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Theorem ballotlemfrc 26924
Description: Express the value of  ( F `
 ( R `  C ) ) in terms of the newly defined  .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)
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
) ) ) ) )
ballotth.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
ballotth.mgtn  |-  N  < 
M
ballotth.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
ballotth.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
ballotth.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
ballotlemg  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
Assertion
Ref Expression
ballotlemfrc  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( F `
 ( R `  C ) ) `  J )  =  ( C  .^  ( (
( S `  C
) `  J ) ... ( I `  C
) ) ) )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O    k, M    k, N    k, O    i, c, F, k    C, i, k    i, E, k    C, k    k, I, c    E, c    i, I, c    k, J    S, k, i, c    R, i   
v, u, C    u, I, v    u, J, v   
u, R, v    u, S, v    i, J
Allowed substitution hints:    C( x, c)    P( x, v, u, i, k, c)    R( x, k, c)    S( x)    E( x, v, u)    .^ ( x, v, u, i, k, c)    F( x, v, u)    I( x)    J( x, c)    M( x, v, u)    N( x, v, u)    O( x, v, u)

Proof of Theorem ballotlemfrc
StepHypRef Expression
1 ballotth.m . . . . . . . . 9  |-  M  e.  NN
2 ballotth.n . . . . . . . . 9  |-  N  e.  NN
3 ballotth.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotth.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotth.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 ballotth.e . . . . . . . . 9  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
7 ballotth.mgtn . . . . . . . . 9  |-  N  < 
M
8 ballotth.i . . . . . . . . 9  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
9 ballotth.s . . . . . . . . 9  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 26911 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1110simpld 459 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
12 f1of1 5655 . . . . . . 7  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
1311, 12syl 16 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-> ( 1 ... ( M  +  N ) ) )
1413adantr 465 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
151, 2, 3, 4, 5, 6, 7, 8ballotlemiex 26899 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1615simpld 459 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
1716adantr 465 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
18 elfzuz3 11465 . . . . . . . . 9  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  ( M  +  N )  e.  ( ZZ>= `  ( I `  C ) ) )
1917, 18syl 16 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  ( I `  C ) ) )
20 elfzuz3 11465 . . . . . . . . 9  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  (
I `  C )  e.  ( ZZ>= `  J )
)
2120adantl 466 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  (
ZZ>= `  J ) )
22 uztrn 10892 . . . . . . . 8  |-  ( ( ( M  +  N
)  e.  ( ZZ>= `  ( I `  C
) )  /\  (
I `  C )  e.  ( ZZ>= `  J )
)  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
2319, 21, 22syl2anc 661 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
24 fzss2 11513 . . . . . . 7  |-  ( ( M  +  N )  e.  ( ZZ>= `  J
)  ->  ( 1 ... J )  C_  ( 1 ... ( M  +  N )
) )
2523, 24syl 16 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( 1 ... J )  C_  (
1 ... ( M  +  N ) ) )
26 ssinss1 3593 . . . . . 6  |-  ( ( 1 ... J ) 
C_  ( 1 ... ( M  +  N
) )  ->  (
( 1 ... J
)  i^i  ( R `  C ) )  C_  ( 1 ... ( M  +  N )
) )
2725, 26syl 16 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  i^i  ( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
28 f1ores 5670 . . . . 5  |-  ( ( ( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) )  /\  ( ( 1 ... J )  i^i  ( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )  ->  ( ( S `
 C )  |`  ( ( 1 ... J )  i^i  ( R `  C )
) ) : ( ( 1 ... J
)  i^i  ( R `  C ) ) -1-1-onto-> ( ( S `  C )
" ( ( 1 ... J )  i^i  ( R `  C
) ) ) )
2914, 27, 28syl2anc 661 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C )  |`  ( ( 1 ... J )  i^i  ( R `  C )
) ) : ( ( 1 ... J
)  i^i  ( R `  C ) ) -1-1-onto-> ( ( S `  C )
" ( ( 1 ... J )  i^i  ( R `  C
) ) ) )
30 ovex 6131 . . . . . 6  |-  ( 1 ... J )  e. 
