ph /\ STAB ps ) -> STAB ( ph /\ ps ) )
 
24-Nov-2023bj-stim 14434 A conjunction with a stable consequent is stable. See stabnot 833 for negation , bj-stan 14435 for conjunction , and bj-stal 14437 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
|- (STAB ps -> STAB ( ph -> ps ) )
 
24-Nov-2023bj-nnbist 14432 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 14445). (Contributed by BJ, 24-Nov-2023.)
|- ( -. -. ph -> (STAB ph <-> ph ) )
 
24-Nov-2023bj-fast 14429 A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)
|- ( -. ph -> STAB ph )
 
24-Nov-2023bj-trst 14427 A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)
|- ( ph -> STAB ph )
 
24-Nov-2023bj-nnan 14424 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-2023bj-nnim 14423 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-2023bj-nnsn 14421 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-2023nnal 1649 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-2023ofvalg 6091 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-2023exmidac 7207 The axiom of choice implies excluded middle. See acexmid 5873 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-2023exmidaclem 7206 Lemma for exmidac 7207. 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-2023exmid1dc 4200 A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4193 or ordtriexmid 4520. 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-2023acfun 7205 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-2023condc 853 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-2023stdcn 847 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 842. (Contributed by BJ, 18-Nov-2023.)
|- (STAB ph <-> (DECID -. ph -> DECID ph ) )
 
17-Nov-2023cnplimclemr 14074 Lemma for cnplimccntop 14075. 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-2023cnplimclemle 14073 Lemma for cnplimccntop 14075. 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-2023limccnp2cntop 14082 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-2023rpmaxcl 11231 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-2023limccnp2lem 14081 Lemma for limccnp2cntop 14082. 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-2023ellimc3apf 14065 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-2023limcmpted 14068 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-2023unct 12442 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-2023ctiunct 12440 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 12444 (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 12442, 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 12395) 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 7109 and ctssdc 7111.

(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-2023ctssdccl 7109 A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7111 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-2023ctiunctlemfo 12439 Lemma for ctiunct 12440. (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-2023ctiunctlemf 12438 Lemma for ctiunct 12440. (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-2023ctiunctlemudc 12437 Lemma for ctiunct 12440. (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-2023ctiunctlemuom 12436 Lemma for ctiunct 12440. (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-2023ctiunctlemu2nd 12435 Lemma for ctiunct 12440. (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-2023ctiunctlemu1st 12434 Lemma for ctiunct 12440. (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-2023pm2.521gdc 868 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-2023stdcndc 845 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-2023conax1k 654 Weakening of conax1 653. General instance of pm2.51 655 and of pm2.52 656. (Contributed by BJ, 28-Oct-2023.)
|- ( -. ( ph -> ps ) -> ( ch -> -. ps ) )
 
28-Oct-2023conax1 653 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
|- ( -. ( ph -> ps ) -> -. ps )
 
25-Oct-2023divcnap 13991 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-2023cnm 7830 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-2023oprssdmm 6171 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-2023addcncntoplem 13987 Lemma for addcncntop 13988, subcncntop 13989, and mulcncntop 13990. (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-2023txmetcnp 13954 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-2023xmetxpbl 13944 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 ) ) )
 
15-Oct-2023xmettxlem 13945 Lemma for xmettx 13946. (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-2023xmettx 13946 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-2023xmetxp 13943 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 ) ) )
 
7-Oct-2023df-iress 12469 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-2023syl2anc2 412 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
|- ( ph -> ps )   &    |- ( ps -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ph -> th )
 
27-Sep-2023fnpr2ob 12758 Biconditional version of fnpr2o 12757. (Contributed by Jim Kingdon, 27-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> { <. (/) , A >. , <. 1o , B >. } Fn 2o )
 
25-Sep-2023xpsval 12770 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-2023fvpr1o 12760 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-2023fvpr0o 12759 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-2023fnpr2o 12757 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-2023df-xps 12724 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-2023pwntru 4199 A slight strengthening of pwtrufal 14683. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
|- ( ( A C_ { (/) } /\ A =/= { (/) } ) -> A = (/) )
 
11-Sep-2023pwtrufal 14683 A subset of the singleton { (/) } cannot be anything other than (/) or { (/) }. Removing the double negation would change the meaning, as seen at exmid01 4198. If we view a subset of a singleton as a truth value (as seen in theorems like exmidexmid 4196), 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-2023mathbox 14420 (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-2023djuexb 7042 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-2023pwf1oexmid 14685 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-2023pwle2 14684 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-2023isomninn 14715 Omniscience stated in terms of natural numbers. Similar to isomnimap 7134 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-2023isomninnlem 14714 Lemma for isomninn 14715. 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 ) ) )
 
28-Aug-2023trilpolemisumle 14722 Lemma for trilpo 14727. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. NN )   =>    |- ( ph -> sum_ i e. Z ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ sum_ i e. Z ( 1 / ( 2 ^ i ) ) )
 
25-Aug-2023cvgcmp2n 14717 A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.)
|- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. NN ) -> 0 <_ ( G ` k ) )   &    |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )   =>    |- ( ph -> seq 1 ( + , G ) e. dom ~~> )
 
25-Aug-2023cvgcmp2nlemabs 14716 Lemma for cvgcmp2n 14717. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting ( seq 1 ( + , G ) ` N ) as the sum of ( seq 1 ( + , G ) ` M ) and a term which gets smaller as M gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)
|- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. NN ) -> 0 <_ ( G ` k ) )   &    |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` ( ( seq 1 ( + , G ) ` N ) - ( seq 1 ( + , G ) ` M ) ) ) < ( 2 / M ) )
 
24-Aug-2023trilpolemclim 14720 Lemma for trilpo 14727. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- G = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )   =>    |- ( ph -> seq 1 ( + , G ) e. dom ~~> )
 
23-Aug-2023trilpo 14727 Real number trichotomy implies the Limited Principle of Omniscience (LPO). 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 contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 14725 (which means the sequence contains a zero), trilpolemeq1 14724 (which means the sequence is all ones), and trilpolemgt1 14723 (which is not possible).

Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 14713) or that the real numbers are a discrete field (see trirec0 14728).

LPO 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 qtri3or 10242 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.)

|- ( A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) -> om e. Omni )
 
23-Aug-2023trilpolemres 14726 Lemma for trilpo 14727. The result. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ph -> ( A < 1 \/ A = 1 \/ 1 < A ) )   =>    |- ( ph -> ( E. x e. NN ( F ` x ) = 0 \/ A. x e. NN ( F ` x ) = 1 ) )
 
23-Aug-2023trilpolemlt1 14725 Lemma for trilpo 14727. The A < 1 case. We can use the distance between A and one (that is, 1 - A) to find a position in the sequence n where terms after that point will not add up to as much as 1 - A. By finomni 7137 we know the terms up to n either contain a zero or are all one. But if they are all one that contradicts the way we constructed n, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ph -> A < 1 )   =>    |- ( ph -> E. x e. NN ( F ` x ) = 0 )
 
23-Aug-2023trilpolemeq1 14724 Lemma for trilpo 14727. The A = 1 case. This is proved by noting that if any ( F ` x ) is zero, then the infinite sum A is less than one based on the term which is zero. We are using the fact that the F sequence is decidable (in the sense that each element is either zero or one). (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   &    |- ( ph -> A = 1 )   =>    |- ( ph -> A. x e. NN ( F ` x ) = 1 )
 
23-Aug-2023trilpolemgt1 14723 Lemma for trilpo 14727. The 1 < A case. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   =>    |- ( ph -> -. 1 < A )
 
23-Aug-2023trilpolemcl 14721 Lemma for trilpo 14727. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ph -> F : NN --> { 0 , 1 } )   &    |- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )   =>    |- ( ph -> A e. RR )
 
23-Aug-2023triap 14713 Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A < B \/ A = B \/ B < A ) <-> DECID A # B ) )
 
19-Aug-2023djuenun 7210 Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.)
|- ( ( A ~~ B /\ C ~~ D /\ ( B i^i D ) = (/) ) -> ( A ⊔ C ) ~~ ( B u. D ) )
 
16-Aug-2023ctssdclemr 7110 Lemma for ctssdc 7111. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
|- ( E. f f : om -onto-> ( A ⊔ 1o ) -> E. s ( s C_ om /\ E. f f : s -onto-> A /\ A. n e. om DECID n e. s ) )
 
16-Aug-2023ctssdclemn0 7108 Lemma for ctssdc 7111. The -. (/) e. S case. (Contributed by Jim Kingdon, 16-Aug-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ph -> -. (/) e. S )   =>    |- ( ph -> E. g g : om -onto-> ( A ⊔ 1o ) )
 
15-Aug-2023ctssexmid 7147 The decidability condition in ctssdc 7111 is needed. More specifically, ctssdc 7111 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
|- ( ( y C_ om /\ E. f f : y -onto-> x ) -> E. f f : om -onto-> ( x ⊔ 1o ) )   &    |- om e. Omni   =>    |- ( ph \/ -. ph )
 
15-Aug-2023ctssdc 7111 A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7147. (Contributed by Jim Kingdon, 15-Aug-2023.)
|- ( E. s ( s C_ om /\ E. f f : s -onto-> A /\ A. n e. om DECID n e. s ) <-> E. f f : om -onto-> ( A ⊔ 1o ) )
 
14-Aug-2023mpoexw 6213 Weak version of mpoex 6214 that holds without ax-coll 4118. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.)
|- A e. _V   &    |- B e. _V   &    |- D e. _V   &    |- A. x e. A A. y e. B C e. D   =>    |- ( x e. A , y e. B |-> C ) e. _V
 
13-Aug-2023grpinvfvalg 12914 The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- .0. = ( 0g ` G )   &    |- N = ( invg ` G )   =>    |- ( G e. V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
 
13-Aug-2023ltntri 8084 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy, A < B \/ A = B \/ B < A. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
|- ( ( A e. RR /\ B e. RR ) -> -. ( -. A < B /\ -. A = B /\ -. B < A ) )
 
13-Aug-2023mptexw 6113 Weak version of mptex 5742 that holds without ax-coll 4118. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|- A e. _V   &    |- C e. _V   &    |- A. x e. A B e. C   =>    |- ( x e. A |-> B ) e. _V
 
13-Aug-2023funexw 6112 Weak version of funex 5739 that holds without ax-coll 4118. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )
 
11-Aug-2023qnnen 12431 The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.)
|- QQ ~~ NN
 
10-Aug-2023ctinfomlemom 12427 Lemma for ctinfom 12428. Converting between om and NN0. (Contributed by Jim Kingdon, 10-Aug-2023.)
|- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- G = ( F o. `' N )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om -. ( F ` k ) e. ( F " n ) )   =>    |- ( ph -> ( G : NN0 -onto-> A /\ A. m e. NN0 E. j e. NN0 A. i e. ( 0 ... m ) ( G ` j ) =/= ( G ` i ) ) )
 
9-Aug-2023difinfsnlem 7097 Lemma for difinfsn 7098. The case where we need to swap B and (inr ` (/) ) in building the mapping G. (Contributed by Jim Kingdon, 9-Aug-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> B e. A )   &    |- ( ph -> F : ( om ⊔ 1o ) -1-1-> A )   &    |- ( ph -> ( F ` (inr ` (/) ) ) =/= B )   &    |- G = ( n e. om |-> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) )   =>    |- ( ph -> G : om -1-1-> ( A \ { B } ) )
 
8-Aug-2023difinfinf 7099 An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
|- ( ( ( A. x e. A A. y e. A DECID x = y /\ om ~<_ A ) /\ ( B C_ A /\ B e. Fin ) ) -> om ~<_ ( A \ B ) )
 
8-Aug-2023difinfsn 7098 An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
|- ( ( A. x e. A A. y e. A DECID x = y /\ om ~<_ A /\ B e. A ) -> om ~<_ ( A \ { B } ) )
 
7-Aug-2023ctinf 12430 A set is countably infinite if and only if it has decidable equality, is countable, and is infinite. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f f : om -onto-> A /\ om ~<_ A ) )
 
7-Aug-2023inffinp1 12429 An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> om ~<_ A )   &    |- ( ph -> B C_ A )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> E. x e. A -. x e. B )
 
7-Aug-2023ctinfom 12428 A condition for a set being countably infinite. Restates ennnfone 12425 in terms of om and function image. Like ennnfone 12425 the condition can be summarized as A being countable, infinite, and having decidable equality. (Contributed by Jim Kingdon, 7-Aug-2023.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : om -onto-> A /\ A. n e. om E. k e. om -. ( f ` k ) e. ( f " n ) ) ) )
 
6-Aug-2023rerestcntop 13986 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   &    |- R = ( topGen ` ran (,) )   =>    |- ( A C_ RR -> ( J ↾t A ) = ( R ↾t A ) )
 
6-Aug-2023tgioo2cntop 13985 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( topGen ` ran (,) ) = ( J ↾t RR )
 
4-Aug-2023nninffeq 14705 Equality of two functions on ℕ∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, |- ( ph -> A. n e. suc om ... ). (Contributed by Jim Kingdon, 4-Aug-2023.)
|- ( ph -> F :ℕ∞ --> NN0 )   &    |- ( ph -> G :ℕ∞ --> NN0 )   &    |- ( ph -> ( F ` ( x e. om |-> 1o ) ) = ( G ` ( x e. om |-> 1o ) ) )   &    |- ( ph -> A. n e. om ( F ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) = ( G ` ( i e. om |-> if ( i e. n , 1o , (/) ) ) ) )   =>    |- ( ph -> F = G )
 
