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Theorem ballotlemfval 24752
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 445 . . . . . . . 8  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  b  =  C )
32ineq2d 3544 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  i^i  b
)  =  ( ( 1 ... i )  i^i  C ) )
43fveq2d 5735 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  i^i  b )
)  =  ( # `  ( ( 1 ... i )  i^i  C
) ) )
52difeq2d 3467 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  \  b
)  =  ( ( 1 ... i ) 
\  C ) )
65fveq2d 5735 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  \  b )
)  =  ( # `  ( ( 1 ... i )  \  C
) ) )
74, 6oveq12d 6102 . . . . 5  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) )  =  ( ( # `  (
( 1 ... i
)  i^i  C )
)  -  ( # `  ( ( 1 ... i )  \  C
) ) ) )
87mpteq2dva 4298 . . . 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 3538 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  i^i  b )  =  ( ( 1 ... i )  i^i  c ) )
1110fveq2d 5735 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  i^i  b ) )  =  ( # `  (
( 1 ... i
)  i^i  c )
) )
12 difeq2 3461 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  \  b )  =  ( ( 1 ... i )  \ 
c ) )
1312fveq2d 5735 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  \ 
b ) )  =  ( # `  (
( 1 ... i
)  \  c )
) )
1411, 13oveq12d 6102 . . . . . . 7  |-  ( b  =  c  ->  (
( # `  ( ( 1 ... i )  i^i  b ) )  -  ( # `  (
( 1 ... i
)  \  b )
) )  =  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) )
1514mpteq2dv 4299 . . . . . 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 4303 . . . . 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 2461 . . . 4  |-  F  =  ( b  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) ) )
18 zex 10296 . . . . 5  |-  ZZ  e.  _V
1918mptex 5969 . . . 4  |-  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) )  e.  _V
208, 17, 19fvmpt 5809 . . 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 6092 . . . . . 6  |-  ( i  =  J  ->  (
1 ... i )  =  ( 1 ... J
) )
2322ineq1d 3543 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  i^i  C )  =  ( ( 1 ... J )  i^i 
C ) )
2423fveq2d 5735 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  i^i 
C ) )  =  ( # `  (
( 1 ... J
)  i^i  C )
) )
2522difeq1d 3466 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  \  C )  =  ( ( 1 ... J )  \  C ) )
2625fveq2d 5735 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  \  C ) )  =  ( # `  (
( 1 ... J
)  \  C )
) )
2724, 26oveq12d 6102 . . 3  |-  ( i  =  J  ->  (
( # `  ( ( 1 ... i )  i^i  C ) )  -  ( # `  (
( 1 ... i
)  \  C )
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
2827adantl 454 . 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 6109 . . 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 5813 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321   ~Pcpw 3801    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   1c1 8996    + caddc 8998    - cmin 9296    / cdiv 9682   NNcn 10005   ZZcz 10287   ...cfz 11048   #chash 11623
This theorem is referenced by:  ballotlemfelz  24753  ballotlemfp1  24754  ballotlemfmpn  24757  ballotlemfval0  24758  ballotlemfg  24788  ballotlemfrc  24789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-cnex 9051  ax-resscn 9052
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-neg 9299  df-z 10288
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