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Theorem ballotlemi 24758
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 5728 . . . . . 6  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5730 . . . . 5  |-  ( d  =  C  ->  (
( F `  d
) `  k )  =  ( ( F `
 C ) `  k ) )
32eqeq1d 2444 . . . 4  |-  ( d  =  C  ->  (
( ( F `  d ) `  k
)  =  0  <->  (
( F `  C
) `  k )  =  0 ) )
43rabbidv 2948 . . 3  |-  ( d  =  C  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
54supeq1d 7451 . 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 5728 . . . . . . . 8  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
87fveq1d 5730 . . . . . . 7  |-  ( c  =  d  ->  (
( F `  c
) `  k )  =  ( ( F `
 d ) `  k ) )
98eqeq1d 2444 . . . . . 6  |-  ( c  =  d  ->  (
( ( F `  c ) `  k
)  =  0  <->  (
( F `  d
) `  k )  =  0 ) )
109rabbidv 2948 . . . . 5  |-  ( c  =  d  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } )
1110supeq1d 7451 . . . 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 4300 . . 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 2456 . 2  |-  I  =  ( d  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 } ,  RR ,  `'  <  ) )
14 ltso 9156 . . . 4  |-  <  Or  RR
15 cnvso 5411 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1614, 15mpbi 200 . . 3  |-  `'  <  Or  RR
1716supex 7468 . 2  |-  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  _V
185, 13, 17fvmpt 5806 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 1652    e. wcel 1725   A.wral 2705   {crab 2709    \ cdif 3317    i^i cin 3319   ~Pcpw 3799   class class class wbr 4212    e. cmpt 4266    Or wor 4502   `'ccnv 4877   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   ZZcz 10282   ...cfz 11043   #chash 11618
This theorem is referenced by:  ballotlemiex  24759  ballotlemimin  24763  ballotlemfrcn0  24787  ballotlemirc  24789
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-ltxr 9125
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