3-Aug-2023txvalex 13690 Existence of the binary topological product. If R and S are known to be topologies, see txtop 13696. (Contributed by Jim Kingdon, 3-Aug-2023.)
|- ( ( R e. V /\ S e. W ) -> ( R tX S ) e. _V )
 
3-Aug-2023ablgrpd 13092 An Abelian group is a group, deduction form of ablgrp 13091. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> G e. Abel )   =>    |- ( ph -> G e. Grp )
 
3-Aug-20231nsgtrivd 13077 A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> (NrmSGrp ` G ) ~~ 1o )   =>    |- ( ph -> B = { .0. } )
 
3-Aug-2023triv1nsgd 13076 A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (NrmSGrp ` G ) ~~ 1o )
 
3-Aug-2023trivnsgd 13075 The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (NrmSGrp ` G ) = { B } )
 
3-Aug-20230idnsgd 13074 The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   =>    |- ( ph -> { { .0. } , B } C_ (NrmSGrp ` G ) )
 
3-Aug-2023trivsubgsnd 13059 The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   =>    |- ( ph -> (SubGrp ` G ) = { B } )
 
3-Aug-2023trivsubgd 13058 The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B = { .0. } )   &    |- ( ph -> A e. (SubGrp ` G ) )   =>    |- ( ph -> A = B )
 
3-Aug-2023mulgcld 13003 Deduction associated with mulgcl 12999. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N .x. X ) e. B )
 
3-Aug-2023hashfingrpnn 12908 A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- ( ph -> G e. Grp )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> (♯ ` B ) e. NN )
 
3-Aug-2023hashfinmndnn 12832 A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- B = ( Base ` G )   &    |- ( ph -> G e. Mnd )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> (♯ ` B ) e. NN )
 
3-Aug-2023dvdsgcdidd 11994 The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> M e. NN )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> M || N )   =>    |- ( ph -> ( M gcd N ) = M )
 
3-Aug-2023gcdmultipled 11993 The greatest common divisor of a nonnegative integer M and a multiple of it is M itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> M e. NN0 )   &    |- ( ph -> N e. ZZ )   =>    |- ( ph -> ( M gcd ( N x. M ) ) = M )
 
3-Aug-2023fihashelne0d 10776 A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> B e. A )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> -. (♯ ` A ) = 0 )
 
3-Aug-2023phpeqd 6931 Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6864 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B C_ A )   &    |- ( ph -> A ~~ B )   =>    |- ( ph -> A = B )
 
3-Aug-2023enpr2d 6816 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. D )   &    |- ( ph -> -. A = B )   =>    |- ( ph -> { A , B } ~~ 2o )
 
3-Aug-2023elrnmpt2d 4882 Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- F = ( x e. A |-> B )   &    |- ( ph -> C e. ran F )   =>    |- ( ph -> E. x e. A C = B )
 
3-Aug-2023elrnmptdv 4881 Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- F = ( x e. A |-> B )   &    |- ( ph -> C e. A )   &    |- ( ph -> D e. V )   &    |- ( ( ph /\ x = C ) -> D = B )   =>    |- ( ph -> D e. ran F )
 
3-Aug-2023rspcime 2848 Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ( ph /\ x = A ) -> ps )   &    |- ( ph -> A e. B )   =>    |- ( ph -> E. x e. B ps )
 
3-Aug-2023neqcomd 2182 Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> -. A = B )   =>    |- ( ph -> -. B = A )
 
2-Aug-2023dvid 14098 Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( CC _D ( _I |` CC ) ) = ( CC X. { 1 } )
 
2-Aug-2023dvconst 14097 Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( A e. CC -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
 
2-Aug-2023dvidlemap 14096 Lemma for dvid 14098 and dvconst 14097. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( ph -> F : CC --> CC )   &    |- ( ( ph /\ ( x e. CC /\ z e. CC /\ z # x ) ) -> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) = B )   &    |- B e. CC   =>    |- ( ph -> ( CC _D F ) = ( CC X. { B } ) )
 
2-Aug-2023diveqap1bd 8792 If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8661. Converse of diveqap1d 8754. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.)
|- ( ph -> B e. CC )   &    |- ( ph -> B # 0 )   &    |- ( ph -> A = B )   =>    |- ( ph -> ( A / B ) = 1 )
 
31-Jul-2023mul0inf 11248 Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11070 and mulap0bd 8613 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = 0 <-> inf ( { ( abs ` A ) , ( abs ` B ) } , RR , < ) = 0 ) )
 
31-Jul-2023mul0eqap 8626 If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> A # B )   &    |- ( ph -> ( A x. B ) = 0 )   =>    |- ( ph -> ( A = 0 \/ B = 0 ) )
 
31-Jul-2023apcon4bid 8580 Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> D e. CC )   &    |- ( ph -> ( A # B <-> C # D ) )   =>    |- ( ph -> ( A = B <-> C = D ) )
 
30-Jul-2023uzennn 10435 An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( M e. ZZ -> ( ZZ>= ` M ) ~~ NN )
 
30-Jul-2023djuen 7209 Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( ( A ~~ B /\ C ~~ D ) -> ( A ⊔ C ) ~~ ( B ⊔ D ) )
 
30-Jul-2023endjudisj 7208 Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A ⊔ B ) ~~ ( A u. B ) )
 
30-Jul-2023eninr 7096 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( A e. V -> (inr " A ) ~~ A )
 
30-Jul-2023eninl 7095 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( A e. V -> (inl " A ) ~~ A )
 
29-Jul-2023exmidunben 12426 If any unbounded set of positive integers is equinumerous to NN, then the Limited Principle of Omniscience (LPO) implies excluded middle. (Contributed by Jim Kingdon, 29-Jul-2023.)
|- ( ( A. x ( ( x C_ NN /\ A. m e. NN E. n e. x m < n ) -> x ~~ NN ) /\ om e. Omni ) -> EXMID )
 
29-Jul-2023exmidsssnc 4203 Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4198 but lets you choose any set as the element of the singleton rather than just (/). It is similar to exmidsssn 4202 but for a particular set B rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
|- ( B e. V -> (EXMID <-> A. x ( x C_ { B } -> ( x = (/) \/ x = { B } ) ) ) )
 
28-Jul-2023dvfcnpm 14095 The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.)
|- ( F e. ( CC ^pm CC ) -> ( CC _D F ) : dom ( CC _D F ) --> CC )
 
28-Jul-2023dvfpm 14094 The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.)
|- ( F e. ( CC ^pm RR ) -> ( RR _D F ) : dom ( RR _D F ) --> CC )
 
23-Jul-2023ennnfonelemhdmp1 12409 Lemma for ennnfone 12425. Domain at a successor where we need to add an element to the sequence. (Contributed by Jim Kingdon, 23-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   &    |- ( ph -> -. ( F ` ( `' N ` P ) ) e. ( F " ( `' N ` P ) ) )   =>    |- ( ph -> dom ( H ` ( P + 1 ) ) = suc dom ( H ` P ) )
 
23-Jul-2023ennnfonelemp1 12406 Lemma for ennnfone 12425. Value of H at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` ( P + 1 ) ) = if ( ( F ` ( `' N ` P ) ) e. ( F " ( `' N ` P ) ) , ( H ` P ) , ( ( H ` P ) u. { <. dom ( H ` P ) , ( F ` ( `' N ` P ) ) >. } ) ) )
 
22-Jul-2023nntr2 6503 Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.)
|- ( ( A e. om /\ C e. om ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )
 
22-Jul-2023nnsssuc 6502 A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.)
|- ( ( A e. om /\ B e. om ) -> ( A C_ B <-> A e. suc B ) )
 
21-Jul-2023ennnfoneleminc 12411 Lemma for ennnfone 12425. We only add elements to H as the index increases. (Contributed by Jim Kingdon, 21-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   &    |- ( ph -> Q e. NN0 )   &    |- ( ph -> P <_ Q )   =>    |- ( ph -> ( H ` P ) C_ ( H ` Q ) )
 
20-Jul-2023ennnfonelemg 12403 Lemma for ennnfone 12425. Closure for G. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ( ph /\ ( f e. { g e. ( A ^pm om ) | dom g e. om } /\ j e. om ) ) -> ( f G j ) e. { g e. ( A ^pm om ) | dom g e. om } )
 
20-Jul-2023ennnfonelemjn 12402 Lemma for ennnfone 12425. Non-initial state for J. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ( ph /\ f e. ( ZZ>= ` ( 0 + 1 ) ) ) -> ( J ` f ) e. om )
 
20-Jul-2023ennnfonelemj0 12401 Lemma for ennnfone 12425. Initial state for J. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( J ` 0 ) e. { g e. ( A ^pm om ) | dom g e. om } )
 
20-Jul-2023seqp1cd 10465 Value of the sequence builder function at a successor. A version of seq3p1 10461 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( N + 1 ) ) = ( ( seq M ( .+ , F ) ` N ) .+ ( F ` ( N + 1 ) ) ) )
 
20-Jul-2023seqovcd 10462 A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10463 and seq1cd 10464 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.)
|- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   =>    |- ( ( ph /\ ( x e. ( ZZ>= ` M ) /\ y e. C ) ) -> ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) e. C )
 
19-Jul-2023ennnfonelemhom 12415 Lemma for ennnfone 12425. The sequences in H increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> M e. om )   =>    |- ( ph -> E. i e. NN0 M e. dom ( H ` i ) )
 
19-Jul-2023ennnfonelemex 12414 Lemma for ennnfone 12425. Extending the sequence ( H ` P ) to include an additional element. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> E. i e. NN0 dom ( H ` P ) e. dom ( H ` i ) )
 
19-Jul-2023ennnfonelemkh 12412 Lemma for ennnfone 12425. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> dom ( H ` P ) C_ ( `' N ` P ) )
 
19-Jul-2023ennnfonelemom 12408 Lemma for ennnfone 12425. H yields finite sequences. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> dom ( H ` P ) e. om )
 
19-Jul-2023ennnfonelem1 12407 Lemma for ennnfone 12425. Second value. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( H ` 1 ) = { <. (/) , ( F ` (/) ) >. } )
 
19-Jul-2023seq1cd 10464 Initial value of the recursive sequence builder. A version of seq3-1 10459 which provides two classes D and C for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.)
|- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> ( seq M ( .+ , F ) ` M ) = ( F ` M ) )
 
17-Jul-2023ennnfonelemhf1o 12413 Lemma for ennnfone 12425. Each of the functions in H is one to one and onto an image of F. (Contributed by Jim Kingdon, 17-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` P ) : dom ( H ` P ) -1-1-onto-> ( F " ( `' N ` P ) ) )
 
16-Jul-2023ennnfonelemen 12421 Lemma for ennnfone 12425. The result. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> A ~~ NN )
 
16-Jul-2023ennnfonelemdm 12420 Lemma for ennnfone 12425. The function L is defined everywhere. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> dom L = om )
 
16-Jul-2023ennnfonelemrn 12419 Lemma for ennnfone 12425. L is onto A. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> ran L = A )
 
16-Jul-2023ennnfonelemf1 12418 Lemma for ennnfone 12425. L is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> L : dom L -1-1-> A )
 
16-Jul-2023ennnfonelemfun 12417 Lemma for ennnfone 12425. L is a function. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- L = U_ i e. NN0 ( H ` i )   =>    |- ( ph -> Fun L )
 
16-Jul-2023ennnfonelemrnh 12416 Lemma for ennnfone 12425. A consequence of ennnfonelemss 12410. (Contributed by Jim Kingdon, 16-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> X e. ran H )   &    |- ( ph -> Y e. ran H )   =>    |- ( ph -> ( X C_ Y \/ Y C_ X ) )
 
15-Jul-2023ennnfonelemss 12410 Lemma for ennnfone 12425. We only add elements to H as the index increases. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   &    |- ( ph -> P e. NN0 )   =>    |- ( ph -> ( H ` P ) C_ ( H ` ( P + 1 ) ) )
 
15-Jul-2023ennnfonelem0 12405 Lemma for ennnfone 12425. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> ( H ` 0 ) = (/) )
 
15-Jul-2023ennnfonelemk 12400 Lemma for ennnfone 12425. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> F : om -onto-> A )   &    |- ( ph -> K e. om )   &    |- ( ph -> N e. om )   &    |- ( ph -> A. j e. suc N ( F ` K ) =/= ( F ` j ) )   =>    |- ( ph -> N e. K )
 
15-Jul-2023ennnfonelemdc 12399 Lemma for ennnfone 12425. A direct consequence of fidcenumlemrk 6952. (Contributed by Jim Kingdon, 15-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> P e. om )   =>    |- ( ph -> DECID ( F ` P ) e. ( F " P ) )
 
14-Jul-2023djur 7067 A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
|- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )
 
13-Jul-2023sbthomlem 14709 Lemma for sbthom 14710. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.)
|- ( ph -> om e. Omni )   &    |- ( ph -> Y C_ { (/) } )   &    |- ( ph -> F : om -1-1-onto-> ( Y ⊔ om ) )   =>    |- ( ph -> ( Y = (/) \/ Y = { (/) } ) )
 
12-Jul-2023caseinr 7090 Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
|- ( ph -> Fun F )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. B )   =>    |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( G ` A ) )
 
12-Jul-2023inl11 7063 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> A = B ) )
 
11-Jul-2023djudomr 7218 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> B ~<_ ( A ⊔ B ) )
 
11-Jul-2023djudoml 7217 A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> A ~<_ ( A ⊔ B ) )
 
