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Theorem ballotlemfrc 24786
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 24773 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1110simpld 447 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
12 f1of1 5675 . . . . . . 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 453 . . . . 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 24761 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1615simpld 447 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
1716adantr 453 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
18 elfzuz3 11058 . . . . . . . . 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 11058 . . . . . . . . 9  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  (
I `  C )  e.  ( ZZ>= `  J )
)
2120adantl 454 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  (
ZZ>= `  J ) )
22 uztrn 10504 . . . . . . . 8  |-  ( ( ( M  +  N
)  e.  ( ZZ>= `  ( I `  C
) )  /\  (
I `  C )  e.  ( ZZ>= `  J )
)  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
2319, 21, 22syl2anc 644 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
24 fzss2 11094 . . . . . . 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 3571 . . . . . 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 5691 . . . . 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 644 . . . 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 6108 . . . . . 6  |-  ( 1 ... J )  e. 
_V
3130inex1 4346 . . . . 5  |-  ( ( 1 ... J )  i^i  ( R `  C ) )  e. 
_V
3231f1oen 7130 . . . 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 11634 . . . 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 19 . . 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 3487 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  \ 
( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
36 f1ores 5691 . . . . 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 644 . . . 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 4353 . . . . . 6  |-  ( ( 1 ... J )  e.  _V  ->  (
( 1 ... J
)  \  ( R `  C ) )  e. 
_V )
3930, 38ax-mp 8 . . . . 5  |-  ( ( 1 ... J ) 
\  ( R `  C ) )  e. 
_V
4039f1oen 7130 . . . 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 11634 . . . 4  |-  ( ( ( 1 ... J
)  \  ( R `  C ) )  ~~  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4237, 40, 413syl 19 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4334, 42oveq12d 6101 . 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 24782 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
4645adantr 453 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( R `  C )  e.  O
)
47 elfzelz 11061 . . . 4  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  J  e.  ZZ )
4847adantl 454 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  J  e.  ZZ )
491, 2, 3, 4, 5, 46, 48ballotlemfval 24749 . 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 11313 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
51 eldifi 3471 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
521, 2, 3ballotlemelo 24747 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
5352simplbi 448 . . . . . . 7  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
5451, 53syl 16 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
5554adantr 453 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  C_  (
1 ... ( M  +  N ) ) )
56 ssfi 7331 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  ->  C  e.  Fin )
5750, 55, 56sylancr 646 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  e.  Fin )
58 fzfid 11314 . . . 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 24783 . . . 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 644 . . 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 5682 . . . . . . . . 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 452 . . . . . . . 8  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  Fun  `' ( S `  C ) )
64 imain 5531 . . . . . . . 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 19 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) ) )
6665adantr 453 . . . . . 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 24775 . . . . . . 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 24778 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( R `
 C ) )  =  C )
6968adantr 453 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( R `  C
) )  =  C )
7067, 69ineq12d 3545 . . . . . 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 2470 . . . . 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 5734 . . . 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 5530 . . . . . . . 8  |-  ( Fun  `' ( S `  C )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7411, 63, 733syl 19 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7574adantr 453 . . . . . 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 3468 . . . . . 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 2470 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  \  ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  \  C )
)
7877fveq2d 5734 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) )  =  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  \  C
) ) )
7972, 78oveq12d 6101 . . 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 2473 . 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 2480 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( F `
 ( R `  C ) ) `  J )  =  ( C  .^  ( (
( S `  C
) `  J ) ... ( I `  C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   ifcif 3741   ~Pcpw 3801   class class class wbr 4214    e. cmpt 4268   `'ccnv 4879    |` cres 4882   "cima 4883   Fun wfun 5450   -1-1->wf1 5453   -onto->wfo 5454   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085    ~~ cen 7108   Fincfn 7111   supcsup 7447   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    < clt 9122    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045   #chash 11620
This theorem is referenced by:  ballotlemfrci  24787  ballotlemfrceq  24788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  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-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-hash 11621
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