Theorem ballotlemrval

Description: Value of R . (Contributed by Thierry Arnoux, 14-Apr-2017)

	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	ballotlemrval	|- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " 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		fveq2	|- ( d = C -> ( S ` d ) = ( S ` C ) )
12		id	|- ( d = C -> d = C )
13	11 12	imaeq12d	|- ( d = C -> ( ( S ` d ) " d ) = ( ( S ` C ) " C ) )
14		fveq2	|- ( c = d -> ( S ` c ) = ( S ` d ) )
15		id	|- ( c = d -> c = d )
16	14 15	imaeq12d	|- ( c = d -> ( ( S ` c ) " c ) = ( ( S ` d ) " d ) )
17	16	cbvmptv	|- ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) ) = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) )
18	10 17	eqtri	|- R = ( d e. ( O \ E ) |-> ( ( S ` d ) " d ) )
19		fvex	|- ( S ` C ) e. _V
20		imaexg	|- ( ( S ` C ) e. _V -> ( ( S ` C ) " C ) e. _V )
21	19 20	ax-mp	|- ( ( S ` C ) " C ) e. _V
22	13 18 21	fvmpt	|- ( C e. ( O \ E ) -> ( R ` C ) = ( ( S ` C ) " C ) )