Theorem ballotlem8

Description: 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)

	Ref	Expression
Hypotheses	ballotth.m	|- M e. NN
	ballotth.n	|- N e. NN
	ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
	ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
	ballotth.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
	ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
	ballotth.mgtn	|- N < M
	ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
	ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
	ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
Assertion	ballotlem8	|- ( # ` { c e. ( O \ E ) | 1 e. c } ) = ( # ` { c e. ( O \ E ) | -. 1 e. c } )

Proof

Step	Hyp	Ref	Expression
1		ballotth.m	|- M e. NN
2		ballotth.n	|- N e. NN
3		ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5		ballotth.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotth.mgtn	|- N < M
8		ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9		ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10		ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
11	1 2 3 4 5 6 7 8 9 10	ballotlem7	|- ( 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 }
12	1 2 3	ballotlemoex	|- O e. _V
13		difexg	|- ( O e. _V -> ( O \ E ) e. _V )
14	12 13	ax-mp	|- ( O \ E ) e. _V
15	14	rabex	|- { c e. ( O \ E ) | 1 e. c } e. _V
16	15	f1oen	|- ( ( 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 } -> { c e. ( O \ E ) | 1 e. c } ~~ { c e. ( O \ E ) | -. 1 e. c } )
17		hasheni	|- ( { c e. ( O \ E ) | 1 e. c } ~~ { c e. ( O \ E ) | -. 1 e. c } -> ( # ` { c e. ( O \ E ) | 1 e. c } ) = ( # ` { c e. ( O \ E ) | -. 1 e. c } ) )
18	11 16 17	mp2b	|- ( # ` { c e. ( O \ E ) | 1 e. c } ) = ( # ` { c e. ( O \ E ) | -. 1 e. c } )