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Theorem ballotlemrval 23092
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 5541 . . 3  |-  ( d  =  C  ->  ( S `  d )  =  ( S `  C ) )
2 id 19 . . 3  |-  ( d  =  C  ->  d  =  C )
31, 2imaeq12d 5029 . 2  |-  ( d  =  C  ->  (
( S `  d
) " d )  =  ( ( S `
 C ) " C ) )
4 ballotth.r . . 3  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
5 nfcv 2432 . . . 4  |-  F/_ d
( ( S `  c ) " c
)
6 nfcv 2432 . . . 4  |-  F/_ c
( ( S `  d ) " d
)
7 fveq2 5541 . . . . 5  |-  ( c  =  d  ->  ( S `  c )  =  ( S `  d ) )
8 id 19 . . . . 5  |-  ( c  =  d  ->  c  =  d )
97, 8imaeq12d 5029 . . . 4  |-  ( c  =  d  ->  (
( S `  c
) " c )  =  ( ( S `
 d ) "
d ) )
105, 6, 9cbvmpt 4126 . . 3  |-  ( c  e.  ( O  \  E )  |->  ( ( S `  c )
" c ) )  =  ( d  e.  ( O  \  E
)  |->  ( ( S `
 d ) "
d ) )
114, 10eqtri 2316 . 2  |-  R  =  ( d  e.  ( O  \  E ) 
|->  ( ( S `  d ) " d
) )
12 fvex 5555 . . 3  |-  ( S `
 C )  e. 
_V
13 imaexg 5042 . . 3  |-  ( ( S `  C )  e.  _V  ->  (
( S `  C
) " C )  e.  _V )
1412, 13ax-mp 8 . 2  |-  ( ( S `  C )
" C )  e. 
_V
153, 11, 14fvmpt 5618 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164   ifcif 3578   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotlemscr  23093  ballotlemrv  23094  ballotlemro  23097  ballotlemrinv0  23107
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-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  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-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-fv 5279
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