Theorem ballotfileme 13214

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 5690	. . . . 5 |- ( d = C -> ( F ` d ) = ( F ` C ) )
2	1	fveq1d 5692	. . . 4 |- ( d = C -> ( ( F ` d ) ` i ) = ( ( F ` C ) ` i ) )
3	2	breq2d 4137	. . 3 |- ( d = C -> ( 0 < ( ( F ` d ) ` i ) <-> 0 < ( ( F ` C ) ` i ) ) )
4	3	ralbidv 2550	. 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 5690	. . . . . . 7 |- ( c = d -> ( F ` c ) = ( F ` d ) )
7	6	fveq1d 5692	. . . . . 6 |- ( c = d -> ( ( F ` c ) ` i ) = ( ( F ` d ) ` i ) )
8	7	breq2d 4137	. . . . 5 |- ( c = d -> ( 0 < ( ( F ` c ) ` i ) <-> 0 < ( ( F ` d ) ` i ) ) )
9	8	ralbidv 2550	. . . 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 2820	. . 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 2259	. 2 |- E = { d e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` d ) ` i ) }
12	4, 11	elrab2 2985	1 |- ( C e. E <-> ( C e. O /\ A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` C ) ` i ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   ~Pcpw 3685   class class class wbr 4125   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   Fincfn 7012   0cc0 8169   1c1 8170   + caddc 8172   < clt 8350   - cmin 8487   / cdiv 8992   NNcn 9283   ZZcz 9623   ...cfz 10390  ♯chash 11192

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  ballotfilemodife  13218  ballotfilem4  13219