P. 133 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13201-13300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ballotfilemsf1o 13201*	The defined S is a bijection, and an involution. (Contributed by Thierry Arnoux, 14-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   =>    |- ( C e. ( O \ E ) -> ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) /\ `' ( S ` C ) = ( S ` C ) ) )
Theorem	ballotfilemsi 13202*	The image by S of the first tie pick is the first pick. (Contributed by Thierry Arnoux, 14-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   =>    |- ( C e. ( O \ E ) -> ( ( S ` C ) ` ( I ` C ) ) = 1 )
Theorem	ballotfilemsima 13203*	The image by S of an interval before the first pick. (Contributed by Thierry Arnoux, 5-May-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( 1 ... J ) ) = ( ( ( S ` C ) ` J ) ... ( I ` C ) ) )
Theorem	ballotfilemieq 13204*	If two countings share the same first tie, they also have the same swap function. (Contributed by Thierry Arnoux, 18-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   =>    |- ( ( C e. ( O \ E ) /\ D e. ( O \ E ) /\ ( I ` C ) = ( I ` D ) ) -> ( S ` C ) = ( S ` D ) )
Theorem	ballotfilemrval 13205*	Value of R. (Contributed by Thierry Arnoux, 14-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )
Theorem	ballotfilemscr 13206*	The image of ( R ` C ) by ( S ` C ). (Contributed by Thierry Arnoux, 21-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( C e. ( O \ E ) -> ( ( S ` C ) " ( R ` C ) ) = C )
Theorem	ballotfilemrv 13207*	Value of R evaluated at J. (Contributed by Thierry Arnoux, 17-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) ) -> ( J e. ( R ` C ) <-> if ( J <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - J ) , J ) e. C ) )
Theorem	ballotfilemrv1 13208*	Value of R before the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J <_ ( I ` C ) ) -> ( J e. ( R ` C ) <-> ( ( ( I ` C ) + 1 ) - J ) e. C ) )
Theorem	ballotfilemrv2 13209*	Value of R after the tie. (Contributed by Thierry Arnoux, 11-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ ( I ` C ) < J ) -> ( J e. ( R ` C ) <-> J e. C ) )
Theorem	ballotfilemro 13210*	Range of R is included in O. (Contributed by Thierry Arnoux, 17-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( C e. ( O \ E ) -> ( R ` C ) e. O )
Theorem	ballotfilemgval 13211*	Expand the value of .^. (Contributed by Thierry Arnoux, 21-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   &    |- ( ph -> U e. O )   &    |- ( ph -> J e. ZZ )   &    |- ( ph -> K e. ZZ )   &    |- ( ph -> V = ( J ... K ) )   =>    |- ( ph -> ( U .^ V ) = ( (♯ ` ( V i^i U ) ) - (♯ ` ( V \ U ) ) ) )
Theorem	ballotfilemgun 13212*	A property of the defined .^ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.) (Revised by Jim Kingdon, 15-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   &    |- ( ph -> U e. O )   &    |- ( ph -> L e. ( J ... K ) )   =>    |- ( ph -> ( U .^ ( J ... K ) ) = ( ( U .^ ( J ... ( L - 1 ) ) ) + ( U .^ ( L ... K ) ) ) )
Theorem	ballotfilemfg 13213*	Express the value of ( F ` C ) in terms of .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 0 ... ( M + N ) ) ) -> ( ( F ` C ) ` J ) = ( C .^ ( 1 ... J ) ) )
Theorem	ballotfilemfrc 13214*	Express the value of ( F ` ( R ` C ) ) in terms of the newly defined .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( F ` ( R ` C ) ) ` J ) = ( C .^ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) ) )
Theorem	ballotfilemfrci 13215*	Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   =>    |- ( C e. ( O \ E ) -> ( ( F ` ( R ` C ) ) ` ( I ` C ) ) = 0 )
Theorem	ballotfilemfrceq 13216*	Value of F for a reverse counting ( R ` C ). (Contributed by Thierry Arnoux, 27-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   &    |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( F ` C ) ` ( ( ( S ` C ) ` J ) - 1 ) ) = -u ( ( F ` ( R ` C ) ) ` J ) )
Theorem	ballotfilemfrcn0 13217*	Value of F for a reversed counting ( R ` C ), before the first tie, cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2017.) (Revised by AV, 6-Oct-2020.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( M + N ) ) /\ J < ( I ` C ) ) -> ( ( F ` ( R ` C ) ) ` J ) =/= 0 )
