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Theorem ballotlemfval 26802
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 454 . . . . . . . 8  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  b  =  C )
32ineq2d 3549 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  i^i  b
)  =  ( ( 1 ... i )  i^i  C ) )
43fveq2d 5692 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  i^i  b )
)  =  ( # `  ( ( 1 ... i )  i^i  C
) ) )
52difeq2d 3471 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  \  b
)  =  ( ( 1 ... i ) 
\  C ) )
65fveq2d 5692 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  \  b )
)  =  ( # `  ( ( 1 ... i )  \  C
) ) )
74, 6oveq12d 6108 . . . . 5  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) )  =  ( ( # `  (
( 1 ... i
)  i^i  C )
)  -  ( # `  ( ( 1 ... i )  \  C
) ) ) )
87mpteq2dva 4375 . . . 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 3543 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  i^i  b )  =  ( ( 1 ... i )  i^i  c ) )
1110fveq2d 5692 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  i^i  b ) )  =  ( # `  (
( 1 ... i
)  i^i  c )
) )
12 difeq2 3465 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  \  b )  =  ( ( 1 ... i )  \ 
c ) )
1312fveq2d 5692 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  \ 
b ) )  =  ( # `  (
( 1 ... i
)  \  c )
) )
1411, 13oveq12d 6108 . . . . . . 7  |-  ( b  =  c  ->  (
( # `  ( ( 1 ... i )  i^i  b ) )  -  ( # `  (
( 1 ... i
)  \  b )
) )  =  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) )
1514mpteq2dv 4376 . . . . . 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 4380 . . . . 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 2464 . . . 4  |-  F  =  ( b  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) ) )
18 zex 10651 . . . . 5  |-  ZZ  e.  _V
1918mptex 5945 . . . 4  |-  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) )  e.  _V
208, 17, 19fvmpt 5771 . . 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 6098 . . . . . 6  |-  ( i  =  J  ->  (
1 ... i )  =  ( 1 ... J
) )
2322ineq1d 3548 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  i^i  C )  =  ( ( 1 ... J )  i^i 
C ) )
2423fveq2d 5692 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  i^i 
C ) )  =  ( # `  (
( 1 ... J
)  i^i  C )
) )
2522difeq1d 3470 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  \  C )  =  ( ( 1 ... J )  \  C ) )
2625fveq2d 5692 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  \  C ) )  =  ( # `  (
( 1 ... J
)  \  C )
) )
2724, 26oveq12d 6108 . . 3  |-  ( i  =  J  ->  (
( # `  ( ( 1 ... i )  i^i  C ) )  -  ( # `  (
( 1 ... i
)  \  C )
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
2827adantl 463 . 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 6115 . . 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 5776 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   {crab 2717   _Vcvv 2970    \ cdif 3322    i^i cin 3324   ~Pcpw 3857    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   1c1 9279    + caddc 9281    - cmin 9591    / cdiv 9989   NNcn 10318   ZZcz 10642   ...cfz 11433   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-cnex 9334  ax-resscn 9335
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-neg 9594  df-z 10643
This theorem is referenced by:  ballotlemfelz  26803  ballotlemfp1  26804  ballotlemfmpn  26807  ballotlemfval0  26808  ballotlemfg  26838  ballotlemfrc  26839
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