Theorem ballotlemi

Description: Value of I for a given counting C . (Contributed by Thierry Arnoux, 1-Dec-2016) (Revised by AV, 6-Oct-2020)

	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 , < ) )
Assertion	ballotlemi	|- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )

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		fveq2	|- ( d = C -> ( F ` d ) = ( F ` C ) )
10	9	fveq1d	|- ( d = C -> ( ( F ` d ) ` k ) = ( ( F ` C ) ` k ) )
11	10	eqeq1d	|- ( d = C -> ( ( ( F ` d ) ` k ) = 0 <-> ( ( F ` C ) ` k ) = 0 ) )
12	11	rabbidv	|- ( d = C -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } )
13	12	infeq1d	|- ( d = C -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )
14		fveq2	|- ( c = d -> ( F ` c ) = ( F ` d ) )
15	14	fveq1d	|- ( c = d -> ( ( F ` c ) ` k ) = ( ( F ` d ) ` k ) )
16	15	eqeq1d	|- ( c = d -> ( ( ( F ` c ) ` k ) = 0 <-> ( ( F ` d ) ` k ) = 0 ) )
17	16	rabbidv	|- ( c = d -> { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } = { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } )
18	17	infeq1d	|- ( c = d -> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
19	18	cbvmptv	|- ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) ) = ( d e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
20	8 19	eqtri	|- I = ( d e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` d ) ` k ) = 0 } , RR , < ) )
21		ltso	|- < Or RR
22	21	infex	|- inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) e. _V
23	13 20 22	fvmpt	|- ( C e. ( O \ E ) -> ( I ` C ) = inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` C ) ` k ) = 0 } , RR , < ) )