_V
3130inex1 4448 . . . . 5  |-  ( ( 1 ... J )  i^i  ( R `  C ) )  e. 
_V
3231f1oen 7345 . . . 4  |-  ( ( ( S `  C
)  |`  ( ( 1 ... J )  i^i  ( R `  C
) ) ) : ( ( 1 ... J )  i^i  ( R `  C )
)
-1-1-onto-> ( ( S `  C ) " (
( 1 ... J
)  i^i  ( R `  C ) ) )  ->  ( ( 1 ... J )  i^i  ( R `  C
) )  ~~  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) ) )
33 hasheni 12134 . . . 4  |-  ( ( ( 1 ... J
)  i^i  ( R `  C ) )  ~~  ( ( S `  C ) " (
( 1 ... J
)  i^i  ( R `  C ) ) )  ->  ( # `  (
( 1 ... J
)  i^i  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) ) ) )
3429, 32, 333syl 20 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( 1 ... J
)  i^i  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) ) ) )
3525ssdifssd 3509 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  \ 
( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
36 f1ores 5670 . . . . 5  |-  ( ( ( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) )  /\  ( ( 1 ... J )  \ 
( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )  ->  ( ( S `
 C )  |`  ( ( 1 ... J )  \  ( R `  C )
) ) : ( ( 1 ... J
)  \  ( R `  C ) ) -1-1-onto-> ( ( S `  C )
" ( ( 1 ... J )  \ 
( R `  C
) ) ) )
3714, 35, 36syl2anc 661 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C )  |`  ( ( 1 ... J )  \  ( R `  C )
) ) : ( ( 1 ... J
)  \  ( R `  C ) ) -1-1-onto-> ( ( S `  C )
" ( ( 1 ... J )  \ 
( R `  C
) ) ) )
38 difexg 4455 . . . . . 6  |-  ( ( 1 ... J )  e.  _V  ->  (
( 1 ... J
)  \  ( R `  C ) )  e. 
_V )
3930, 38ax-mp 5 . . . . 5  |-  ( ( 1 ... J ) 
\  ( R `  C ) )  e. 
_V
4039f1oen 7345 . . . 4  |-  ( ( ( S `  C
)  |`  ( ( 1 ... J )  \ 
( R `  C
) ) ) : ( ( 1 ... J )  \  ( R `  C )
)
-1-1-onto-> ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) )  ->  ( ( 1 ... J )  \ 
( R `  C
) )  ~~  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) )
41 hasheni 12134 . . . 4  |-  ( ( ( 1 ... J
)  \  ( R `  C ) )  ~~  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4237, 40, 413syl 20 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4334, 42oveq12d 6124 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( # `  ( ( 1 ... J )  i^i  ( R `  C )
) )  -  ( # `
 ( ( 1 ... J )  \ 
( R `  C
) ) ) )  =  ( ( # `  ( ( S `  C ) " (
( 1 ... J
)  i^i  ( R `  C ) ) ) )  -  ( # `  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) ) ) ) )
44 ballotth.r . . . . 5  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
451, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemro 26920 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
4645adantr 465 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( R `  C )  e.  O
)
47 elfzelz 11468 . . . 4  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  J  e.  ZZ )
4847adantl 466 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  J  e.  ZZ )
491, 2, 3, 4, 5, 46, 48ballotlemfval 26887 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( F `
 ( R `  C ) ) `  J )  =  ( ( # `  (
( 1 ... J
)  i^i  ( R `  C ) ) )  -  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) ) ) )
50 fzfi 11809 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
51 eldifi 3493 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
521, 2, 3ballotlemelo 26885 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
5352simplbi 460 . . . . . . 7  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
5451, 53syl 16 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
5554adantr 465 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  C_  (
1 ... ( M  +  N ) ) )
56 ssfi 7548 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  ->  C  e.  Fin )
5750, 55, 56sylancr 663 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  e.  Fin )
58 fzfid 11810 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  e.  Fin )
59 ballotlemg . . . . 5  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
601, 2, 3, 4, 5, 6, 7, 8, 9, 44, 59ballotlemgval 26921 . . . 4  |-  ( ( C  e.  Fin  /\  ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  e.  Fin )  ->  ( C  .^  (
( ( S `  C ) `  J
) ... ( I `  C ) ) )  =  ( ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  i^i  C
