Theorem ballotfileme 13180

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

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfilem.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` 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
ballotfileme	|- ( 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 ballotfileme

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5675	. . . . 5 |- ( d = C -> ( F ` d ) = ( F ` C ) )
2	1	fveq1d 5677	. . . 4 |- ( d = C -> ( ( F ` d ) ` i ) = ( ( F ` C ) ` i ) )
3	2	breq2d 4126	. . 3 |- ( d = C -> ( 0 < ( ( F ` d ) ` i ) <-> 0 < ( ( F ` C ) ` i ) ) )
4	3	ralbidv 2544	. 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 5675	. . . . . . 7 |- ( c = d -> ( F ` c ) = ( F ` d ) )
7	6	fveq1d 5677	. . . . . 6 |- ( c = d -> ( ( F ` c ) ` i ) = ( ( F ` d ) ` i ) )
8	7	breq2d 4126	. . . . 5 |- ( c = d -> ( 0 < ( ( F ` c ) ` i ) <-> 0 < ( ( F ` d ) ` i ) ) )
9	8	ralbidv 2544	. . . 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 2814	. . 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 2255	. 2 |- E = { d e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) }
12	4, 11	elrab2 2979	1 |- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   \ cdif 3211   i^i cin 3213   ~Pcpw 3674   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   / cdiv 8963   NNcn 9254   ZZcz 9594   ...cfz 10361  ♯chash 11163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365

This theorem is referenced by:  ballotfilemodife  13184  ballotfilem4  13185