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Theorem ballotlemfval 23996
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 3446 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  i^i  b
)  =  ( ( 1 ... i )  i^i  C ) )
43fveq2d 5609 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  i^i  b )
)  =  ( # `  ( ( 1 ... i )  i^i  C
) ) )
52difeq2d 3370 . . . . . . 7  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( 1 ... i )  \  b
)  =  ( ( 1 ... i ) 
\  C ) )
65fveq2d 5609 . . . . . 6  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( # `  (
( 1 ... i
)  \  b )
)  =  ( # `  ( ( 1 ... i )  \  C
) ) )
74, 6oveq12d 5960 . . . . 5  |-  ( ( b  =  C  /\  i  e.  ZZ )  ->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) )  =  ( ( # `  (
( 1 ... i
)  i^i  C )
)  -  ( # `  ( ( 1 ... i )  \  C
) ) ) )
87mpteq2dva 4185 . . . 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 3440 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  i^i  b )  =  ( ( 1 ... i )  i^i  c ) )
1110fveq2d 5609 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  i^i  b ) )  =  ( # `  (
( 1 ... i
)  i^i  c )
) )
12 difeq2 3364 . . . . . . . . 9  |-  ( b  =  c  ->  (
( 1 ... i
)  \  b )  =  ( ( 1 ... i )  \ 
c ) )
1312fveq2d 5609 . . . . . . . 8  |-  ( b  =  c  ->  ( # `
 ( ( 1 ... i )  \ 
b ) )  =  ( # `  (
( 1 ... i
)  \  c )
) )
1411, 13oveq12d 5960 . . . . . . 7  |-  ( b  =  c  ->  (
( # `  ( ( 1 ... i )  i^i  b ) )  -  ( # `  (
( 1 ... i
)  \  b )
) )  =  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) )
1514mpteq2dv 4186 . . . . . 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 4190 . . . . 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 2381 . . . 4  |-  F  =  ( b  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  b )
)  -  ( # `  ( ( 1 ... i )  \  b
) ) ) ) )
18 zex 10122 . . . . 5  |-  ZZ  e.  _V
1918mptex 5829 . . . 4  |-  ( i  e.  ZZ  |->  ( (
# `  ( (
1 ... i )  i^i 
C ) )  -  ( # `  ( ( 1 ... i ) 
\  C ) ) ) )  e.  _V
208, 17, 19fvmpt 5682 . . 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 5950 . . . . . 6  |-  ( i  =  J  ->  (
1 ... i )  =  ( 1 ... J
) )
2322ineq1d 3445 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  i^i  C )  =  ( ( 1 ... J )  i^i 
C ) )
2423fveq2d 5609 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  i^i 
C ) )  =  ( # `  (
( 1 ... J
)  i^i  C )
) )
2522difeq1d 3369 . . . . 5  |-  ( i  =  J  ->  (
( 1 ... i
)  \  C )  =  ( ( 1 ... J )  \  C ) )
2625fveq2d 5609 . . . 4  |-  ( i  =  J  ->  ( # `
 ( ( 1 ... i )  \  C ) )  =  ( # `  (
( 1 ... J
)  \  C )
) )
2724, 26oveq12d 5960 . . 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 5967 . . 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 5686 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 1642    e. wcel 1710   {crab 2623   _Vcvv 2864    \ cdif 3225    i^i cin 3227   ~Pcpw 3701    e. cmpt 4156   ` cfv 5334  (class class class)co 5942   1c1 8825    + caddc 8827    - cmin 9124    / cdiv 9510   NNcn 9833   ZZcz 10113   ...cfz 10871   #chash 11427
This theorem is referenced by:  ballotlemfelz  23997  ballotlemfp1  23998  ballotlemfmpn  24001  ballotlemfval0  24002  ballotlemfg  24032  ballotlemfrc  24033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-cnex 8880  ax-resscn 8881
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-neg 9127  df-z 10114
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