Theorem	ballotfilemrc 13218*	Range of R. (Contributed by Thierry Arnoux, 19-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( C e. ( O \ E ) -> ( R ` C ) e. ( O \ E ) )
Theorem	ballotfilemirc 13219*	Applying R does not change first ties. (Contributed by Thierry Arnoux, 19-Apr-2017.) (Revised by AV, 6-Oct-2020.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( C e. ( O \ E ) -> ( I ` ( R ` C ) ) = ( I ` C ) )
Theorem	ballotfilemrinv0 13220*	Lemma for ballotfilemrinv 13221. (Contributed by Thierry Arnoux, 18-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( ( C e. ( O \ E ) /\ D = ( ( S ` C ) " C ) ) -> ( D e. ( O \ E ) /\ C = ( ( S ` D ) " D ) ) )
Theorem	ballotfilemrinv 13221*	R is its own inverse : it is an involution. (Contributed by Thierry Arnoux, 10-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- `' R = R
Theorem	ballotfilem1ri 13222*	When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-Apr-2017.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( C e. ( O \ E ) -> ( 1 e. ( R ` C ) <-> ( I ` C ) e. C ) )
Theorem	ballotfilem7 13223*	R is a bijection between two subsets of ( O \ E ): one where a vote for A is picked first, and one where a vote for B is picked first. (Contributed by Thierry Arnoux, 12-Dec-2016.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( R |` { c e. ( O \ E ) | 1 e. c } ) : { c e. ( O \ E ) | 1 e. c } -1-1-onto-> { c e. ( O \ E ) | -. 1 e. c }
Theorem	ballotfilem8 13224*	There are as many countings with ties starting with a ballot for A as there are starting with a ballot for B. (Contributed by Thierry Arnoux, 7-Dec-2016.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- (♯ ` { c e. ( O \ E ) | 1 e. c } ) = (♯ ` { c e. ( O \ E ) | -. 1 e. c } )
Theorem	ballotfilemth 13225*	Lemma for ballotfi 13226. The result, with several additional hypotheses which are for use during the proof. (Contributed by Thierry Arnoux, 7-Dec-2016.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   &    |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )   &    |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )   &    |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )   =>    |- ( P ` E ) = ( ( M - N ) / ( M + N ) )
Theorem	ballotfi 13226*	Bertrand's ballot problem : the probability that A is ahead throughout the counting. The proof formalized here is a proof "by reflection", as opposed to other known proofs "by induction" or "by permutation". This is Metamath 100 proof #30. (Contributed by Thierry Arnoux, 7-Dec-2016.) (Revised by Jim Kingdon, 17-Jun-2026.)
|- M e. NN   &    |- N e. NN   &    |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }   &    |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )   &    |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )   &    |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }   &    |- N < M   =>    |- ( P ` E ) = ( ( M - N ) / ( M + N ) )
5.3  Cardinality of real and complex number subsets
5.3.1  Countability of integers and rationals
Theorem	oddennn 13227	There are as many odd positive integers as there are positive integers. (Contributed by Jim Kingdon, 11-May-2022.)
|- { z e. NN | -. 2 || z } ~~ NN
Theorem	evenennn 13228	There are as many even positive integers as there are positive integers. (Contributed by Jim Kingdon, 12-May-2022.)
|- { z e. NN | 2 || z } ~~ NN
Theorem	xpnnen 13229	The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. (Contributed by NM, 1-Aug-2004.)
|- ( NN X. NN ) ~~ NN
Theorem	xpomen 13230	The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.)
|- ( om X. om ) ~~ om
Theorem	xpct 13231	The cartesian product of two sets dominated by om is dominated by om. (Contributed by Thierry Arnoux, 24-Sep-2017.)
|- ( ( A ~<_ om /\ B ~<_ om ) -> ( A X. B ) ~<_ om )
Theorem	unennn 13232	The union of two disjoint countably infinite sets is countably infinite. (Contributed by Jim Kingdon, 13-May-2022.)
|- ( ( A ~~ NN /\ B ~~ NN /\ ( A i^i B ) = (/) ) -> ( A u. B ) ~~ NN )
Theorem	znnen 13233	The set of integers and the set of positive integers are equinumerous. Corollary 8.1.23 of [AczelRathjen], p. 75. (Contributed by NM, 31-Jul-2004.)
