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Theorem ballotlemi 28067
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 5859 . . . . . 6  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5861 . . . . 5  |-  ( d  =  C  ->  (
( F `  d
) `  k )  =  ( ( F `
 C ) `  k ) )
32eqeq1d 2464 . . . 4  |-  ( d  =  C  ->  (
( ( F `  d ) `  k
)  =  0  <->  (
( F `  C
) `  k )  =  0 ) )
43rabbidv 3100 . . 3  |-  ( d  =  C  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
54supeq1d 7897 . 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 5859 . . . . . . . 8  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
87fveq1d 5861 . . . . . . 7  |-  ( c  =  d  ->  (
( F `  c
) `  k )  =  ( ( F `
 d ) `  k ) )
98eqeq1d 2464 . . . . . 6  |-  ( c  =  d  ->  (
( ( F `  c ) `  k
)  =  0  <->  (
( F `  d
) `  k )  =  0 ) )
109rabbidv 3100 . . . . 5  |-  ( c  =  d  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } )
1110supeq1d 7897 . . . 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 4533 . . 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 2491 . 2  |-  I  =  ( d  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 } ,  RR ,  `'  <  ) )
14 ltso 9656 . . . 4  |-  <  Or  RR
15 cnvso 5539 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1614, 15mpbi 208 . . 3  |-  `'  <  Or  RR
1716supex 7914 . 2  |-  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  _V
185, 13, 17fvmpt 5943 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 1374    e. wcel 1762   A.wral 2809   {crab 2813    \ cdif 3468    i^i cin 3470   ~Pcpw 4005   class class class wbr 4442    |-> cmpt 4500    Or wor 4794   `'ccnv 4993   ` cfv 5581  (class class class)co 6277   supcsup 7891   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    < clt 9619    - cmin 9796    / cdiv 10197   NNcn 10527   ZZcz 10855   ...cfz 11663   #chash 12362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-pre-lttri 9557  ax-pre-lttrn 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-ltxr 9624
This theorem is referenced by:  ballotlemiex  28068  ballotlemimin  28072  ballotlemfrcn0  28096  ballotlemirc  28098
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