) )  -  ( # `
 ( ( ( ( S `  C
) `  J ) ... ( I `  C
) )  \  C
) ) ) )
6157, 58, 60syl2anc 661 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( C  .^  ( ( ( S `
 C ) `  J ) ... (
I `  C )
) )  =  ( ( # `  (
( ( ( S `
 C ) `  J ) ... (
I `  C )
)  i^i  C )
)  -  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  \  C
) ) ) )
62 dff1o3 5662 . . . . . . . . 9  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  <->  ( ( S `
 C ) : ( 1 ... ( M  +  N )
) -onto-> ( 1 ... ( M  +  N
) )  /\  Fun  `' ( S `  C
) ) )
6362simprbi 464 . . . . . . . 8  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  Fun  `' ( S `  C ) )
64 imain 5509 . . . . . . . 8  |-  ( Fun  `' ( S `  C )  ->  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) ) )
6511, 63, 643syl 20 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) ) )
6665adantr 465 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  i^i  ( R `  C )
) )  =  ( ( ( S `  C ) " (
1 ... J ) )  i^i  ( ( S `
 C ) "
( R `  C
) ) ) )
671, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsima 26913 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( 1 ... J
) )  =  ( ( ( S `  C ) `  J
) ... ( I `  C ) ) )
681, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemscr 26916 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( R `
 C ) )  =  C )
6968adantr 465 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( R `  C
) )  =  C )
7067, 69ineq12d 3568 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  i^i  C )
)
7166, 70eqtrd 2475 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  i^i  ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  i^i  C )
)
7271fveq2d 5710 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) ) )  =  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  i^i  C
) ) )
73 imadif 5508 . . . . . . . 8  |-  ( Fun  `' ( S `  C )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7411, 63, 733syl 20 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7574adantr 465 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  \  ( R `  C )
) )  =  ( ( ( S `  C ) " (
1 ... J ) ) 
\  ( ( S `
 C ) "
( R `  C
) ) ) )
7667, 69difeq12d 3490 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  \  C )
)
7775, 76eqtrd 2475 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  \  ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  \  C )
)
7877fveq2d 5710 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) )  =  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  \  C
) ) )
7972, 78oveq12d 6124 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( # `  ( ( S `  C ) " (
( 1 ... J
)  i^i  ( R `  C ) ) ) )  -  ( # `  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) ) ) )  =  ( ( # `  (
( ( ( S `
 C ) `  J ) ... (
I `  C )
)  i^i  C )
)  -  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  \  C
) ) ) )
8061, 79eqtr4d 2478 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( C  .^  ( ( ( S `
 C ) `  J ) ... (
I `  C )
) )  =  ( ( # `  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) ) )  -  ( # `  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) ) ) ) )
8143, 49, 803eqtr4d 2485 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( F `
 ( R `  C ) ) `  J )  =  ( C  .^  ( (
( S `  C
) `  J ) ... ( I `  C
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2730   {crab 2734   _Vcvv 2987    \ cdif 3340    i^i cin 3342    C_ wss 3343   ifcif 3806   ~Pcpw 3875   class class class wbr 4307    e. cmpt 4365   `'ccnv 4854    |` cres 4857   "cima 4858   Fun wfun 5427   -1-1->wf1 5430   -onto->wfo 5431   -1-1-onto->wf1o 5432   ` cfv 5433  (class class class)co 6106    e. cmpt2 6108    ~~ cen 7322   Fincfn 7325   supcsup 7705   RRcr 9296   0cc0 9297   1c1 9298    + caddc 9300    < clt 9433    <_ cle 9434    - cmin 9610    / cdiv 10008   NNcn 10337   ZZcz 10661   ZZ>=cuz 10876   ...cfz 11452   #chash 12118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-fz 11453  df-hash 12119
This theorem is referenced by:  ballotlemfrci  26925  ballotlemfrceq  26926
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