|- ZZ ~~ NN
Theorem	ennnfonelemdc 13234*	Lemma for ennnfone 13260. A direct consequence of fidcenumlemrk 7237. (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 ) )
Theorem	ennnfonelemk 13235*	Lemma for ennnfone 13260. (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 )
Theorem	ennnfonelemj0 13236*	Lemma for ennnfone 13260. 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 } )
Theorem	ennnfonelemjn 13237*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemg 13238*	Lemma for ennnfone 13260. 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 } )
Theorem	ennnfonelemh 13239*	Lemma for ennnfone 13260. (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 ) )
Theorem	ennnfonelem0 13240*	Lemma for ennnfone 13260. 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 ) = (/) )
Theorem	ennnfonelemp1 13241*	Lemma for ennnfone 13260. 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 ) ) >. } ) ) )
Theorem	ennnfonelem1 13242*	Lemma for ennnfone 13260. 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 ` (/) ) >. } )
Theorem	ennnfonelemom 13243*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemhdmp1 13244*	Lemma for ennnfone 13260. 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 ) )
Theorem	ennnfonelemss 13245*	Lemma for ennnfone 13260. 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 ) ) )
Theorem	ennnfoneleminc 13246*	Lemma for ennnfone 13260. 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 ) )
Theorem	ennnfonelemkh 13247*	Lemma for ennnfone 13260. 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 ) )
Theorem	ennnfonelemhf1o 13248*	Lemma for ennnfone 13260. 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 ) ) )
Theorem	ennnfonelemex 13249*	Lemma for ennnfone 13260. 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 ) )
Theorem	ennnfonelemhom 13250*	Lemma for ennnfone 13260. 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 ) )
Theorem	ennnfonelemrnh 13251*	Lemma for ennnfone 13260. A consequence of ennnfonelemss 13245. (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 ) )
Theorem	ennnfonelemfun 13252*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemf1 13253*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemrn 13254*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemdm 13255*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemen 13256*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemnn0 13257*	Lemma for ennnfone 13260. A version of ennnfonelemen 13256 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 )
Theorem	ennnfonelemr 13258*	Lemma for ennnfone 13260. 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 )
Theorem	ennnfonelemim 13259*	Lemma for ennnfone 13260. 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 ) ) ) )
Theorem	ennnfone 13260*	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 7413), 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 ) ) ) )
Theorem	exmidunben 13261*	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 )
Theorem	ctinfomlemom 13262*	Lemma for ctinfom 13263. 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 ) ) )
Theorem	ctinfom 13263*	A condition for a set being countably infinite. Restates ennnfone 13260 in terms of om and function image. Like ennnfone 13260 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 ) ) ) )
Theorem	inffinp1 13264*	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 )
Theorem	ctinf 13265*	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 ) )
Theorem	qnnen 13266	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
Theorem	enctlem 13267*	Lemma for enct 13268. 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 ) ) )
Theorem	enct 13268*	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 ) ) )
Theorem	ctiunctlemu1st 13269*	Lemma for ctiunct 13275. (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 )
Theorem	ctiunctlemu2nd 13270*	Lemma for ctiunct 13275. (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 )
Theorem	ctiunctlemuom 13271	Lemma for ctiunct 13275. (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 )
Theorem	ctiunctlemudc 13272*	Lemma for ctiunct 13275. (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 )
Theorem	ctiunctlemf 13273*	Lemma for ctiunct 13275. (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 )
Theorem	ctiunctlemfo 13274*	Lemma for ctiunct 13275. (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 )
Theorem	ctiunct 13275*	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 13279 (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 13277, 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 13230) 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 7415 and ctssdc 7417. (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 ) )
Theorem	ctiunctal 13276*	Variation of ctiunct 13275 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 ) )
Theorem	unct 13277*	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 ) )
Theorem	omctfn 13278*	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 ) ) )
Theorem	omiunct 13279*	The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 13275 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 ) )
Theorem	ssomct 13280*	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 ) )
Theorem	ssnnctlemct 13281*	Lemma for ssnnct 13282. 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 ) )
Theorem	ssnnct 13282*	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 ) )
Theorem	nninfdclemcl 13283*	Lemma for nninfdc 13288. (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 )
Theorem	nninfdclemf 13284*	Lemma for nninfdc 13288. 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 )
Theorem	nninfdclemp1 13285*	Lemma for nninfdc 13288. 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 ) ) )
Theorem	nninfdclemlt 13286*	Lemma for nninfdc 13288. The function from nninfdclemf 13284 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 ) )
Theorem	nninfdclemf1 13287*	Lemma for nninfdc 13288. The function from nninfdclemf 13284 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 )
Theorem	nninfdc 13288*	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 )
Theorem	unbendc 13289*	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 )
Theorem	prminf 13290	There are an infinite number of primes. Theorem 1.7 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 28-Nov-2012.)
