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Theorem ballotlemi 23075
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 5541 . . . . . 6  |-  ( d  =  C  ->  ( F `  d )  =  ( F `  C ) )
21fveq1d 5543 . . . . 5  |-  ( d  =  C  ->  (
( F `  d
) `  k )  =  ( ( F `
 C ) `  k ) )
32eqeq1d 2304 . . . 4  |-  ( d  =  C  ->  (
( ( F `  d ) `  k
)  =  0  <->  (
( F `  C
) `  k )  =  0 ) )
43rabbidv 2793 . . 3  |-  ( d  =  C  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
54supeq1d 7215 . 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 2432 . . . 4  |-  F/_ d sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  c ) `
 k )  =  0 } ,  RR ,  `'  <  )
8 nfcv 2432 . . . 4  |-  F/_ c sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  d ) `
 k )  =  0 } ,  RR ,  `'  <  )
9 fveq2 5541 . . . . . . . 8  |-  ( c  =  d  ->  ( F `  c )  =  ( F `  d ) )
109fveq1d 5543 . . . . . . 7  |-  ( c  =  d  ->  (
( F `  c
) `  k )  =  ( ( F `
 d ) `  k ) )
1110eqeq1d 2304 . . . . . 6  |-  ( c  =  d  ->  (
( ( F `  c ) `  k
)  =  0  <->  (
( F `  d
) `  k )  =  0 ) )
1211rabbidv 2793 . . . . 5  |-  ( c  =  d  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 }  =  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 d ) `  k )  =  0 } )
1312supeq1d 7215 . . . 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 4126 . . 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 2316 . 2  |-  I  =  ( d  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  d
) `  k )  =  0 } ,  RR ,  `'  <  ) )
16 ltso 8919 . . . 4  |-  <  Or  RR
17 cnvso 5230 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1816, 17mpbi 199 . . 3  |-  `'  <  Or  RR
1918supex 7230 . 2  |-  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  _V
205, 15, 19fvmpt 5618 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093    Or wor 4329   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotlemiex  23076  ballotlemimin  23080  ballotlemfrcn0  23104  ballotlemirc  23106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-ltxr 8888
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