Theorem ballotleme 30558

Description: Elements of E. (Contributed by Thierry Arnoux, 14-Dec-2016.)

Hypotheses

Ref	Expression
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

Ref	Expression
ballotleme	|- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O, c   F, c, i   C, i

Allowed substitution hints:   C( x, c)   P( x, i, c)   E( x, i, c)   F( x)   M( x)   N( x)   O( x)

Proof of Theorem ballotleme

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( d = C -> ( F ` d ) = ( F ` C ) )
2	1	fveq1d 6193	. . . 4 |- ( d = C -> ( ( F ` d ) ` i ) = ( ( F ` C ) ` i ) )
3	2	breq2d 4665	. . 3 |- ( d = C -> ( 0 < ( ( F ` d ) ` i ) <-> 0 < ( ( F ` C ) ` i ) ) )
4	3	ralbidv 2986	. 2 |- ( d = C -> ( A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) <-> A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )
5		ballotth.e	. . 3 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
6		fveq2 6191	. . . . . . 7 |- ( c = d -> ( F ` c ) = ( F ` d ) )
7	6	fveq1d 6193	. . . . . 6 |- ( c = d -> ( ( F ` c ) ` i ) = ( ( F ` d ) ` i ) )
8	7	breq2d 4665	. . . . 5 |- ( c = d -> ( 0 < ( ( F ` c ) ` i ) <-> 0 < ( ( F ` d ) ` i ) ) )
9	8	ralbidv 2986	. . . 4 |- ( c = d -> ( A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) <-> A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) ) )
10	9	cbvrabv 3199	. . 3 |- { 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 ) }
11	5, 10	eqtri 2644	. 2 |- E = { d e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) }
12	4, 11	elrab2 3366	1 |- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  ballotlemodife  30559  ballotlem4  30560