11-Jul-2023omp1eomlem 7092 Lemma for omp1eom 7093. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- F = ( x e. om |-> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )   &    |- S = ( x e. om |-> suc x )   &    |- G = case ( S , ( _I |` 1o ) )   =>    |- F : om -1-1-onto-> ( om ⊔ 1o )
 
11-Jul-2023xp01disjl 6434 Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- ( ( { (/) } X. A ) i^i ( { 1o } X. C ) ) = (/)
 
10-Jul-2023sbthom 14710 Schroeder-Bernstein is not possible even for om. We know by exmidsbth 14708 that full Schroeder-Bernstein will not be provable but what about the case where one of the sets is om? That case plus the Limited Principle of Omniscience (LPO) implies excluded middle, so we will not be able to prove it. (Contributed by Mario Carneiro and Jim Kingdon, 10-Jul-2023.)
|- ( ( A. x ( ( x ~<_ om /\ om ~<_ x ) -> x ~~ om ) /\ om e. Omni ) -> EXMID )
 
10-Jul-2023endjusym 7094 Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )
 
10-Jul-2023omp1eom 7093 Adding one to om. (Contributed by Jim Kingdon, 10-Jul-2023.)
|- ( om ⊔ 1o ) ~~ om
 
9-Jul-2023refeq 14712 Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.)
|- ( ph -> F : RR --> RR )   &    |- ( ph -> G : RR --> RR )   &    |- ( ph -> A. x e. RR ( x < 0 -> ( F ` x ) = ( G ` x ) ) )   &    |- ( ph -> A. x e. RR ( 0 < x -> ( F ` x ) = ( G ` x ) ) )   &    |- ( ph -> ( F ` 0 ) = ( G ` 0 ) )   =>    |- ( ph -> F = G )
 
9-Jul-2023seqvalcd 10458 Value of the sequence builder function. Similar to seq3val 10457 but the classes D (type of each term) and C (type of the value we are accumulating) do not need to be the same. (Contributed by Jim Kingdon, 9-Jul-2023.)
|- ( ph -> M e. ZZ )   &    |- R = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. C |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )   &    |- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> seq M ( .+ , F ) = ran R )
 
9-Jul-2023djuun 7065 The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
|- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
 
9-Jul-2023djuin 7062 The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
|- ( (inl " A ) i^i (inr " B ) ) = (/)
 
8-Jul-2023limcimo 14070 Conditions which ensure there is at most one limit value of F at B. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 8-Jul-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B e. C )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. ( K ↾t S ) )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> { q e. C | q # B } C_ A )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> E* x x e. ( F lim CC B ) )
 
8-Jul-2023ennnfonelemh 12404 Lemma for ennnfone 12425. (Contributed by Jim Kingdon, 8-Jul-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : om -onto-> A )   &    |- ( ph -> A. n e. om E. k e. om A. j e. suc n ( F ` k ) =/= ( F ` j ) )   &    |- G = ( x e. ( A ^pm om ) , y e. om |-> if ( ( F ` y ) e. ( F " y ) , x , ( x u. { <. dom x , ( F ` y ) >. } ) ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   &    |- J = ( x e. NN0 |-> if ( x = 0 , (/) , ( `' N ` ( x - 1 ) ) ) )   &    |- H = seq 0 ( G , J )   =>    |- ( ph -> H : NN0 --> ( A ^pm om ) )
 
7-Jul-2023seqf2 10463 Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.)
|- ( ph -> ( F ` M ) e. C )   &    |- ( ( ph /\ ( x e. C /\ y e. D ) ) -> ( x .+ y ) e. C )   &    |- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` x ) e. D )   =>    |- ( ph -> seq M ( .+ , F ) : Z --> C )
 
3-Jul-2023limcimolemlt 14069 Lemma for limcimo 14070. (Contributed by Jim Kingdon, 3-Jul-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> B e. C )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. ( K ↾t S ) )   &    |- ( ph -> S e. { RR , CC } )   &    |- ( ph -> { q e. C | q # B } C_ A )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- ( ph -> D e. RR+ )   &    |- ( ph -> X e. ( F lim CC B ) )   &    |- ( ph -> Y e. ( F lim CC B ) )   &    |- ( ph -> A. z e. A ( ( z # B /\ ( abs ` ( z - B ) ) < D ) -> ( abs ` ( ( F ` z ) - X ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> A. w e. A ( ( w # B /\ ( abs ` ( w - B ) ) < G ) -> ( abs ` ( ( F ` w ) - Y ) ) < ( ( abs ` ( X - Y ) ) / 2 ) ) )   =>    |- ( ph -> ( abs ` ( X - Y ) ) < ( abs ` ( X - Y ) ) )
 
28-Jun-2023dvfgg 14093 Explicitly write out the functionality condition on derivative for S = RR and CC. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
 
28-Jun-2023dvbsssg 14091 The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.)
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> dom ( S _D F ) C_ S )
 
27-Jun-2023dvbssntrcntop 14089 The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- ( ph -> S C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ S )   &    |- J = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ph -> dom ( S _D F ) C_ ( ( int ` J ) ` A ) )
 
27-Jun-2023eldvap 14087 The differentiable predicate. A function F is differentiable at B with derivative C iff F is defined in a neighborhood of B and the difference quotient has limit C at B. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- T = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- G = ( z e. { w e. A | w # B } |-> ( ( ( F ` z ) - ( F ` B ) ) / ( z - B ) ) )   &    |- ( ph -> S C_ CC )   &    |- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ S )   =>    |- ( ph -> ( B ( S _D F ) C <-> ( B e. ( ( int ` T ) ` A ) /\ C e. ( G lim CC B ) ) ) )
 
27-Jun-2023dvfvalap 14086 Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- T = ( K ↾t S )   &    |- K = ( MetOpen ` ( abs o. - ) )   =>    |- ( ( S C_ CC /\ F : A --> CC /\ A C_ S ) -> ( ( S _D F ) = U_ x e. ( ( int ` T ) ` A ) ( { x } X. ( ( z e. { w e. A | w # x } |-> ( ( ( F ` z ) - ( F ` x ) ) / ( z - x ) ) ) lim CC x ) ) /\ ( S _D F ) C_ ( ( ( int ` T ) ` A ) X. CC ) ) )
 
27-Jun-2023dvlemap 14085 Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
|- ( ph -> F : D --> CC )   &    |- ( ph -> D C_ CC )   &    |- ( ph -> B e. D )   =>    |- ( ( ph /\ A e. { w e. D | w # B } ) -> ( ( ( F ` A ) - ( F ` B ) ) / ( A - B ) ) e. CC )
 
25-Jun-2023reldvg 14084 The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
|- ( ( S C_ CC /\ F e. ( CC ^pm S ) ) -> Rel ( S _D F ) )
 
25-Jun-2023df-dvap 14062 Define the derivative operator. This acts on functions to produce a function that is defined where the original function is differentiable, with value the derivative of the function at these points. The set s here is the ambient topological space under which we are evaluating the continuity of the difference quotient. Although the definition is valid for any subset of CC and is well-behaved when s contains no isolated points, we will restrict our attention to the cases s = RR or s = CC for the majority of the development, these corresponding respectively to real and complex differentiation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.)
|- _D = ( s e. ~P CC , f e. ( CC ^pm s ) |-> U_ x e. ( ( int ` ( ( MetOpen ` ( abs o. - ) ) ↾t s ) ) ` dom f ) ( { x } X. ( ( z e. { w e. dom f | w # x } |-> ( ( ( f ` z ) - ( f ` x ) ) / ( z - x ) ) ) lim CC x ) ) )
 
18-Jun-2023limccnpcntop 14080 If the limit of F at B is C and G is continuous at C, then the limit of G o. F at B is G ( C ). (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 18-Jun-2023.)
|- ( ph -> F : A --> D )   &    |- ( ph -> D C_ CC )   &    |- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t D )   &    |- ( ph -> C e. ( F lim CC B ) )   &    |- ( ph -> G e. ( ( J CnP K ) ` C ) )   =>    |- ( ph -> ( G ` C ) e. ( ( G o. F ) lim CC B ) )
 
18-Jun-2023r19.30dc 2624 Restricted quantifier version of 19.30dc 1627. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
|- ( ( A. x e. A ( ph \/ ps ) /\ DECID E. x e. A ps ) -> ( A. x e. A ph \/ E. x e. A ps ) )
 
17-Jun-2023r19.28v 2605 Restricted quantifier version of one direction of 19.28 1563. (The other direction holds when A is inhabited, see r19.28mv 3515.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
|- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )
 
17-Jun-2023r19.27v 2604 Restricted quantitifer version of one direction of 19.27 1561. (The other direction holds when A is inhabited, see r19.27mv 3519.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
|- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )
 
16-Jun-2023cnlimcim 14076 If F is a continuous function, the limit of the function at each point equals the value of the function. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 16-Jun-2023.)
|- ( A C_ CC -> ( F e. ( A -cn-> CC ) -> ( F : A --> CC /\ A. x e. A ( F ` x ) e. ( F lim CC x ) ) ) )
 
16-Jun-2023cncfcn1cntop 14017 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- ( CC -cn-> CC ) = ( J Cn J )
 
14-Jun-2023cnplimcim 14072 If a function is continuous at B, its limit at B equals the value of the function there. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Jun-2023.)
|- K = ( MetOpen ` ( abs o. - ) )   &    |- J = ( K ↾t A )   =>    |- ( ( A C_ CC /\ B e. A ) -> ( F e. ( ( J CnP K ) ` B ) -> ( F : A --> CC /\ ( F ` B ) e. ( F lim CC B ) ) ) )
 
14-Jun-2023metcnpd 13956 Two ways to say a mapping from metric C to metric D is continuous at point P. (Contributed by Jim Kingdon, 14-Jun-2023.)
|- ( ph -> J = ( MetOpen ` C ) )   &    |- ( ph -> K = ( MetOpen ` D ) )   &    |- ( ph -> C e. ( *Met ` X ) )   &    |- ( ph -> D e. ( *Met ` Y ) )   &    |- ( ph -> P e. X )   =>    |- ( ph -> ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. RR+ E. z e. RR+ A. w e. X ( ( P C w ) < z -> ( ( F ` P ) D ( F ` w ) ) < y ) ) ) )
 
6-Jun-2023cntoptop 13969 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- J e. Top
 
6-Jun-2023cntoptopon 13968 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
|- J = ( MetOpen ` ( abs o. - ) )   =>    |- J e. (TopOn ` CC )
 
3-Jun-2023limcdifap 14067 It suffices to consider functions which are not defined at B to define the limit of a function. In particular, the value of the original function F at B does not affect the limit of F. (Contributed by Mario Carneiro, 25-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   =>    |- ( ph -> ( F lim CC B ) = ( ( F |` { x e. A | x # B } ) lim CC B ) )
 
3-Jun-2023ellimc3ap 14066 Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.)
|- ( ph -> F : A --> CC )   &    |- ( ph -> A C_ CC )   &    |- ( ph -> B e. CC )   =>    |- ( 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-Jun-2023df-limced 14061 Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.)
|- lim CC = ( f e. ( CC ^pm CC ) , x e. CC |-> { y e. CC | ( ( f : dom f --> CC /\ dom f C_ CC ) /\ ( x e. CC /\ A. e e. RR+ E. d e. RR+ A. z e. dom f ( ( z # x /\ ( abs ` ( z - x ) ) < d ) -> ( abs ` ( ( f ` z ) - y ) ) < e ) ) ) } )
 
30-May-2023modprm1div 12246 A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.)
|- ( ( P e. Prime /\ A e. ZZ ) -> ( ( A mod P ) = 1 <-> P || ( A - 1 ) ) )
 
30-May-2023modm1div 11806 An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.)
|- ( ( N e. ( ZZ>= ` 2 ) /\ A e. ZZ ) -> ( ( A mod N ) = 1 <-> N || ( A - 1 ) ) )
 
30-May-2023eluz4nn 9567 An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. NN )
 
30-May-2023eluz4eluz2 9566 An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.)
|- ( X e. ( ZZ>= ` 4 ) -> X e. ( ZZ>= ` 2 ) )
 
29-May-2023mulcncflem 14026 Lemma for mulcncf 14027. (Contributed by Jim Kingdon, 29-May-2023.)
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( X -cn-> CC ) )   &    |- ( ph -> V e. X )   &    |- ( ph -> E e. RR+ )   &    |- ( ph -> F e. RR+ )   &    |- ( ph -> G e. RR+ )   &    |- ( ph -> S e. RR+ )   &    |- ( ph -> T e. RR+ )   &    |- ( ph -> A. u e. X ( ( abs ` ( u - V ) ) < S -> ( abs ` ( ( ( x e. X |-> A ) ` u ) - ( ( x e. X |-> A ) ` V ) ) ) < F ) )   &    |- ( ph -> A. u e. X ( ( abs ` ( u - V ) ) < T -> ( abs ` ( ( ( x e. X |-> B ) ` u ) - ( ( x e. X |-> B ) ` V ) ) ) < G ) )   &    |- ( ph -> A. u e. X ( ( ( abs ` ( [_ u / x ]_ A - [_ V / x ]_ A ) ) < F /\ ( abs ` ( [_ u / x ]_ B - [_ V / x ]_ B ) ) < G ) -> ( abs ` ( ( [_ u / x ]_ A x. [_ u / x ]_ B ) - ( [_ V / x ]_ A x. [_ V / x ]_ B ) ) ) < E ) )   =>    |- ( ph -> E. d e. RR+ A. u e. X ( ( abs ` ( u - V ) ) < d -> ( abs ` ( ( ( x e. X |-> ( A x. B ) ) ` u ) - ( ( x e. X |-> ( A x. B ) ) ` V ) ) ) < E ) )
 
