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Theorem ballotlemrval 26830
Description: Value of  R. (Contributed by Thierry Arnoux, 14-Apr-2017.)
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 ,  `'  <  ) )
ballotth.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
ballotth.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
Assertion
Ref Expression
ballotlemrval  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
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, c    E, c    i, I, c    S, k, i, c
Allowed substitution hints:    C( x, c)    P( x, i, k, c)    R( x, i, k, c)    S( x)    E( x)    F( x)    I( x)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemrval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 fveq2 5688 . . 3  |-  ( d  =  C  ->  ( S `  d )  =  ( S `  C ) )
2 id 22 . . 3  |-  ( d  =  C  ->  d  =  C )
31, 2imaeq12d 5167 . 2  |-  ( d  =  C  ->  (
( S `  d
) " d )  =  ( ( S `
 C ) " C ) )
4 ballotth.r . . 3  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
5 fveq2 5688 . . . . 5  |-  ( c  =  d  ->  ( S `  c )  =  ( S `  d ) )
6 id 22 . . . . 5  |-  ( c  =  d  ->  c  =  d )
75, 6imaeq12d 5167 . . . 4  |-  ( c  =  d  ->  (
( S `  c
) " c )  =  ( ( S `
 d ) "
d ) )
87cbvmptv 4380 . . 3  |-  ( c  e.  ( O  \  E )  |->  ( ( S `  c )
" c ) )  =  ( d  e.  ( O  \  E
)  |->  ( ( S `
 d ) "
d ) )
94, 8eqtri 2461 . 2  |-  R  =  ( d  e.  ( O  \  E ) 
|->  ( ( S `  d ) " d
) )
10 fvex 5698 . . 3  |-  ( S `
 C )  e. 
_V
11 imaexg 6514 . . 3  |-  ( ( S `  C )  e.  _V  ->  (
( S `  C
) " C )  e.  _V )
1210, 11ax-mp 5 . 2  |-  ( ( S `  C )
" C )  e. 
_V
133, 9, 12fvmpt 5771 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   A.wral 2713   {crab 2717   _Vcvv 2970    \ cdif 3322    i^i cin 3324   ifcif 3788   ~Pcpw 3857   class class class wbr 4289    e. cmpt 4347   `'ccnv 4835   "cima 4839   ` cfv 5415  (class class class)co 6090   supcsup 7686   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    <_ cle 9415    - cmin 9591    / cdiv 9989   NNcn 10318   ZZcz 10642   ...cfz 11433   #chash 12099
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fv 5423
This theorem is referenced by:  ballotlemscr  26831  ballotlemrv  26832  ballotlemro  26835  ballotlemrinv0  26845
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