Theorem ballotleme

Description: Elements of E . (Contributed by Thierry Arnoux, 14-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 ) }
Assertion	ballotleme	|- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )

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		fveq2	|- ( d = C -> ( F ` d ) = ( F ` C ) )
8	7	fveq1d	|- ( d = C -> ( ( F ` d ) ` i ) = ( ( F ` C ) ` i ) )
9	8	breq2d	|- ( d = C -> ( 0 < ( ( F ` d ) ` i ) <-> 0 < ( ( F ` C ) ` i ) ) )
10	9	ralbidv	|- ( d = C -> ( A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) <-> A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )
11		fveq2	|- ( c = d -> ( F ` c ) = ( F ` d ) )
12	11	fveq1d	|- ( c = d -> ( ( F ` c ) ` i ) = ( ( F ` d ) ` i ) )
13	12	breq2d	|- ( c = d -> ( 0 < ( ( F ` c ) ` i ) <-> 0 < ( ( F ` d ) ` i ) ) )
14	13	ralbidv	|- ( c = d -> ( A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) <-> A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) ) )
15	14	cbvrabv	|- { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) } = { d e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) }
16	6 15	eqtri	|- E = { d e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) }
17	10 16	elrab2	|- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )