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Theorem ballotlemrval 28124
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 5866 . . 3  |-  ( d  =  C  ->  ( S `  d )  =  ( S `  C ) )
2 id 22 . . 3  |-  ( d  =  C  ->  d  =  C )
31, 2imaeq12d 5338 . 2  |-  ( d  =  C  ->  (
( S `  d
) " d )  =  ( ( S `
 C ) " C ) )
4 ballotth.r . . 3  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
5 fveq2 5866 . . . . 5  |-  ( c  =  d  ->  ( S `  c )  =  ( S `  d ) )
6 id 22 . . . . 5  |-  ( c  =  d  ->  c  =  d )
75, 6imaeq12d 5338 . . . 4  |-  ( c  =  d  ->  (
( S `  c
) " c )  =  ( ( S `
 d ) "
d ) )
87cbvmptv 4538 . . 3  |-  ( c  e.  ( O  \  E )  |->  ( ( S `  c )
" c ) )  =  ( d  e.  ( O  \  E
)  |->  ( ( S `
 d ) "
d ) )
94, 8eqtri 2496 . 2  |-  R  =  ( d  e.  ( O  \  E ) 
|->  ( ( S `  d ) " d
) )
10 fvex 5876 . . 3  |-  ( S `
 C )  e. 
_V
11 imaexg 6721 . . 3  |-  ( ( S `  C )  e.  _V  ->  (
( S `  C
) " C )  e.  _V )
1210, 11ax-mp 5 . 2  |-  ( ( S `  C )
" C )  e. 
_V
133, 9, 12fvmpt 5950 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475   ifcif 3939   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   "cima 5002   ` cfv 5588  (class class class)co 6284   supcsup 7900   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   ZZcz 10864   ...cfz 11672   #chash 12373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  ballotlemscr  28125  ballotlemrv  28126  ballotlemro  28129  ballotlemrinv0  28139
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