26-May-2023cdivcncfap 14023 Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
|- F = ( x e. { y e. CC | y # 0 } |-> ( A / x ) )   =>    |- ( A e. CC -> F e. ( { y e. CC | y # 0 } -cn-> CC ) )
 
26-May-2023reccn2ap 11320 The reciprocal function is continuous. The class T is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2177. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
|- T = (inf ( { 1 , ( ( abs ` A ) x. B ) } , RR , < ) x. ( ( abs ` A ) / 2 ) )   =>    |- ( ( A e. CC /\ A # 0 /\ B e. RR+ ) -> E. y e. RR+ A. z e. { w e. CC | w # 0 } ( ( abs ` ( z - A ) ) < y -> ( abs ` ( ( 1 / z ) - ( 1 / A ) ) ) < B ) )
 
23-May-2023iooretopg 13964 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A (,) B ) e. ( topGen ` ran (,) ) )
 
23-May-2023minclpr 11244 The minimum of two real numbers is one of those numbers if and only if dichotomy ( A <_ B \/ B <_ A) holds. For example, this can be combined with zletric 9296 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
|- ( ( A e. RR /\ B e. RR ) -> (inf ( { A , B } , RR , < ) e. { A , B } <-> ( A <_ B \/ B <_ A ) ) )
 
22-May-2023qtopbasss 13957 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
|- S C_ RR*   &    |- ( ( x e. S /\ y e. S ) -> sup ( { x , y } , RR* , < ) e. S )   &    |- ( ( x e. S /\ y e. S ) -> inf ( { x , y } , RR* , < ) e. S )   =>    |- ( (,) " ( S X. S ) ) e. TopBases
 
22-May-2023iooinsup 11284 Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A (,) B ) i^i ( C (,) D ) ) = ( sup ( { A , C } , RR* , < ) (,)inf ( { B , D } , RR* , < ) ) )
 
21-May-2023df-sumdc 11361 Define the sum of a series with an index set of integers A. The variable k is normally a free variable in B, i.e., B can be thought of as B ( k ). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. In both cases we have an if expression so that we only need B to be defined where k e. A. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). Examples: sum_ k e. { 1 , 2 , 4 } k means 1 + 2 + 4 = 7, and sum_ k e. NN ( 1 / ( 2 ^ k ) ) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11529). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)
|- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
 
19-May-2023bdmopn 13940 The standard bounded metric corresponding to C generates the same topology as C. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   &    |- J = ( MetOpen ` C )   =>    |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> J = ( MetOpen ` D ) )
 
19-May-2023bdbl 13939 The standard bounded metric corresponding to C generates the same balls as C for radii less than R. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) /\ ( P e. X /\ S e. RR* /\ S <_ R ) ) -> ( P ( ball ` D ) S ) = ( P ( ball ` C ) S ) )
 
19-May-2023bdmet 13938 The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( C e. ( *Met ` X ) /\ R e. RR+ ) -> D e. ( Met ` X ) )
 
19-May-2023xrminltinf 11279 Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> (inf ( { B , C } , RR* , < ) < A <-> ( B < A \/ C < A ) ) )
 
19-May-2023clel5 2874 Alternate definition of class membership: a class X is an element of another class A iff there is an element of A equal to X. (Contributed by AV, 13-Nov-2020.) (Revised by Steven Nguyen, 19-May-2023.)
|- ( X e. A <-> E. x e. A X = x )
 
18-May-2023xrminrecl 11280 The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR* , < ) = inf ( { A , B } , RR , < ) )
 
18-May-2023ralnex2 2616 Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
|- ( A. x e. A A. y e. B -. ph <-> -. E. x e. A E. y e. B ph )
 
17-May-2023bdtrilem 11246 Lemma for bdtri 11247. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( ( abs ` ( A - C ) ) + ( abs ` ( B - C ) ) ) <_ ( C + ( abs ` ( ( A + B ) - C ) ) ) )
 
15-May-2023xrbdtri 11283 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( ( A e. RR* /\ 0 <_ A ) /\ ( B e. RR* /\ 0 <_ B ) /\ ( C e. RR* /\ 0 < C ) ) -> inf ( { ( A +e B ) , C } , RR* , < ) <_ (inf ( { A , C } , RR* , < ) +einf ( { B , C } , RR* , < ) ) )
 
15-May-2023bdtri 11247 Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> inf ( { ( A + B ) , C } , RR , < ) <_ (inf ( { A , C } , RR , < ) + inf ( { B , C } , RR , < ) ) )
 
15-May-2023minabs 11243 The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) = ( ( ( A + B ) - ( abs ` ( A - B ) ) ) / 2 ) )
 
11-May-2023xrmaxadd 11268 Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
 
11-May-2023xrmaxaddlem 11267 Lemma for xrmaxadd 11268. The case where A is real. (Contributed by Jim Kingdon, 11-May-2023.)
|- ( ( A e. RR /\ B e. RR* /\ C e. RR* ) -> sup ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +e sup ( { B , C } , RR* , < ) ) )
 
10-May-2023xrminadd 11282 Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> inf ( { ( A +e B ) , ( A +e C ) } , RR* , < ) = ( A +einf ( { B , C } , RR* , < ) ) )
 
10-May-2023xrmaxlesup 11266 Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) <_ C <-> ( A <_ C /\ B <_ C ) ) )
 
10-May-2023xrltmaxsup 11264 The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( C < sup ( { A , B } , RR* , < ) <-> ( C < A \/ C < B ) ) )
 
9-May-2023bdxmet 13937 The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( C e. ( *Met ` X ) /\ R e. RR* /\ 0 < R ) -> D e. ( *Met ` X ) )
 
9-May-2023bdmetval 13936 Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
|- D = ( x e. X , y e. X |-> inf ( { ( x C y ) , R } , RR* , < ) )   =>    |- ( ( ( C : ( X X. X ) --> RR* /\ R e. RR* ) /\ ( A e. X /\ B e. X ) ) -> ( A D B ) = inf ( { ( A C B ) , R } , RR* , < ) )
 
7-May-2023setsmstsetg 13917 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
|- ( ph -> X = ( Base ` M ) )   &    |- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )   &    |- ( ph -> K = ( M sSet <. (TopSet ` ndx ) , ( MetOpen ` D ) >. ) )   &    |- ( ph -> M e. V )   &    |- ( ph -> ( MetOpen ` D ) e. W )   =>    |- ( ph -> ( MetOpen ` D ) = (TopSet ` K ) )
 
6-May-2023dsslid 12667 Slot property of dist. (Contributed by Jim Kingdon, 6-May-2023.)
|- ( dist = Slot ( dist ` ndx ) /\ ( dist ` ndx ) e. NN )
 
5-May-2023mopnrel 13877 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
|- Rel MetOpen
 
5-May-2023fsumsersdc 11402 Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.)
|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> sum_ k e. A B = ( seq M ( + , F ) ` N ) )
 
4-May-2023blex 13823 A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.)
|- ( D e. ( *Met ` X ) -> ( ball ` D ) e. _V )
 
4-May-2023summodc 11390 A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   =>    |- ( ph -> E* x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , G ) ` m ) ) ) )
 
4-May-2023summodclem2 11389 Lemma for summodc 11390. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   =>    |- ( ( ph /\ E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , F ) ~~> x ) ) -> ( E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ y = ( seq 1 ( + , G ) ` m ) ) -> x = y ) )
 
4-May-2023xrminrpcl 11281 The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR* , < ) e. RR+ )
 
4-May-2023xrlemininf 11278 Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ inf ( { B , C } , RR* , < ) <-> ( A <_ B /\ A <_ C ) ) )
 
3-May-2023xrltmininf 11277 Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < inf ( { B , C } , RR* , < ) <-> ( A < B /\ A < C ) ) )
 
3-May-2023xrmineqinf 11276 The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ B <_ A ) -> inf ( { A , B } , RR* , < ) = B )
 
3-May-2023xrmin2inf 11275 The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) <_ B )
 
3-May-2023xrmin1inf 11274 The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) <_ A )
 
3-May-2023xrmincl 11273 The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) e. RR* )
 
2-May-2023xrminmax 11272 Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> inf ( { A , B } , RR* , < ) = -e sup ( { -e A , -e B } , RR* , < ) )
 
2-May-2023xrnegcon1d 11271 Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   =>    |- ( ph -> ( -e A = B <-> -e B = A ) )
 
2-May-2023infxrnegsupex 11270 The infimum of a set of extended reals A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
|- ( 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 -> inf ( A , RR* , < ) = -e sup ( { z e. RR* | -e z e. A } , RR* , < ) )
 
2-May-2023xrnegiso 11269 Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
|- F = ( x e. RR* |-> -e x )   =>    |- ( F Isom < , `' < ( RR* , RR* ) /\ `' F = F )
 
30-Apr-2023xrmaxltsup 11265 Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( sup ( { A , B } , RR* , < ) < C <-> ( A < C /\ B < C ) ) )
 
30-Apr-2023xrmaxrecl 11262 The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR* , < ) = sup ( { A , B } , RR , < ) )
 
30-Apr-2023xrmax2sup 11261 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> B <_ sup ( { A , B } , RR* , < ) )
 
30-Apr-2023xrmax1sup 11260 An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> A <_ sup ( { A , B } , RR* , < ) )
 
29-Apr-2023xrmaxcl 11259 The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> sup ( { A , B } , RR* , < ) e. RR* )
 
29-Apr-2023xrmaxiflemval 11257 Lemma for xrmaxif 11258. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- M = if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )   =>    |- ( ( A e. RR* /\ B e. RR* ) -> ( M e. RR* /\ A. x e. { A , B } -. M < x /\ A. x e. RR* ( x < M -> E. z e. { A , B } x < z ) ) )
 
29-Apr-2023xrmaxiflemcom 11256 Lemma for xrmaxif 11258. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = if ( A = +oo , +oo , if ( A = -oo , B , if ( B = +oo , +oo , if ( B = -oo , A , sup ( { B , A } , RR , < ) ) ) ) ) )
 
29-Apr-2023xrmaxiflemcl 11252 Lemma for xrmaxif 11258. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) e. RR* )
 
29-Apr-2023sbco2v 1948 Version of sbco2 1965 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)
|- F/ z ph   =>    |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
 
28-Apr-2023xrmaxiflemlub 11255 Lemma for xrmaxif 11258. A least upper bound for { A , B }. (Contributed by Jim Kingdon, 28-Apr-2023.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> C e. RR* )   &    |- ( ph -> C < if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )   =>    |- ( ph -> ( C < A \/ C < B ) )
 
26-Apr-2023xrmaxif 11258 Maximum of two extended reals in terms of if expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> sup ( { A , B } , RR* , < ) = if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
 
26-Apr-2023xrmaxiflemab 11254 Lemma for xrmaxif 11258. A variation of xrmaxleim 11251- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
|- ( ph -> A e. RR* )   &    |- ( ph -> B e. RR* )   &    |- ( ph -> A < B )   =>    |- ( ph -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = B )
 
26-Apr-2023xrmaxifle 11253 An upper bound for { A , B } in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
 
25-Apr-2023xrmaxleim 11251 Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )
 
25-Apr-2023rpmincl 11245 The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> inf ( { A , B } , RR , < ) e. RR+ )
 
25-Apr-2023mincl 11238 The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
|- ( ( A e. RR /\ B e. RR ) -> inf ( { A , B } , RR , < ) e. RR )
 
24-Apr-2023psmetrel 13758 The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.)
|- Rel PsMet
 
23-Apr-2023bcval5 10742 Write out the top and bottom parts of the binomial coefficient ( N _C K ) = ( N x. ( N - 1 ) x. ... x. ( ( N - K ) + 1 ) ) / K ! explicitly. In this form, it is valid even for N < K, although it is no longer valid for nonpositive K. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( N e. NN0 /\ K e. NN ) -> ( N _C K ) = ( ( seq ( ( N - K ) + 1 ) ( x. , _I ) ` N ) / ( ! ` K ) ) )
 
23-Apr-2023ser3le 10517 Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. RR )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) <_ ( G ` k ) )   =>    |- ( ph -> ( seq M ( + , F ) ` N ) <_ ( seq M ( + , G ) ` N ) )
 
23-Apr-2023seq3z 10510 If the operation .+ has an absorbing element Z (a.k.a. zero element), then any sequence containing a Z evaluates to Z. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z )   &    |- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z )   &    |- ( ph -> K e. ( M ... N ) )   &    |- ( ph -> ( F ` K ) = Z )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )
 
23-Apr-2023seq3caopr 10482 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( 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 /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) .+ ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) .+ ( seq M ( .+ , G ) ` N ) ) )
 
23-Apr-2023seq3caopr2 10481 The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ( ph /\ ( ( x e. S /\ y e. S ) /\ ( z e. S /\ w e. S ) ) ) -> ( ( x Q z ) .+ ( y Q w ) ) = ( ( x .+ y ) Q ( z .+ w ) ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
 
22-Apr-2023ser3sub 10505 The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( 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 ) - ( G ` k ) ) )   =>    |- ( ph -> ( seq M ( + , H ) ` N ) = ( ( seq M ( + , F ) ` N ) - ( seq M ( + , G ) ` N ) ) )
 