|- Prime ~~ NN
Theorem	infpn2 13291*	There exist infinitely many prime numbers: the set of all primes S is unbounded by infpn 13084, so by unbendc 13289 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)
|- S = { n e. NN | ( 1 < n /\ A. m e. NN ( ( n / m ) e. NN -> ( m = 1 \/ m = n ) ) ) }   =>    |- S ~~ NN
PART 6  BASIC STRUCTURES
6.1  Extensible structures
6.1.1  Basic definitions An "extensible structure" (or "structure" in short, at least in this section) is used to define a specific group, ring, poset, and so on. An extensible structure can contain many components. For example, a group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This extensible structure approach allows theorems from more general structures (such as groups) to be reused for more specialized structures (such as rings) without having to reprove anything. Structures are common in mathematics, but in informal (natural language) proofs the details are assumed in ways that we must make explicit. An extensible structure is implemented as a function (a set of ordered pairs) on a finite (and not necessarily sequential) subset of NN. The function's argument is the index of a structure component (such as 1 for the base set of a group), and its value is the component (such as the base set). By convention, we normally avoid direct reference to the hard-coded numeric index and instead use structure component extractors such as ndxid 13320 and strslfv 13341. Using extractors makes it easier to change numeric indices and also makes the components' purpose clearer. See the comment of basendx 13351 for more details on numeric indices versus the structure component extractors. There are many other possible ways to handle structures. We chose this extensible structure approach because this approach (1) results in simpler notation than other approaches we are aware of, and (2) is easier to do proofs with. We cannot use an approach that uses "hidden" arguments; Metamath does not support hidden arguments, and in any case we want nothing hidden. It would be possible to use a categorical approach (e.g., something vaguely similar to Lean's mathlib). However, instances (the chain of proofs that an X is a Y via a bunch of forgetful functors) can cause serious performance problems for automated tooling, and the resulting proofs would be painful to look at directly (in the case of Lean, they are long past the level where people would find it acceptable to look at them directly). Metamath is working under much stricter conditions than this, and it has still managed to achieve about the same level of flexibility through this "extensible structure" approach. To create a substructure of a given extensible structure, you can simply use the multifunction restriction operator for extensible structures ↾s as defined in df-iress 13304. This 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. Extensible structures only work well when they represent concrete categories, where there is a "base set", morphisms are functions, and subobjects are subsets with induced operations. In short, they primarily work well for "sets with (some) extra structure". Extensible structures may not suffice for more complicated situations. For example, in manifolds, ↾s would not work. That said, extensible structures are sufficient for many of the structures that set.mm currently considers, and offer a good compromise for a goal-oriented formalization.
Syntax	cstr 13292	Extend class notation with the class of structures with components numbered below A.
class Struct
Syntax	cnx 13293	Extend class notation with the structure component index extractor.
class ndx
Syntax	csts 13294	Set components of a structure.
class sSet
Syntax	cslot 13295	Extend class notation with the slot function.
class Slot A
Syntax	cbs 13296	Extend class notation with the class of all base set extractors.
class Base
Syntax	cress 13297	Extend class notation with the extensible structure builder restriction operator.
class ↾s
Definition	df-struct 13298*	Define a structure with components in M ... N. This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. As mentioned in the section header, an "extensible structure should be implemented as a function (a set of ordered pairs)". The current definition, however, is less restrictive: it allows for classes which contain the empty set (/) to be extensible structures. Because of 0nelfun 5375, such classes cannot be functions. Without the empty set, however, a structure must be a function, see structn0fun 13309: F Struct X -> Fun ( F \ { (/) } ). Allowing an extensible structure to contain the empty set ensures that expressions like { <. A , B >. , <. C , D >. } are structures without asserting or implying that A, B, C and D are sets (if A or B is a proper class, then <. A , B >. = (/), see opprc 3909). (Contributed by Mario Carneiro, 29-Aug-2015.)
|- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
Definition	df-ndx 13299	Define the structure component index extractor. See Theorem ndxarg 13319 to understand its purpose. The restriction to NN ensures that ndx is a set. The restriction to some set is necessary since _I is a proper class. In principle, we could have chosen CC or (if we revise all structure component definitions such as df-base 13302) another set such as the set of finite ordinals om (df-iom 4718). (Contributed by NM, 4-Sep-2011.)
|- ndx = ( _I |` NN )
Definition	df-slot 13300*	Define the slot extractor for extensible structures. The class Slot A is a function whose argument can be any set, although it is meaningful only if that set is a member of an extensible structure (such as a partially ordered set or a group). Note that Slot A is implemented as "evaluation at A". That is, (Slot A ` S ) is defined to be ( S ` A ), where A will typically be a small nonzero natural number. Each extensible structure S is a function defined on specific natural number "slots", and this function extracts the value at a particular slot. The special "structure" ndx, defined as the identity function restricted to NN, can be used to extract the number A from a slot, since (Slot A ` ndx ) = A (see ndxarg 13319). This is typically used to refer to the number of a slot when defining structures without having to expose the detail of what that number is (for instance, we use the expression ( Base ` ndx ) in theorems and proofs instead of its value 1). The class Slot cannot be defined as ( x e. _V |-> ( f e. _V |-> ( f ` x ) ) ) because each Slot A is a function on the proper class _V so is itself a proper class, and the values of functions are sets (fvex 5695). It is necessary to allow proper classes as values of Slot A since for instance the class of all (base sets of) groups is proper. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- Slot A = ( x e. _V |-> ( x ` A ) )