Theorem ballotlemsi

Description: The image by S of the first tie pick is the first pick. (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 ) ) )
Assertion	ballotlemsi	|- ( C e. ( O \ E ) -> ( ( S ` C ) ` ( I ` C ) ) = 1 )

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	1 2 3 4 5 6 7 8	ballotlemiex	|- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
11	10	simpld	|- ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
12	1 2 3 4 5 6 7 8 9	ballotlemsv	|- ( ( C e. ( O \ E ) /\ ( I ` C ) e. ( 1 ... ( M + N ) ) ) -> ( ( S ` C ) ` ( I ` C ) ) = if ( ( I ` C ) <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - ( I ` C ) ) , ( I ` C ) ) )
13	11 12	mpdan	|- ( C e. ( O \ E ) -> ( ( S ` C ) ` ( I ` C ) ) = if ( ( I ` C ) <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - ( I ` C ) ) , ( I ` C ) ) )
14		elfzelz	|- ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( I ` C ) e. ZZ )
15	14	zred	|- ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( I ` C ) e. RR )
16	11 15	syl	|- ( C e. ( O \ E ) -> ( I ` C ) e. RR )
17	16	leidd	|- ( C e. ( O \ E ) -> ( I ` C ) <_ ( I ` C ) )
18	17	iftrued	|- ( C e. ( O \ E ) -> if ( ( I ` C ) <_ ( I ` C ) , ( ( ( I ` C ) + 1 ) - ( I ` C ) ) , ( I ` C ) ) = ( ( ( I ` C ) + 1 ) - ( I ` C ) ) )
19	16	recnd	|- ( C e. ( O \ E ) -> ( I ` C ) e. CC )
20		1cnd	|- ( C e. ( O \ E ) -> 1 e. CC )
21	19 20	pncan2d	|- ( C e. ( O \ E ) -> ( ( ( I ` C ) + 1 ) - ( I ` C ) ) = 1 )
22	13 18 21	3eqtrd	|- ( C e. ( O \ E ) -> ( ( S ` C ) ` ( I ` C ) ) = 1 )