22-Apr-2023seq3caopr3 10480 Lemma for seq3caopr2 10481. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x Q y ) e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) Q ( G ` k ) ) )   &    |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( ( seq M ( .+ , F ) ` n ) Q ( seq M ( .+ , G ) ` n ) ) .+ ( ( F ` ( n + 1 ) ) Q ( G ` ( n + 1 ) ) ) ) = ( ( ( seq M ( .+ , F ) ` n ) .+ ( F ` ( n + 1 ) ) ) Q ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( ( seq M ( .+ , F ) ` N ) Q ( seq M ( .+ , G ) ` N ) ) )
 
22-Apr-2023ser3mono 10477 The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
|- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. RR )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> 0 <_ ( F ` x ) )   =>    |- ( ph -> ( seq M ( + , F ) ` K ) <_ ( seq M ( + , F ) ` N ) )
 
21-Apr-2023metrtri 13813 Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.)
|- ( ( D e. ( Met ` X ) /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( abs ` ( ( A D C ) - ( B D C ) ) ) <_ ( A D B ) )
 
21-Apr-2023sqxpeq0 5052 A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)
|- ( ( A X. A ) = (/) <-> A = (/) )
 
20-Apr-2023xmetrel 13779 The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel *Met
 
20-Apr-2023metrel 13778 The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.)
|- Rel Met
 
19-Apr-2023psmetge0 13767 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.)
|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> 0 <_ ( A D B ) )
 
18-Apr-2023xleaddadd 9886 Cancelling a factor of two in <_ (expressed as addition rather than as a factor to avoid extended real multiplication). (Contributed by Jim Kingdon, 18-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> ( A +e A ) <_ ( B +e B ) ) )
 
17-Apr-2023xposdif 9881 Extended real version of posdif 8411. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> 0 < ( B +e -e A ) ) )
 
17-Apr-2023nmnfgt 9817 An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
|- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )
 
17-Apr-2023npnflt 9814 An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
|- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
 
16-Apr-2023xltadd1 9875 Extended real version of ltadd1 8385. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.)
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR ) -> ( A < B <-> ( A +e C ) < ( B +e C ) ) )
 
13-Apr-2023xrmnfdc 9842 An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = -oo )
 
13-Apr-2023xrpnfdc 9841 An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
|- ( A e. RR* -> DECID A = +oo )
 
11-Apr-2023dmxpid 4848 The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.)
|- dom ( A X. A ) = A
 
9-Apr-2023isumz 11396 Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- ( ( ( M e. ZZ /\ A C_ ( ZZ>= ` M ) /\ A. j e. ( ZZ>= ` M )DECID j e. A ) \/ A e. Fin ) -> sum_ k e. A 0 = 0 )
 
9-Apr-2023summodclem2a 11388 Lemma for summodc 11390. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- G = ( n e. NN |-> if ( n <_ (♯ ` A ) , [_ ( f ` n ) / k ]_ B , 0 ) )   &    |- H = ( n e. NN |-> if ( n <_ N , [_ ( K ` n ) / k ]_ B , 0 ) )   &    |- ( 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 ( + , F ) ~~> ( seq 1 ( + , G ) ` N ) )
 
9-Apr-2023summodclem3 11387 Lemma for summodc 11390. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> ( M e. NN /\ N e. NN ) )   &    |- ( ph -> f : ( 1 ... M ) -1-1-onto-> A )   &    |- ( ph -> K : ( 1 ... N ) -1-1-onto-> A )   &    |- G = ( n e. NN |-> if ( n <_ M , [_ ( f ` n ) / k ]_ B , 0 ) )   &    |- H = ( n e. NN |-> if ( n <_ N , [_ ( K ` n ) / k ]_ B , 0 ) )   =>    |- ( ph -> ( seq 1 ( + , G ) ` M ) = ( seq 1 ( + , H ) ` N ) )
 
9-Apr-2023sumrbdc 11386 Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( ZZ>= ` N ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> DECID k e. A )   =>    |- ( ph -> ( seq M ( + , F ) ~~> C <-> seq N ( + , F ) ~~> C ) )
 
9-Apr-2023seq3coll 10821 The function F contains a sparse set of nonzero values to be summed. The function G is an order isomorphism from the set of nonzero values of F to a 1-based finite sequence, and H collects these nonzero values together. Under these conditions, the sum over the values in H yields the same result as the sum over the original set F. (Contributed by Mario Carneiro, 2-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)
|- ( ( ph /\ k e. S ) -> ( Z .+ k ) = k )   &    |- ( ( ph /\ k e. S ) -> ( k .+ Z ) = k )   &    |- ( ( ph /\ ( k e. S /\ n e. S ) ) -> ( k .+ n ) e. S )   &    |- ( ph -> Z e. S )   &    |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` A ) ) , A ) )   &    |- ( ph -> N e. ( 1 ... (♯ ` A ) ) )   &    |- ( ph -> A C_ ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` 1 ) ) -> ( H ` k ) e. S )   &    |- ( ( ph /\ k e. ( ( M ... ( G ` (♯ ` A ) ) ) \ A ) ) -> ( F ` k ) = Z )   &    |- ( ( ph /\ n e. ( 1 ... (♯ ` A ) ) ) -> ( H ` n ) = ( F ` ( G ` n ) ) )   =>    |- ( ph -> ( seq M ( .+ , F ) ` ( G ` N ) ) = ( seq 1 ( .+ , H ) ` N ) )
 
8-Apr-2023zsumdc 11391 Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A C_ Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ph -> A. x e. Z DECID x e. A )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> sum_ k e. A B = ( ~~> ` seq M ( + , F ) ) )
 
8-Apr-2023sumrbdclem 11384 Lemma for sumrbdc 11386. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( 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 ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )
 
8-Apr-2023isermulc2 11347 Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> C e. CC )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( C x. ( F ` k ) ) )   =>    |- ( ph -> seq M ( + , G ) ~~> ( C x. A ) )
 
8-Apr-2023seq3id 10507 Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for .+) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = x )   &    |- ( ph -> Z e. S )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> ( F ` N ) e. S )   &    |- ( ( ph /\ x e. ( M ... ( N - 1 ) ) ) -> ( F ` x ) = Z )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) |` ( ZZ>= ` N ) ) = seq N ( .+ , F ) )
 
8-Apr-2023seq3id3 10506 A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a .+ -idempotent sums (or " .+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Jim Kingdon, 8-Apr-2023.)
|- ( ph -> ( Z .+ Z ) = Z )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) = Z )   &    |- ( ph -> Z e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = Z )
 
7-Apr-2023seq3shft2 10472 Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> K e. ZZ )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = ( G ` ( k + K ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` ( M + K ) ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq ( M + K ) ( .+ , G ) ` ( N + K ) ) )
 
7-Apr-2023seq3feq 10471 Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ph -> M e. ZZ )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( G ` k ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) = seq M ( .+ , G ) )
 
7-Apr-2023r19.2m 3509 Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.)
|- ( ( E. y y e. A /\ A. x e. A ph ) -> E. x e. A ph )
 
6-Apr-2023lmtopcnp 13686 The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
|- ( ph -> F ( ~~> t ` J ) P )   &    |- ( ph -> K e. Top )   &    |- ( ph -> G e. ( ( J CnP K ) ` P ) )   =>    |- ( ph -> ( G o. F ) ( ~~> t ` K ) ( G ` P ) )
 
6-Apr-2023cnptoprest2 13676 Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : X --> B /\ B C_ Y ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> F e. ( ( J CnP ( K ↾t B ) ) ` P ) ) )
 
5-Apr-2023cnptoprest 13675 Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( ( J e. Top /\ K e. Top /\ A C_ X ) /\ ( P e. ( ( int ` J ) ` A ) /\ F : X --> Y ) ) -> ( F e. ( ( J CnP K ) ` P ) <-> ( F |` A ) e. ( ( ( J ↾t A ) CnP K ) ` P ) ) )
 
4-Apr-2023exmidmp 7154 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
|- (EXMID -> om e. Markov )
 
2-Apr-2023sup3exmid 8913 If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
|- ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) -> E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) )   =>    |- DECID ph
 
31-Mar-2023cnptopresti 13674 One direction of cnptoprest 13675 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. Top ) /\ ( A C_ X /\ P e. A /\ F e. ( ( J CnP K ) ` P ) ) ) -> ( F |` A ) e. ( ( ( J ↾t A ) CnP K ) ` P ) )
 
30-Mar-2023cncnp2m 13667 A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.)
|- X = U. J   &    |- Y = U. K   =>    |- ( ( J e. Top /\ K e. Top /\ E. y y e. X ) -> ( F e. ( J Cn K ) <-> A. x e. X F e. ( ( J CnP K ) ` x ) ) )
 
29-Mar-2023exmidlpo 7140 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
|- (EXMID -> om e. Omni )
 
28-Mar-2023icnpimaex 13647 Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ P e. X ) /\ ( F e. ( ( J CnP K ) ` P ) /\ A e. K /\ ( F ` P ) e. A ) ) -> E. x e. J ( P e. x /\ ( F " x ) C_ A ) )
 
28-Mar-2023cnpf2 13643 A continuous function at point P is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` Y ) /\ F e. ( ( J CnP K ) ` P ) ) -> F : X --> Y )
 
28-Mar-2023cnprcl2k 13642 Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.)
|- ( ( J e. (TopOn ` X ) /\ K e. Top /\ F e. ( ( J CnP K ) ` P ) ) -> P e. X )
 
27-Mar-2023mptrcl 5598 Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
|- F = ( x e. A |-> B )   =>    |- ( I e. ( F ` X ) -> X e. A )
 
25-Mar-2023lmreltop 13629 The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.)
|- ( J e. Top -> Rel ( ~~> t ` J ) )
 
25-Mar-2023fodjumkv 7157 A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
|- ( ph -> M e. Markov )   &    |- ( ph -> F : M -onto-> ( A ⊔ B ) )   =>    |- ( ph -> ( A =/= (/) -> E. x x e. A ) )
 
25-Mar-2023fodjumkvlemres 7156 Lemma for fodjumkv 7157. The final result with P expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
|- ( ph -> M e. Markov )   &    |- ( ph -> F : M -onto-> ( A ⊔ B ) )   &    |- P = ( y e. M |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   =>    |- ( ph -> ( A =/= (/) -> E. x x e. A ) )
 
25-Mar-2023fodju0 7144 Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
|- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   &    |- ( ph -> A. w e. O ( P ` w ) = 1o )   =>    |- ( ph -> A = (/) )
 
25-Mar-2023fodjum 7143 Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that A is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
|- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   &    |- ( ph -> E. w e. O ( P ` w ) = (/) )   =>    |- ( ph -> E. x x e. A )
 
25-Mar-2023fodjuf 7142 Lemma for fodjuomni 7146 and fodjumkv 7157. Domain and range of P. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
|- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   &    |- ( ph -> O e. V )   =>    |- ( ph -> P e. ( 2o ^m O ) )
 
23-Mar-2023restrcl 13603 Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.)
|- ( ( J ↾t A ) e. Top -> ( J e. _V /\ A e. _V ) )
 
22-Mar-2023neipsm 13590 A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ E. x x e. S ) -> ( N e. ( ( nei ` J ) ` S ) <-> A. p e. S N e. ( ( nei ` J ) ` { p } ) ) )
 
19-Mar-2023mkvprop 7155 Markov's Principle expressed in terms of propositions (or more precisely, the A = om case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
|- ( ( A e. Markov /\ A. n e. A DECID ph /\ -. A. n e. A -. ph ) -> E. n e. A ph )
 
18-Mar-2023omnimkv 7153 An omniscient set is Markov. In particular, the case where A is om means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
|- ( A e. Omni -> A e. Markov )
 
18-Mar-2023ismkvmap 7151 The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
|- ( A e. V -> ( A e. Markov <-> A. f e. ( 2o ^m A ) ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) )
 
18-Mar-2023ismkv 7150 The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
|- ( A e. V -> ( A e. Markov <-> A. f ( f : A --> 2o -> ( -. A. x e. A ( f ` x ) = 1o -> E. x e. A ( f ` x ) = (/) ) ) ) )
 
18-Mar-2023df-markov 7149 A Markov set is one where if a predicate (here represented by a function f) on that set does not hold (where hold means is equal to 1o) for all elements, then there exists an element where it fails (is equal to (/)). Generalization of definition 2.5 of [Pierik], p. 9.

