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Theorem ballotlemfval 24700
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 444 . . . . . . . 8  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  b  =  C )
32ineq2d 3502 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  i^i  b
)  =  ( ( 1 ... i )  i^i  C ) )
43fveq2d 5691 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  i^i  b )
)  =  ( # `  ( ( 1 ... i )  i^i  C
) ) )
52difeq2d 3425 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  \  b
)  =  ( ( 1 ... i ) 
\  C ) )
65fveq2d 5691 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  \  b )
)  =  ( # `  ( ( 1 ... i )  \  C
) ) )
74, 6oveq12d 6058 . . . . 5  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) )  =  ( ( # `  (
( 1 ... i
)  i^i  C )
)  -  ( # `  ( ( 1 ... i )  \  C
) ) ) )
87mpteq2dva 4255 . . . 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 3496 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  i^i  b )  =  ( ( 1 ... i )  i^i  c ) )
1110fveq2d 5691 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  i^i  b ) )  =  ( # `  (
( 1 ... i
)  i^i  c )
) )
12 difeq2 3419 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  \  b )  =  ( ( 1 ... i )  \ 
c ) )
1312fveq2d 5691 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  \ 
b ) )  =  ( # `  (
( 1 ... i
)  \  c )
) )
1411, 13oveq12d 6058 . . . . . . 7  |-  ( b  =  c  ->  (
( # `  ( ( 1 ... i )  i^i  b ) )  -  ( # `  (
( 1 ... i
)  \  b )
) )  =  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) )
1514mpteq2dv 4256 . . . . . 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 4260 . . . . 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 2427 . . . 4  |-  F  =  ( b  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) ) )
18 zex 10247 . . . . 5  |-  ZZ  e.  _V
1918mptex 5925 . . . 4  |-  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) )  e.  _V
208, 17, 19fvmpt 5765 . . 3  |-  ( C  e.  O  ->  ( F `  C )  =  ( i  e.  ZZ  |->  ( ( # `  ( ( 1 ... i )  i^i  C
) )  -  ( # `
 ( ( 1 ... i )  \  C ) ) ) ) )
211, 20syl 16 . 2  |-  ( ph  ->  ( F `  C
)  =  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) ) )
22 oveq2 6048 . . . . . 6  |-  ( i  =  J  ->  (
1 ... i )  =  ( 1 ... J
) )
2322ineq1d 3501 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  i^i  C )  =  ( ( 1 ... J )  i^i 
C ) )
2423fveq2d 5691 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  i^i 
C ) )  =  ( # `  (
( 1 ... J
)  i^i  C )
) )
2522difeq1d 3424 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  \  C )  =  ( ( 1 ... J )  \  C ) )
2625fveq2d 5691 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  \  C ) )  =  ( # `  (
( 1 ... J
)  \  C )
) )
2724, 26oveq12d 6058 . . 3  |-  ( i  =  J  ->  (
( # `  ( ( 1 ... i )  i^i  C ) )  -  ( # `  (
( 1 ... i
)  \  C )
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
2827adantl 453 . 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 6065 . . 3  |-  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) )  e.  _V
3130a1i 11 . 2  |-  ( ph  ->  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) )  e. 
_V )
3221, 28, 29, 31fvmptd 5769 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    \ cdif 3277    i^i cin 3279   ~Pcpw 3759    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   1c1 8947    + caddc 8949    - cmin 9247    / cdiv 9633   NNcn 9956   ZZcz 10238   ...cfz 10999   #chash 11573
This theorem is referenced by:  ballotlemfelz  24701  ballotlemfp1  24702  ballotlemfmpn  24705  ballotlemfval0  24706  ballotlemfg  24736  ballotlemfrc  24737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-neg 9250  df-z 10239
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