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Theorem ballotleme 24746
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 5720 . . . . 5  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5722 . . . 4  |-  ( d  =  C  ->  (
( F `  d
) `  i )  =  ( ( F `
 C ) `  i ) )
32breq2d 4216 . . 3  |-  ( d  =  C  ->  (
0  <  ( ( F `  d ) `  i )  <->  0  <  ( ( F `  C
) `  i )
) )
43ralbidv 2717 . 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 5720 . . . . . . 7  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
76fveq1d 5722 . . . . . 6  |-  ( c  =  d  ->  (
( F `  c
) `  i )  =  ( ( F `
 d ) `  i ) )
87breq2d 4216 . . . . 5  |-  ( c  =  d  ->  (
0  <  ( ( F `  c ) `  i )  <->  0  <  ( ( F `  d
) `  i )
) )
98ralbidv 2717 . . . 4  |-  ( c  =  d  ->  ( A. i  e.  (
1 ... ( M  +  N ) ) 0  <  ( ( F `
 c ) `  i )  <->  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  d
) `  i )
) )
109cbvrabv 2947 . . 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 ) }
115, 10eqtri 2455 . 2  |-  E  =  { d  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  d ) `
 i ) }
124, 11elrab2 3086 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309    i^i cin 3311   ~Pcpw 3791   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    - cmin 9283    / cdiv 9669   NNcn 9992   ZZcz 10274   ...cfz 11035   #chash 11610
This theorem is referenced by:  ballotlemodife  24747  ballotlem4  24748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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