In particular, om e. Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)

|- Markov = { y | A. f ( f : y --> 2o -> ( -. A. x e. y ( f ` x ) = 1o -> E. x e. y ( f ` x ) = (/) ) ) }
 
17-Mar-2023finct 7114 A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
|- ( A e. Fin -> E. g g : om -onto-> ( A ⊔ 1o ) )
 
16-Mar-2023ctmlemr 7106 Lemma for ctm 7107. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
|- ( E. x x e. A -> ( E. f f : om -onto-> A -> E. f f : om -onto-> ( A ⊔ 1o ) ) )
 
15-Mar-2023caseinl 7089 Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
|- ( ph -> F Fn B )   &    |- ( ph -> Fun G )   &    |- ( ph -> A e. B )   =>    |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )
 
13-Mar-2023enumct 7113 A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as E. n e. om E. f f : n -onto-> A per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as E. g g : om -onto-> ( A ⊔ 1o ) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( E. n e. om E. f f : n -onto-> A -> E. g g : om -onto-> ( A ⊔ 1o ) )
 
13-Mar-2023enumctlemm 7112 Lemma for enumct 7113. The case where N is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( ph -> F : N -onto-> A )   &    |- ( ph -> N e. om )   &    |- ( ph -> (/) e. N )   &    |- G = ( k e. om |-> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) )   =>    |- ( ph -> G : om -onto-> A )
 
13-Mar-2023ctm 7107 Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( E. x x e. A -> ( E. f f : om -onto-> ( A ⊔ 1o ) <-> E. f f : om -onto-> A ) )
 
13-Mar-20230ct 7105 The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- E. f f : om -onto-> ( (/) ⊔ 1o )
 
13-Mar-2023ctex 6752 A class dominated by om is a set. See also ctfoex 7116 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
|- ( A ~<_ om -> A e. _V )
 
12-Mar-2023cls0 13569 The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.)
|- ( J e. Top -> ( ( cls ` J ) ` (/) ) = (/) )
 
12-Mar-2023algrp1 12045 The value of the algorithm iterator R at ( K + 1 ). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   =>    |- ( ( ph /\ K e. Z ) -> ( R ` ( K + 1 ) ) = ( F ` ( R ` K ) ) )
 
12-Mar-2023ialgr0 12043 The value of the algorithm iterator R at 0 is the initial state A. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.)
|- Z = ( ZZ>= ` M )   &    |- R = seq M ( ( F o. 1st ) , ( Z X. { A } ) )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. S )   &    |- ( ph -> F : S --> S )   =>    |- ( ph -> ( R ` M ) = A )
 
11-Mar-2023ntreq0 13568 Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X ) -> ( ( ( int ` J ) ` S ) = (/) <-> A. x e. J ( x C_ S -> x = (/) ) ) )
 
11-Mar-2023clstop 13563 The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)
|- X = U. J   =>    |- ( J e. Top -> ( ( cls ` J ) ` X ) = X )
 
11-Mar-2023ntrss 13555 Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )
 
10-Mar-2023iuncld 13551 A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.)
|- X = U. J   =>    |- ( ( J e. Top /\ A e. Fin /\ A. x e. A B e. ( Clsd ` J ) ) -> U_ x e. A B e. ( Clsd ` J ) )
 
5-Mar-20232basgeng 13518 Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
|- ( ( B e. V /\ B C_ C /\ C C_ ( topGen ` B ) ) -> ( topGen ` B ) = ( topGen ` C ) )
 
5-Mar-2023exmidsssn 4202 Excluded middle is equivalent to the biconditionalized version of sssnr 3753 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
|- (EXMID <-> A. x A. y ( x C_ { y } <-> ( x = (/) \/ x = { y } ) ) )
 
5-Mar-2023exmidn0m 4201 Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
|- (EXMID <-> A. x ( x = (/) \/ E. y y e. x ) )
 
4-Mar-2023eltg3 13493 Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> E. x ( x C_ B /\ A = U. x ) ) )
 
4-Mar-2023tgvalex 12711 The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.)
|- ( B e. V -> ( topGen ` B ) e. _V )
 
4-Mar-2023biadanii 613 Inference associated with biadani 612. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.)
|- ( ph -> ps )   &    |- ( ps -> ( ph <-> ch ) )   =>    |- ( ph <-> ( ps /\ ch ) )
 
4-Mar-2023biadani 612 An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.)
|- ( ph -> ps )   =>    |- ( ( ps -> ( ph <-> ch ) ) <-> ( ph <-> ( ps /\ ch ) ) )
 
16-Feb-2023ixp0 6730 The infinite Cartesian product of a family B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)
|- ( E. x e. A B = (/) -> X_ x e. A B = (/) )
 
16-Feb-2023ixpm 6729 If an infinite Cartesian product of a family B ( x ) is inhabited, every B ( x ) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
|- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
 
16-Feb-2023exmidundifim 4207 Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4206 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
|- (EXMID <-> A. x A. y ( x C_ y -> ( x u. ( y \ x ) ) = y ) )
 
15-Feb-2023ixpintm 6724 The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- ( E. z z e. B -> X_ x e. A |^| B = |^|_ y e. B X_ x e. A y )
 
15-Feb-2023ixpiinm 6723 The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- ( E. z z e. B -> X_ x e. A |^|_ y e. B C = |^|_ y e. B X_ x e. A C )
 
15-Feb-2023ixpexgg 6721 The existence of an infinite Cartesian product. x is normally a free-variable parameter in B. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- ( ( A e. W /\ A. x e. A B e. V ) -> X_ x e. A B e. _V )
 
15-Feb-2023nfixpxy 6716 Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
|- F/_ y A   &    |- F/_ y B   =>    |- F/_ y X_ x e. A B
 
13-Feb-2023topnpropgd 12701 The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.)
|- ( ph -> ( Base ` K ) = ( Base ` L ) )   &    |- ( ph -> (TopSet ` K ) = (TopSet ` L ) )   &    |- ( ph -> K e. V )   &    |- ( ph -> L e. W )   =>    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )
 
12-Feb-2023slotex 12488 Existence of slot value. A corollary of slotslfn 12487. (Contributed by Jim Kingdon, 12-Feb-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- ( A e. V -> ( E ` A ) e. _V )
 
11-Feb-2023topnvalg 12699 Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
|- B = ( Base ` W )   &    |- J = (TopSet ` W )   =>    |- ( W e. V -> ( J ↾t B ) = ( TopOpen ` W ) )
 
10-Feb-2023slotslfn 12487 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- E Fn _V
 
9-Feb-2023pleslid 12656 Slot property of le. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- ( le = Slot ( le ` ndx ) /\ ( le ` ndx ) e. NN )
 
9-Feb-2023topgrptsetd 12653 The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> J = (TopSet ` W ) )
 
9-Feb-2023topgrpplusgd 12652 The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> .+ = ( +g ` W ) )
 
9-Feb-2023topgrpbasd 12651 The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> B = ( Base ` W ) )
 
9-Feb-2023topgrpstrd 12650 A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.)
|- W = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , J >. }   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> J e. X )   =>    |- ( ph -> W Struct <. 1 , 9 >. )
 
9-Feb-2023tsetslid 12642 Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.)
|- (TopSet = Slot (TopSet ` ndx ) /\ (TopSet ` ndx ) e. NN )
 
8-Feb-2023ipsipd 12639 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> I = ( .i ` A ) )
 
8-Feb-2023ipsvscad 12638 The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> .x. = ( .s ` A ) )
 
8-Feb-2023ipsscad 12637 The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> S = (Scalar ` A ) )
 
7-Feb-2023ipsmulrd 12636 The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> .X. = ( .r ` A ) )
 
7-Feb-2023ipsaddgd 12635 The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> .+ = ( +g ` A ) )
 
7-Feb-2023ipsbased 12634 The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> B = ( Base ` A ) )
 
7-Feb-2023ipsstrd 12633 A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .X. e. X )   &    |- ( ph -> S e. Y )   &    |- ( ph -> .x. e. Q )   &    |- ( ph -> I e. Z )   =>    |- ( ph -> A Struct <. 1 , 8 >. )
 
7-Feb-2023ipslid 12628 Slot property of .i. (Contributed by Jim Kingdon, 7-Feb-2023.)
|- ( .i = Slot ( .i ` ndx ) /\ ( .i ` ndx ) e. NN )
 
7-Feb-2023lmodvscad 12625 The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> .x. = ( .s ` W ) )
 
6-Feb-2023lmodscad 12624 The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> F = (Scalar ` W ) )
 
6-Feb-2023lmodplusgd 12623 The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> .+ = ( +g ` W ) )
 
6-Feb-2023lmodbased 12622 The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> B = ( Base ` W ) )
 
5-Feb-2023lmodstrd 12621 A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. X )   &    |- ( ph -> F e. Y )   &    |- ( ph -> .x. e. Z )   =>    |- ( ph -> W Struct <. 1 , 6 >. )
 
5-Feb-2023vscaslid 12620 Slot property of .s. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- ( .s = Slot ( .s ` ndx ) /\ ( .s ` ndx ) e. NN )
 
5-Feb-2023scaslid 12610 Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- (Scalar = Slot (Scalar ` ndx ) /\ (Scalar ` ndx ) e. NN )
 
5-Feb-2023srngplusgd 12605 The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> .+ = ( +g ` R ) )
 
5-Feb-2023srngbased 12604 The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> B = ( Base ` R ) )
 
5-Feb-2023srngstrd 12603 A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.)
|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )   &    |- ( ph -> B e. V )   &    |- ( ph -> .+ e. W )   &    |- ( ph -> .x. e. X )   &    |- ( ph -> .* e. Y )   =>    |- ( ph -> R Struct <. 1 , 4 >. )
 
5-Feb-2023opelstrsl 12572 The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( ph -> S Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> <. ( E ` ndx ) , V >. e. S )   =>    |- ( ph -> V = ( E ` S ) )
 
4-Feb-2023starvslid 12598 Slot property of *r. (Contributed by Jim Kingdon, 4-Feb-2023.)
|- ( *r = Slot ( *r ` ndx ) /\ ( *r ` ndx ) e. NN )
 
3-Feb-2023rngbaseg 12593 The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }   =>    |- ( ( B e. V /\ .+ e. W /\ .x. e. X ) -> B = ( Base ` R ) )
 
3-Feb-2023rngstrg 12592 A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.)
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }   =>    |- ( ( B e. V /\ .+ e. W /\ .x. e. X ) -> R Struct <. 1 , 3 >. )
 
3-Feb-2023mulrslid 12589 Slot property of .r. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
 
3-Feb-2023plusgslid 12570 Slot property of +g. (Contributed by Jim Kingdon, 3-Feb-2023.)
|- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
 
2-Feb-20232strop1g 12581 The other slot of a constructed two-slot structure. Version of 2stropg 12578 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   &    |- E = Slot N   =>    |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )
 
2-Feb-20232strbas1g 12580 The base set of a constructed two-slot structure. Version of 2strbasg 12577 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )
 
2-Feb-20232strstr1g 12579 A constructed two-slot structure. Version of 2strstrg 12576 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. N , .+ >. }   &    |- ( Base ` ndx ) < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. ( Base ` ndx ) , N >. )
 
31-Jan-2023baseslid 12518 The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.)
|- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
 
31-Jan-2023strsl0 12510 All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- (/) = ( E ` (/) )
 
31-Jan-2023strslss 12509 Propagate component extraction to a structure T from a subset structure S. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Jim Kingdon, 31-Jan-2023.)
|- T e. _V   &    |- Fun T   &    |- S C_ T   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- <. ( E ` ndx ) , C >. e. S   =>    |- ( E ` T ) = ( E ` S )
 
31-Jan-2023strslssd 12508 Deduction version of strslss 12509. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( ph -> T e. V )   &    |- ( ph -> Fun T )   &    |- ( ph -> S C_ T )   &    |- ( ph -> <. ( E ` ndx ) , C >. e. S )   =>    |- ( ph -> ( E ` T ) = ( E ` S ) )
 
30-Jan-2023strslfv3 12507 Variant on strslfv 12506 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- ( ph -> U = S )   &    |- S Struct X   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- { <. ( E ` ndx ) , C >. } C_ S   &    |- ( ph -> C e. V )   &    |- A = ( E ` U )   =>    |- ( ph -> A = C )
 
30-Jan-2023strslfv 12506 Extract a structure component C (such as the base set) from a structure S with a component extractor E (such as the base set extractor df-base 12467). By virtue of ndxslid 12486, this can be done without having to refer to the hard-coded numeric index of E. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- S Struct X   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- { <. ( E ` ndx ) , C >. } C_ S   =>    |- ( C e. V -> C = ( E ` S ) )
 
30-Jan-2023strslfv2 12505 A variation on strslfv 12506 to avoid asserting that S itself is a function, which involves sethood of all the ordered pair components of S. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- S e. _V   &    |- Fun `' `' S   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- <. ( E ` ndx ) , C >. e. S   =>    |- ( C e. V -> C = ( E ` S ) )
 
30-Jan-2023strslfv2d 12504 Deduction version of strslfv 12506. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( ph -> S e. V )   &    |- ( ph -> Fun `' `' S )   &    |- ( ph -> <. ( E ` ndx ) , C >. e. S )   &    |- ( ph -> C e. W )   =>    |- ( ph -> C = ( E ` S ) )
 
30-Jan-2023strslfvd 12503 Deduction version of strslfv 12506. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( ph -> S e. V )   &    |- ( ph -> Fun S )   &    |- ( ph -> <. ( E ` ndx ) , C >. e. S )   =>    |- ( ph -> C = ( E ` S ) )
 
30-Jan-2023strsetsid 12494 Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- E = Slot ( E ` ndx )   &    |- ( ph -> S Struct <. M , N >. )   &    |- ( ph -> Fun S )   &    |- ( ph -> ( E ` ndx ) e. dom S )   =>    |- ( ph -> S = ( S sSet <. ( E ` ndx ) , ( E ` S ) >. ) )
 
30-Jan-2023funresdfunsndc 6506 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.)
|- ( ( A. x e. dom F A. y e. dom FDECID x = y /\ Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) = F )
 
29-Jan-2023ndxslid 12486 A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12506. (Contributed by Jim Kingdon, 29-Jan-2023.)
|- E = Slot N   &    |- N e. NN   =>    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
 
29-Jan-2023fnsnsplitdc 6505 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
|- ( ( A. x e. A A. y e. A DECID x = y /\ F Fn A /\ X e. A ) -> F = ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) )
 
28-Jan-20232stropg 12578 The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }   &    |- E = Slot N   &    |- 1 < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )
 
28-Jan-20232strbasg 12577 The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }   &    |- E = Slot N   &    |- 1 < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )
 
