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Theorem ballotlemfval 23048
Description: The value of F. (Contributed by Thierry Arnoux, 23-Nov-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
) ) ) ) )
ballotlemfval.c  |-  ( ph  ->  C  e.  O )
ballotlemfval.j  |-  ( ph  ->  J  e.  ZZ )
Assertion
Ref Expression
ballotlemfval  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O, c    F, c, i    C, i    i, J    ph, i
Allowed substitution hints:    ph( x, c)    C( x, c)    P( x, i, c)    F( x)    J( x, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemfval
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 ballotlemfval.c . . 3  |-  ( ph  ->  C  e.  O )
2 simpl 443 . . . . . . . 8  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  b  =  C )
32ineq2d 3370 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  i^i  b
)  =  ( ( 1 ... i )  i^i  C ) )
43fveq2d 5529 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  i^i  b )
)  =  ( # `  ( ( 1 ... i )  i^i  C
) ) )
52difeq2d 3294 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  \  b
)  =  ( ( 1 ... i ) 
\  C ) )
65fveq2d 5529 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  \  b )
)  =  ( # `  ( ( 1 ... i )  \  C
) ) )
74, 6oveq12d 5876 . . . . 5  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) )  =  ( ( # `  (
( 1 ... i
)  i^i  C )
)  -  ( # `  ( ( 1 ... i )  \  C
) ) ) )
87mpteq2dva 4106 . . . 4  |-  ( b  =  C  ->  (
i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) )  =  ( i  e.  ZZ  |->  ( ( # `  ( ( 1 ... i )  i^i  C
) )  -  ( # `
 ( ( 1 ... i )  \  C ) ) ) ) )
9 ballotth.f . . . . 5  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
10 ineq2 3364 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  i^i  b )  =  ( ( 1 ... i )  i^i  c ) )
1110fveq2d 5529 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  i^i  b ) )  =  ( # `  (
( 1 ... i
)  i^i  c )
) )
12 difeq2 3288 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  \  b )  =  ( ( 1 ... i )  \ 
c ) )
1312fveq2d 5529 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  \ 
b ) )  =  ( # `  (
( 1 ... i
)  \  c )
) )
1411, 13oveq12d 5876 . . . . . . 7  |-  ( b  =  c  ->  (
( # `  ( ( 1 ... i )  i^i  b ) )  -  ( # `  (
( 1 ... i
)  \  b )
) )  =  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) )
1514mpteq2dv 4107 . . . . . 6  |-  ( b  =  c  ->  (
i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) )  =  ( i  e.  ZZ  |->  ( ( # `  ( ( 1 ... i )  i^i  c
) )  -  ( # `
 ( ( 1 ... i )  \ 
c ) ) ) ) )
1615cbvmptv 4111 . . . . 5  |-  ( b  e.  O  |->  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i  b ) )  -  ( # `  ( ( 1 ... i ) 
\  b ) ) ) ) )  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
179, 16eqtr4i 2306 . . . 4  |-  F  =  ( b  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) ) )
18 zex 10033 . . . . 5  |-  ZZ  e.  _V
1918mptex 5746 . . . 4  |-  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) )  e.  _V
208, 17, 19fvmpt 5602 . . 3  |-  ( C  e.  O  ->  ( F `  C )  =  ( i  e.  ZZ  |->  ( ( # `  ( ( 1 ... i )  i^i  C
) )  -  ( # `
 ( ( 1 ... i )  \  C ) ) ) ) )
211, 20syl 15 . 2  |-  ( ph  ->  ( F `  C
)  =  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) ) )
22 oveq2 5866 . . . . . 6  |-  ( i  =  J  ->  (
1 ... i )  =  ( 1 ... J
) )
2322ineq1d 3369 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  i^i  C )  =  ( ( 1 ... J )  i^i 
C ) )
2423fveq2d 5529 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  i^i 
C ) )  =  ( # `  (
( 1 ... J
)  i^i  C )
) )
2522difeq1d 3293 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  \  C )  =  ( ( 1 ... J )  \  C ) )
2625fveq2d 5529 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  \  C ) )  =  ( # `  (
( 1 ... J
)  \  C )
) )
2724, 26oveq12d 5876 . . 3  |-  ( i  =  J  ->  (
( # `  ( ( 1 ... i )  i^i  C ) )  -  ( # `  (
( 1 ... i
)  \  C )
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
2827adantl 452 . 2  |-  ( (
ph  /\  i  =  J )  ->  (
( # `  ( ( 1 ... i )  i^i  C ) )  -  ( # `  (
( 1 ... i
)  \  C )
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
29 ballotlemfval.j . 2  |-  ( ph  ->  J  e.  ZZ )
30 ovex 5883 . . 3  |-  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) )  e.  _V
3130a1i 10 . 2  |-  ( ph  ->  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) )  e. 
_V )
3221, 28, 29, 31fvmptd 5606 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemfelz  23049  ballotlemfp1  23050  ballotlemfmpn  23053  ballotlemfval0  23054  ballotlemfg  23084  ballotlemfrc  23085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-neg 9040  df-z 10025
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