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Theorem ballotleme 23055
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.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5527 . . . 4  |-  ( d  =  C  ->  (
( F `  d
) `  i )  =  ( ( F `
 C ) `  i ) )
32breq2d 4035 . . 3  |-  ( d  =  C  ->  (
0  <  ( ( F `  d ) `  i )  <->  0  <  ( ( F `  C
) `  i )
) )
43ralbidv 2563 . 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 nfcv 2419 . . . 4  |-  F/_ c O
7 nfcv 2419 . . . 4  |-  F/_ d O
8 nfv 1605 . . . 4  |-  F/ d A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i )
9 nfv 1605 . . . 4  |-  F/ c A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  d ) `
 i )
10 fveq2 5525 . . . . . . 7  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
1110fveq1d 5527 . . . . . 6  |-  ( c  =  d  ->  (
( F `  c
) `  i )  =  ( ( F `
 d ) `  i ) )
1211breq2d 4035 . . . . 5  |-  ( c  =  d  ->  (
0  <  ( ( F `  c ) `  i )  <->  0  <  ( ( F `  d
) `  i )
) )
1312ralbidv 2563 . . . 4  |-  ( c  =  d  ->  ( A. i  e.  (
1 ... ( M  +  N ) ) 0  <  ( ( F `
 c ) `  i )  <->  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  d
) `  i )
) )
146, 7, 8, 9, 13cbvrab 2786 . . 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 ) }
155, 14eqtri 2303 . 2  |-  E  =  { d  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  d ) `
 i ) }
164, 15elrab2 2925 1  |-  ( C  e.  E  <->  ( C  e.  O  /\  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemodife  23056  ballotlem4  23057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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