28-Jan-20232strstrg 12576 A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }   &    |- E = Slot N   &    |- 1 < N   &    |- N e. NN   =>    |- ( ( B e. V /\ .+ e. W ) -> G Struct <. 1 , N >. )
 
28-Jan-20231strstrg 12574 A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.)
|- G = { <. ( Base ` ndx ) , B >. }   =>    |- ( B e. V -> G Struct <. 1 , 1 >. )
 
27-Jan-2023strle2g 12565 Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|- I e. NN   &    |- A = I   &    |- I < J   &    |- J e. NN   &    |- B = J   =>    |- ( ( X e. V /\ Y e. W ) -> { <. A , X >. , <. B , Y >. } Struct <. I , J >. )
 
27-Jan-2023strle1g 12564 Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|- I e. NN   &    |- A = I   =>    |- ( X e. V -> { <. A , X >. } Struct <. I , I >. )
 
27-Jan-2023strleund 12561 Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
|- ( ph -> F Struct <. A , B >. )   &    |- ( ph -> G Struct <. C , D >. )   &    |- ( ph -> B < C )   =>    |- ( ph -> ( F u. G ) Struct <. A , D >. )
 
24-Jan-2023setsslnid 12513 Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( E ` ndx ) =/= D   &    |- D e. NN   =>    |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )
 
24-Jan-2023setsslid 12512 Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)
|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   =>    |- ( ( W e. A /\ C e. V ) -> C = ( E ` ( W sSet <. ( E ` ndx ) , C >. ) ) )
 
22-Jan-2023setsabsd 12500 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
|- ( ph -> S e. V )   &    |- ( ph -> A e. W )   &    |- ( ph -> B e. X )   &    |- ( ph -> C e. U )   =>    |- ( ph -> ( ( S sSet <. A , B >. ) sSet <. A , C >. ) = ( S sSet <. A , C >. ) )
 
22-Jan-2023setsresg 12499 The structure replacement function does not affect the value of S away from A. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
|- ( ( S e. V /\ A e. W /\ B e. X ) -> ( ( S sSet <. A , B >. ) |` ( _V \ { A } ) ) = ( S |` ( _V \ { A } ) ) )
 
22-Jan-2023setsex 12493 Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.)
|- ( ( S e. V /\ A e. X /\ B e. W ) -> ( S sSet <. A , B >. ) e. _V )
 
22-Jan-20232zsupmax 11233 Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> sup ( { A , B } , RR , < ) = if ( A <_ B , B , A ) )
 
22-Jan-2023elpwpwel 4475 A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
|- ( A e. ~P ~P B <-> U. A e. ~P B )
 
21-Jan-2023funresdfunsnss 5719 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.)
|- ( ( Fun F /\ X e. dom F ) -> ( ( F |` ( _V \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
 
20-Jan-2023setsvala 12492 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.)
|- ( ( S e. V /\ A e. X /\ B e. W ) -> ( S sSet <. A , B >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , B >. } ) )
 
20-Jan-2023fnsnsplitss 5715 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.)
|- ( ( F Fn A /\ X e. A ) -> ( ( F |` ( A \ { X } ) ) u. { <. X , ( F ` X ) >. } ) C_ F )
 
19-Jan-2023strfvssn 12483 A structure component extractor produces a value which is contained in a set dependent on S, but not E. This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 19-Jan-2023.)
|- E = Slot N   &    |- ( ph -> S e. V )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( E ` S ) C_ U. ran S )
 
19-Jan-2023strnfvn 12482 Value of a structure component extractor E. Normally, E is a defined constant symbol such as Base (df-base 12467) and N is a fixed integer such as 1. S is a structure, i.e. a specific member of a class of structures.

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12506. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.)

|- S e. _V   &    |- E = Slot N   &    |- N e. NN   =>    |- ( E ` S ) = ( S ` N )
 
19-Jan-2023strnfvnd 12481 Deduction version of strnfvn 12482. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.)
|- E = Slot N   &    |- ( ph -> S e. V )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( E ` S ) = ( S ` N ) )
 
18-Jan-2023isstructr 12476 The property of being a structure with components in M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
|- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> F Struct <. M , N >. )
 
18-Jan-2023isstructim 12475 The property of being a structure with components in M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
|- ( F Struct <. M , N >. -> ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( M ... N ) ) )
 
18-Jan-2023isstruct2r 12472 The property of being a structure with components in ( 1st ` X ) ... ( 2nd ` X ). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
|- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F Struct X )
 
18-Jan-2023isstruct2im 12471 The property of being a structure with components in ( 1st ` X ) ... ( 2nd ` X ). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)
|- ( F Struct X -> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) )
 
18-Jan-2023sbiev 1792 Conversion of implicit substitution to explicit substitution. Version of sbie 1791 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)
|- F/ x ps   &    |- ( x = y -> ( ph <-> ps ) )   =>    |- ( [ y / x ] ph <-> ps )
 
16-Jan-2023toponsspwpwg 13458 The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
|- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )
 
14-Jan-2023istopfin 13436 Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 13435. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) )
 
14-Jan-2023fiintim 6927 If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as x and y not being equal, or A having decidable equality.

This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)

|- ( A. x e. A A. y e. A ( x i^i y ) e. A -> A. x ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) )
 
9-Jan-2023divccncfap 14013 Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
|- F = ( x e. CC |-> ( x / A ) )   =>    |- ( ( A e. CC /\ A # 0 ) -> F e. ( CC -cn-> CC ) )
 
7-Jan-2023eap1 11792 _e is apart from 1. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 1
 
7-Jan-2023eap0 11790 _e is apart from 0. (Contributed by Jim Kingdon, 7-Jan-2023.)
|- _e # 0
 
7-Jan-2023egt2lt3 11786 Euler's constant _e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.)
|- ( 2 < _e /\ _e < 3 )
 
6-Jan-2023eirr 11785 _e is not rational. In the absence of excluded middle, we can distinguish between this and saying that _e is irrational in the sense of being apart from any rational number, which is eirrap 11784. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 6-Jan-2023.)
|- _e e/ QQ
 
6-Jan-2023eirrap 11784 _e is irrational. That is, for any rational number, _e is apart from it. In the absence of excluded middle, we can distinguish between this and saying that _e is not rational, which is eirr 11785. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( Q e. QQ -> _e # Q )
 
6-Jan-2023btwnapz 9382 A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( ph -> A e. ZZ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> A < B )   &    |- ( ph -> B < ( A + 1 ) )   =>    |- ( ph -> B # C )
 
6-Jan-2023apmul2 8745 Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.)
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C # 0 ) ) -> ( A # B <-> ( C x. A ) # ( C x. B ) ) )
 
1-Jan-2023nnap0i 8949 A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
 
31-Dec-20222logb9irrALT 14328 Alternate proof of 2logb9irr 14325: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( 2 logb 9 ) e. ( RR \ QQ )
 
31-Dec-20222logb3irr 14327 Example for logbprmirr 14326. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
|- ( 2 logb 3 ) e. ( RR \ QQ )
 
31-Dec-2022logbprmirr 14326 The logarithm of a prime to a different prime base is not rational. For example, ( 2 logb 3 ) e. ( RR \ QQ ) (see 2logb3irr 14327). (Contributed by AV, 31-Dec-2022.)
|- ( ( X e. Prime /\ B e. Prime /\ X =/= B ) -> ( B logb X ) e. ( RR \ QQ ) )
 
30-Dec-2022elpqb 9648 A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.)
|- ( ( A e. QQ /\ 0 < A ) <-> E. x e. NN E. y e. NN A = ( x / y ) )
 
29-Dec-2022sqrt2cxp2logb9e3 14329 The square root of two to the power of the logarithm of nine to base two is three. ( sqr ` 2 ) and ( 2 logb 9 ) are not rational (see sqrt2irr0 12163 resp. 2logb9irr 14325), satisfying the statement in 2irrexpq 14330. (Contributed by AV, 29-Dec-2022.)
|- ( ( sqr ` 2 ) ^c ( 2 logb 9 ) ) = 3
 
29-Dec-20222logb9irr 14325 Example for logbgcd1irr 14321. The logarithm of nine to base two is not rational. Also see 2logb9irrap 14331 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.)
|- ( 2 logb 9 ) e. ( RR \ QQ )
 
29-Dec-2022logbgcd1irrap 14324 The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, ( 2 logb 9 ) # Q where Q is rational. (Contributed by AV, 29-Dec-2022.)
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( B logb X ) # Q )
 
29-Dec-2022logbgcd1irr 14321 The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, ( 2 logb 9 ) e. ( RR \ QQ ). (Contributed by AV, 29-Dec-2022.)
|- ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) /\ ( X gcd B ) = 1 ) -> ( B logb X ) e. ( RR \ QQ ) )
 
29-Dec-2022logbgt0b 14320 The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. RR+ /\ ( B e. RR+ /\ 1 < B ) ) -> ( 0 < ( B logb A ) <-> 1 < A ) )
 
29-Dec-2022cxpcom 14293 Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. RR+ /\ B e. RR /\ C e. RR ) -> ( ( A ^c B ) ^c C ) = ( ( A ^c C ) ^c B ) )
 
29-Dec-2022elpq 9647 A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. QQ /\ 0 < A ) -> E. x e. NN E. y e. NN A = ( x / y ) )
 
26-Dec-2022apdivmuld 8769 Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. CC )   &    |- ( ph -> B # 0 )   =>    |- ( ph -> ( ( A / B ) # C <-> ( B x. C ) # A ) )
 
25-Dec-2022tanaddaplem 11745 A useful intermediate step in tanaddap 11746 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.)
|- ( ( ( A e. CC /\ B e. CC ) /\ ( ( cos ` A ) # 0 /\ ( cos ` B ) # 0 ) ) -> ( ( cos ` ( A + B ) ) # 0 <-> ( ( tan ` A ) x. ( tan ` B ) ) # 1 ) )
 
25-Dec-2022subap0 8599 Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) # 0 <-> A # B ) )
 
23-Dec-20222irrexpq 14330 There exist real numbers a and b which are not rational 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 non-rational numbers ( sqr ` 2 ) and ( 2 logb 9 ), see sqrt2irr0 12163, 2logb9irr 14325 and sqrt2cxp2logb9e3 14329. Therefore, this proof is acceptable/usable in intuitionistic logic.

For a theorem which is the same but proves that a and b are irrational (in the sense of being apart from any rational number), see 2irrexpqap 14332. (Contributed by AV, 23-Dec-2022.)

|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ
 
23-Dec-2022rpcxpsqrtth 14286 Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11039. (Contributed by AV, 23-Dec-2022.)
|- ( A e. RR+ -> ( ( sqr ` A ) ^c 2 ) = A )
 
23-Dec-2022sqrt2irr0 12163 The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.)
|- ( sqr ` 2 ) e. ( RR \ QQ )
 
22-Dec-2022tanval3ap 11721 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
|- ( ( A e. CC /\ ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( 2 x. ( _i x. A ) ) ) - 1 ) / ( _i x. ( ( exp ` ( 2 x. ( _i x. A ) ) ) + 1 ) ) ) )
 
22-Dec-2022tanval2ap 11720 Express the tangent function directly in terms of exp. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Jim Kingdon, 22-Dec-2022.)
|- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( _i x. ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) ) ) )
 
22-Dec-2022tanclapd 11719 Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( cos ` A ) # 0 )   =>    |- ( ph -> ( tan ` A ) e. CC )
 
21-Dec-2022tanclap 11716 The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
|- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) e. CC )
 
21-Dec-2022tanvalap 11715 Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.)
|- ( ( A e. CC /\ ( cos ` A ) # 0 ) -> ( tan ` A ) = ( ( sin ` A ) / ( cos ` A ) ) )
 
20-Dec-2022reef11 11706 The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( exp ` A ) = ( exp ` B ) <-> A = B ) )
 
20-Dec-2022efltim 11705 The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B -> ( exp ` A ) < ( exp ` B ) ) )
 
20-Dec-2022eqord1 8439 A strictly increasing real function on a subset of RR is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by Jim Kingdon, 20-Dec-2022.)
|- ( x = y -> A = B )   &    |- ( x = C -> A = M )   &    |- ( x = D -> A = N )   &    |- S C_ RR   &    |- ( ( ph /\ x e. S ) -> A e. RR )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x < y -> A < B ) )   =>    |- ( ( ph /\ ( C e. S /\ D e. S ) ) -> ( C = D <-> M = N ) )
 
14-Dec-2022iserabs 11482 Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> seq M ( + , F ) ~~> A )   &    |- ( ph -> seq M ( + , G ) ~~> B )   &    |- ( ph -> M e. ZZ )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )   =>    |- ( ph -> ( abs ` A ) <_ B )
 
12-Dec-2022efap0 11684 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
|- ( A e. CC -> ( exp ` A ) # 0 )
 
8-Dec-2022efcllem 11666 Lemma for efcl 11671. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( A e. CC -> seq 0 ( + , F ) e. dom ~~> )
 
8-Dec-2022efcllemp 11665 Lemma for efcl 11671. The series that defines the exponential function converges. The ratio test cvgratgt0 11540 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   &    |- ( ph -> A e. CC )   &    |- ( ph -> K e. NN )   &    |- ( ph -> ( 2 x. ( abs ` A ) ) < K )   =>    |- ( ph -> seq 0 ( + , F ) e. dom ~~> )
 
8-Dec-2022eftvalcn 11664 The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )   =>    |- ( ( A e. CC /\ N e. NN0 ) -> ( F ` N ) = ( ( A ^ N ) / ( ! ` N ) ) )
 
