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Theorem ballotlemi 28826
Description: Value of  I for a given counting  C. (Contributed by Thierry Arnoux, 1-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 ) }
ballotth.mgtn  |-  N  < 
M
ballotth.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
Assertion
Ref Expression
ballotlemi  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O    k, M    k, N    k, O    i, c, F, k    C, i, k    i, E, k    C, k    k, I   
k, c, E
Allowed substitution hints:    C( x, c)    P( x, i, k, c)    E( x)    F( x)    I( x, i, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemi
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 fveq2 5803 . . . . . 6  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5805 . . . . 5  |-  ( d  =  C  ->  (
( F `  d
) `  k )  =  ( ( F `
 C ) `  k ) )
32eqeq1d 2402 . . . 4  |-  ( d  =  C  ->  (
( ( F `  d ) `  k
)  =  0  <->  (
( F `  C
) `  k )  =  0 ) )
43rabbidv 3048 . . 3  |-  ( d  =  C  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
54supeq1d 7857 . 2  |-  ( d  =  C  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } ,  RR ,  `'  <  )  =  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  ) )
6 ballotth.i . . 3  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
7 fveq2 5803 . . . . . . . 8  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
87fveq1d 5805 . . . . . . 7  |-  ( c  =  d  ->  (
( F `  c
) `  k )  =  ( ( F `
 d ) `  k ) )
98eqeq1d 2402 . . . . . 6  |-  ( c  =  d  ->  (
( ( F `  c ) `  k
)  =  0  <->  (
( F `  d
) `  k )  =  0 ) )
109rabbidv 3048 . . . . 5  |-  ( c  =  d  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } )
1110supeq1d 7857 . . . 4  |-  ( c  =  d  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 c ) `  k )  =  0 } ,  RR ,  `'  <  )  =  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } ,  RR ,  `'  <  ) )
1211cbvmptv 4484 . . 3  |-  ( c  e.  ( O  \  E )  |->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 c ) `  k )  =  0 } ,  RR ,  `'  <  ) )  =  ( d  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 } ,  RR ,  `'  <  ) )
136, 12eqtri 2429 . 2  |-  I  =  ( d  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 } ,  RR ,  `'  <  ) )
14 gtso 9615 . . 3  |-  `'  <  Or  RR
1514supex 7874 . 2  |-  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  _V
165, 13, 15fvmpt 5886 1  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   A.wral 2751   {crab 2755    \ cdif 3408    i^i cin 3410   ~Pcpw 3952   class class class wbr 4392    |-> cmpt 4450   `'ccnv 4939   ` cfv 5523  (class class class)co 6232   supcsup 7852   RRcr 9439   0cc0 9440   1c1 9441    + caddc 9443    < clt 9576    - cmin 9759    / cdiv 10165   NNcn 10494   ZZcz 10823   ...cfz 11641   #chash 12357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-resscn 9497  ax-pre-lttri 9514  ax-pre-lttrn 9515
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-ltxr 9581
This theorem is referenced by:  ballotlemiex  28827  ballotlemimin  28831  ballotlemfrcn0  28855  ballotlemirc  28857
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