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Theorem ballotlemi 23059
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 5525 . . . . . 6  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5527 . . . . 5  |-  ( d  =  C  ->  (
( F `  d
) `  k )  =  ( ( F `
 C ) `  k ) )
32eqeq1d 2291 . . . 4  |-  ( d  =  C  ->  (
( ( F `  d ) `  k
)  =  0  <->  (
( F `  C
) `  k )  =  0 ) )
43rabbidv 2780 . . 3  |-  ( d  =  C  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
54supeq1d 7199 . 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 nfcv 2419 . . . 4  |-  F/_ d sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  c ) `
 k )  =  0 } ,  RR ,  `'  <  )
8 nfcv 2419 . . . 4  |-  F/_ c sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  d ) `
 k )  =  0 } ,  RR ,  `'  <  )
9 fveq2 5525 . . . . . . . 8  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
109fveq1d 5527 . . . . . . 7  |-  ( c  =  d  ->  (
( F `  c
) `  k )  =  ( ( F `
 d ) `  k ) )
1110eqeq1d 2291 . . . . . 6  |-  ( c  =  d  ->  (
( ( F `  c ) `  k
)  =  0  <->  (
( F `  d
) `  k )  =  0 ) )
1211rabbidv 2780 . . . . 5  |-  ( c  =  d  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } )
1312supeq1d 7199 . . . 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 ,  `'  <  ) )
147, 8, 13cbvmpt 4110 . . 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 ,  `'  <  ) )
156, 14eqtri 2303 . 2  |-  I  =  ( d  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 } ,  RR ,  `'  <  ) )
16 ltso 8903 . . . 4  |-  <  Or  RR
17 cnvso 5214 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1816, 17mpbi 199 . . 3  |-  `'  <  Or  RR
1918supex 7214 . 2  |-  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  _V
205, 15, 19fvmpt 5602 1  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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    Or wor 4313   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   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:  ballotlemiex  23060  ballotlemimin  23064  ballotlemfrcn0  23088  ballotlemirc  23090
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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