8-Dec-2022mertensabs 11544 Mertens' theorem. If A ( j ) is an absolutely convergent series and B ( k ) is convergent, then ( sum_ j e. NN0 A ( j ) x. sum_ k e. NN0 B ( k ) ) = sum_ k e. NN0 sum_ j e. ( 0 ... k ) ( A ( j ) x. B ( k - j ) ) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 8-Dec-2022.)
|- ( ( ph /\ j e. NN0 ) -> ( F ` j ) = A )   &    |- ( ( ph /\ j e. NN0 ) -> ( K ` j ) = ( abs ` A ) )   &    |- ( ( ph /\ j e. NN0 ) -> A e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( H ` k ) = sum_ j e. ( 0 ... k ) ( A x. ( G ` ( k - j ) ) ) )   &    |- ( ph -> seq 0 ( + , K ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , F ) e. dom ~~> )   =>    |- ( ph -> seq 0 ( + , H ) ~~> ( sum_ j e. NN0 A x. sum_ k e. NN0 B ) )
 
3-Dec-2022mertenslemub 11541 Lemma for mertensabs 11544. An upper bound for T. (Contributed by Jim Kingdon, 3-Dec-2022.)
|- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- T = { z | E. n e. ( 0 ... ( S - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }   &    |- ( ph -> X e. T )   &    |- ( ph -> S e. NN )   =>    |- ( ph -> X <_ sum_ n e. ( 0 ... ( S - 1 ) ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) )
 
2-Dec-2022mertenslemi1 11542 Lemma for mertensabs 11544. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|- ( ( ph /\ j e. NN0 ) -> ( F ` j ) = A )   &    |- ( ( ph /\ j e. NN0 ) -> ( K ` j ) = ( abs ` A ) )   &    |- ( ( ph /\ j e. NN0 ) -> A e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( G ` k ) = B )   &    |- ( ( ph /\ k e. NN0 ) -> B e. CC )   &    |- ( ( ph /\ k e. NN0 ) -> ( H ` k ) = sum_ j e. ( 0 ... k ) ( A x. ( G ` ( k - j ) ) ) )   &    |- ( ph -> seq 0 ( + , K ) e. dom ~~> )   &    |- ( ph -> seq 0 ( + , G ) e. dom ~~> )   &    |- ( ph -> E e. RR+ )   &    |- T = { z | E. n e. ( 0 ... ( s - 1 ) ) z = ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) }   &    |- ( ps <-> ( s e. NN /\ A. n e. ( ZZ>= ` s ) ( abs ` sum_ k e. ( ZZ>= ` ( n + 1 ) ) ( G ` k ) ) < ( ( E / 2 ) / ( sum_ j e. NN0 ( K ` j ) + 1 ) ) ) )   &    |- ( ph -> P e. RR )   &    |- ( ph -> ( ps /\ ( t e. NN0 /\ A. m e. ( ZZ>= ` t ) ( K ` m ) < ( ( ( E / 2 ) / s ) / ( P + 1 ) ) ) ) )   &    |- ( ph -> 0 <_ P )   &    |- ( ph -> A. w e. T w <_ P )   =>    |- ( ph -> E. y e. NN0 A. m e. ( ZZ>= ` y ) ( abs ` sum_ j e. ( 0 ... m ) ( A x. sum_ k e. ( ZZ>= ` ( ( m - j ) + 1 ) ) B ) ) < E )
 
2-Dec-2022fsum3cvg3 11403 A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A C_ Z )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   =>    |- ( ph -> seq M ( + , F ) e. dom ~~> )
 
2-Dec-2022fsum3cvg2 11401 The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.)
|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = if ( k e. A , B , 0 ) )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )
 
24-Nov-2022cvgratnnlembern 11530 Lemma for cvgratnn 11538. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ph -> M e. NN )   =>    |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )
 
23-Nov-2022cvgratnnlemfm 11536 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 23-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   =>    |- ( ph -> ( abs ` ( F ` M ) ) < ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) )
 
23-Nov-2022cvgratnnlemsumlt 11535 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 23-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
 
21-Nov-2022cvgratnnlemrate 11537 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 21-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / M ) )
 
21-Nov-2022cvgratnnlemabsle 11534 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 21-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) )
 
21-Nov-2022cvgratnnlemseq 11533 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 21-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
 
15-Nov-2022cvgratnnlemmn 11532 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 15-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> ( abs ` ( F ` N ) ) <_ ( ( abs ` ( F ` M ) ) x. ( A ^ ( N - M ) ) ) )
 
15-Nov-2022cvgratnnlemnexp 11531 Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 15-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( abs ` ( F ` N ) ) <_ ( ( abs ` ( F ` 1 ) ) x. ( A ^ ( N - 1 ) ) ) )
 
12-Nov-2022cvgratnn 11538 Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms, then the infinite sum of the terms of F converges to a complex number. Although this theorem is similar to cvgratz 11539 and cvgratgt0 11540, the decision to index starting at one is not merely cosmetic, as proving convergence using climcvg1n 11357 is sensitive to how a sequence is indexed. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   =>    |- ( ph -> seq 1 ( + , F ) e. dom ~~> )
 
12-Nov-2022fsum3cvg 11385 The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )   &    |- ( ( ph /\ k e. A ) -> B e. CC )   &    |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> DECID k e. A )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   &    |- ( ph -> A C_ ( M ... N ) )   =>    |- ( ph -> seq M ( + , F ) ~~> ( seq M ( + , F ) ` N ) )
 
12-Nov-2022seq3id2 10508 The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)
|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = x )   &    |- ( ph -> K e. ( ZZ>= ` M ) )   &    |- ( ph -> N e. ( ZZ>= ` K ) )   &    |- ( ph -> ( seq M ( .+ , F ) ` K ) e. S )   &    |- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = Z )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` N ) )
 
11-Nov-2022cvgratgt0 11540 Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms beyond some index B, then the infinite sum of the terms of F converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. W ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   =>    |- ( ph -> seq M ( + , F ) e. dom ~~> )
 
11-Nov-2022cvgratz 11539 Ratio test for convergence of a complex infinite series. If the ratio A of the absolute values of successive terms in an infinite sequence F is less than 1 for all terms, then the infinite sum of the terms of F converges to a complex number. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 11-Nov-2022.)
|- Z = ( ZZ>= ` M )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 < A )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )   &    |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )   =>    |- ( ph -> seq M ( + , F ) e. dom ~~> )
 
4-Nov-2022seq3val 10457 Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10460, seq3-1 10459 and seq3p1 10461, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
|- ( ph -> M e. ZZ )   &    |- R = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( x ( z e. ( ZZ>= ` M ) , w e. S |-> ( w .+ ( F ` ( z + 1 ) ) ) ) y ) >. ) , <. M , ( F ` M ) >. )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> seq M ( .+ , F ) = ran R )
 
4-Nov-2022df-seqfrec 10445 Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as NN or NN0) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seqf 10460, seq3-1 10459 and seq3p1 10461. Typically, those are the main theorems that would be used in practice.

The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence F with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq 1 ( + , F ) with values 1, 3/2, 7/4, 15/8,.., so that ( seq 1 ( + , F ) ` 1 ) = 1, ( seq 1 ( + , F ) ` 2 ) = 3/2, etc. In other words, seq M ( + , F ) transforms a sequence F into an infinite series. seq M ( + , F ) ~~> 2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 11300), by climdm 11302 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ~~> ` seq 1 ( + , F ) ) (provided the sequence converges) and evaluates to 2 in this example.

Internally, the frec function generates as its values a set of ordered pairs starting at <. M , ( F ` M ) >., with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain.

(Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

|- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
 
3-Nov-2022seq3f1o 10503 Rearrange a sum via an arbitrary bijection on ( M ... N ). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( 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 -> F : ( M ... N ) -1-1-onto-> ( M ... N ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( G ` x ) e. S )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( H ` x ) e. S )   &    |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( G ` ( F ` k ) ) )   =>    |- ( ph -> ( seq M ( .+ , H ) ` N ) = ( seq M ( .+ , G ) ` N ) )
 
3-Nov-2022seq3m1 10467 Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
|- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` M ) ) -> ( F ` x ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   =>    |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( ( seq M ( .+ , F ) ` ( N - 1 ) ) .+ ( F ` N ) ) )
 
29-Oct-2022absgtap 11517 Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. RR+ )   &    |- ( ph -> B < ( abs ` A ) )   =>    |- ( ph -> A # B )
 
29-Oct-2022absltap 11516 Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> B e. RR )   &    |- ( ph -> ( abs ` A ) < B )   =>    |- ( ph -> A # B )
 
29-Oct-20221ap2 9125 1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- 1 # 2
 
28-Oct-2022expcnv 11511 A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) < 1 )   =>    |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
 
28-Oct-2022expcnvre 11510 A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)
|- ( ph -> A e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> 0 <_ A )   =>    |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
 
27-Oct-2022ennnfone 12425 A condition for a set being countably infinite. Corollary 8.1.13 of [AczelRathjen], p. 73. Roughly speaking, the condition says that A is countable (that's the f : NN0 -onto-> A part, as seen in theorems like ctm 7107), infinite (that's the part about being able to find an element of A distinct from any mapping of a natural number via f), and has decidable equality. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( A ~~ NN <-> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : NN0 -onto-> A /\ A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( f ` k ) =/= ( f ` j ) ) ) )
 
27-Oct-2022ennnfonelemim 12424 Lemma for ennnfone 12425. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( A ~~ NN -> ( A. x e. A A. y e. A DECID x = y /\ E. f ( f : NN0 -onto-> A /\ A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( f ` k ) =/= ( f ` j ) ) ) )
 
27-Oct-2022ennnfonelemr 12423 Lemma for ennnfone 12425. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : NN0 -onto-> A )   &    |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )   =>    |- ( ph -> A ~~ NN )
 
27-Oct-2022ennnfonelemnn0 12422 Lemma for ennnfone 12425. A version of ennnfonelemen 12421 expressed in terms of NN0 instead of om. (Contributed by Jim Kingdon, 27-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : NN0 -onto-> A )   &    |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )   &    |- N = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )   =>    |- ( ph -> A ~~ NN )
 
24-Oct-2022pwm1geoserap1 11515 The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + A ^ 1 + A ^ 2 +... + A ^ ( N - 1 ). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> N e. NN0 )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )
 
24-Oct-2022geoserap 11514 The value of the finite geometric series 1 + A ^ 1 + A ^ 2 +... + A ^ ( N - 1 ). This is Metamath 100 proof #66. (Contributed by NM, 12-May-2006.) (Revised by Jim Kingdon, 24-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 1 )   &    |- ( ph -> N e. NN0 )   =>    |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
 
24-Oct-2022geosergap 11513 The value of the finite geometric series A ^ M + A ^ ( M + 1 ) +... + A ^ ( N - 1 ). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> A # 1 )   &    |- ( ph -> M e. NN0 )   &    |- ( ph -> N e. ( ZZ>= ` M ) )   =>    |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )
 
23-Oct-2022expcnvap0 11509 A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.)
|- ( ph -> A e. CC )   &    |- ( ph -> ( abs ` A ) < 1 )   &    |- ( ph -> A # 0 )   =>    |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
 
22-Oct-2022divcnv 11504 The sequence of reciprocals of positive integers, multiplied by the factor A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)
|- ( A e. CC -> ( n e. NN |-> ( A / n ) ) ~~> 0 )
 
22-Oct-2022impcomd 255 Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ph -> ( ( ch /\ ps ) -> th ) )
 
21-Oct-2022isumsplit 11498 Split off the first N terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 21-Oct-2022.)
|- Z = ( ZZ>= ` M )   &    |- W = ( ZZ>= ` N )   &    |- ( ph -> N e. Z )   &    |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )   &    |- ( ( ph /\ k e. Z ) -> A e. CC )   &    |- ( ph -> seq M ( + , F ) e. dom ~~> )   =>    |- ( ph -> sum_ k e. Z A = ( sum_ k e. ( M ... ( N - 1 ) ) A + sum_ k e. W A ) )
 
21-Oct-2022seq3split 10478 Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )   &    |- ( ph -> N e. ( ZZ>= ` ( M + 1 ) ) )   &    |- ( ph -> M e. ( ZZ>= ` K ) )   &    |- ( ( ph /\ x e. ( ZZ>= ` K ) ) -> ( F ` x ) e. S )   =>    |- ( ph -> ( seq K ( .+ , F ) ` N ) = ( ( seq K ( .+ , F ) ` M ) .+ ( seq ( M + 1 ) ( .+ , F ) ` N ) ) )
 
20-Oct-2022fidcenumlemrk 6952 Lemma for fidcenum 6954. (Contributed by Jim Kingdon, 20-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> K e. om )   &    |- ( ph -> K C_ N )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) )
 
20-Oct-2022fidcenumlemrks 6951 Lemma for fidcenum 6954. Induction step for fidcenumlemrk 6952. (Contributed by Jim Kingdon, 20-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> J e. om )   &    |- ( ph -> suc J C_ N )   &    |- ( ph -> ( X e. ( F " J ) \/ -. X e. ( F " J ) ) )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
 
19-Oct-2022fidcenum 6954 A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as E. n e. om E. f f : n -onto-> A is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( A e. Fin <-> ( A. x e. A A. y e. A DECID x = y /\ E. n e. om E. f f : n -onto-> A ) )
 
19-Oct-2022fidcenumlemr 6953 Lemma for fidcenum 6954. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> N e. om )   =>    |- ( ph -> A e. Fin )
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