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Theorem ballotlemfrc 23085
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 23072 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1110simpld 445 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
12 f1of1 5471 . . . . . . 7  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
1311, 12syl 15 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-> ( 1 ... ( M  +  N ) ) )
1413adantr 451 . . . . 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 23060 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1615simpld 445 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
1716adantr 451 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
18 elfzuz3 10795 . . . . . . . . 9  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  ( M  +  N )  e.  ( ZZ>= `  ( I `  C ) ) )
1917, 18syl 15 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  ( I `  C ) ) )
20 simpr 447 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  J  e.  ( 1 ... ( I `
 C ) ) )
21 elfzuz3 10795 . . . . . . . . 9  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  (
I `  C )  e.  ( ZZ>= `  J )
)
2220, 21syl 15 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  (
ZZ>= `  J ) )
23 uztrn 10244 . . . . . . . 8  |-  ( ( ( M  +  N
)  e.  ( ZZ>= `  ( I `  C
) )  /\  (
I `  C )  e.  ( ZZ>= `  J )
)  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
2419, 22, 23syl2anc 642 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
25 fzss2 10831 . . . . . . 7  |-  ( ( M  +  N )  e.  ( ZZ>= `  J
)  ->  ( 1 ... J )  C_  ( 1 ... ( M  +  N )
) )
2624, 25syl 15 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( 1 ... J )  C_  (
1 ... ( M  +  N ) ) )
27 ssinss1 3397 . . . . . 6  |-  ( ( 1 ... J ) 
C_  ( 1 ... ( M  +  N
) )  ->  (
( 1 ... J
)  i^i  ( R `  C ) )  C_  ( 1 ... ( M  +  N )
) )
2826, 27syl 15 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  i^i  ( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
29 f1ores 5487 . . . . 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
) ) ) )
3014, 28, 29syl2anc 642 . . . 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
) ) ) )
31 ovex 5883 . . . . . 6  |-  ( 1 ... J )  e. 
_V
3231inex1 4155 . . . . 5  |-  ( ( 1 ... J )  i^i  ( R `  C ) )  e. 
_V
3332f1oen 6882 . . . 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 ) ) ) )
34 hasheni 11347 . . . 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 ) ) ) ) )
3530, 33, 343syl 18 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( 1 ... J
)  i^i  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) ) ) )
36 ssdifss 3307 . . . . . 6  |-  ( ( 1 ... J ) 
C_  ( 1 ... ( M  +  N
) )  ->  (
( 1 ... J
)  \  ( R `  C ) )  C_  ( 1 ... ( M  +  N )
) )
3726, 36syl 15 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  \ 
( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
38 f1ores 5487 . . . . 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
) ) ) )
3914, 37, 38syl2anc 642 . . . 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
) ) ) )
40 difexg 4162 . . . . . 6  |-  ( ( 1 ... J )  e.  _V  ->  (
( 1 ... J
)  \  ( R `  C ) )  e. 
_V )
4131, 40ax-mp 8 . . . . 5  |-  ( ( 1 ... J ) 
\  ( R `  C ) )  e. 
_V
4241f1oen 6882 . . . 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 ) ) ) )
43 hasheni 11347 . . . 4  |-  ( ( ( 1 ... J
)  \  ( R `  C ) )  ~~  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4439, 42, 433syl 18 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4535, 44oveq12d 5876 . 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 ) ) ) ) ) )
46 ballotth.r . . . . 5  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
471, 2, 3, 4, 5, 6, 7, 8, 9, 46ballotlemro 23081 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
4847adantr 451 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( R `  C )  e.  O
)
49 elfzelz 10798 . . . 4  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  J  e.  ZZ )
5049adantl 452 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  J  e.  ZZ )
511, 2, 3, 4, 5, 48, 50ballotlemfval 23048 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( F `
 ( R `  C ) ) `  J )  =  ( ( # `  (
( 1 ... J
)  i^i  ( R `  C ) ) )  -  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) ) ) )
52 fzfi 11034 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
53 eldifi 3298 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
541, 2, 3ballotlemelo 23046 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
5554simplbi 446 . . . . . . 7  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
5653, 55syl 15 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
5756adantr 451 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  C_  (
1 ... ( M  +  N ) ) )
58 ssfi 7083 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  ->  C  e.  Fin )
5952, 57, 58sylancr 644 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  e.  Fin )
60 fzfid 11035 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  e.  Fin )
61 ballotlemg . . . . 5  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
621, 2, 3, 4, 5, 6, 7, 8, 9, 46, 61ballotlemgval 23082 . . . 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
) ) ) )
6359, 60, 62syl2anc 642 . . 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
) ) ) )
64 dff1o3 5478 . . . . . . . . 9  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  <->  ( ( S `
 C ) : ( 1 ... ( M  +  N )
) -onto-> ( 1 ... ( M  +  N
) )  /\  Fun  `' ( S `  C
) ) )
6564simprbi 450 . . . . . . . 8  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  Fun  `' ( S `  C ) )
66 imain 5328 . . . . . . . 8  |-  ( Fun  `' ( S `  C )  ->  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) ) )
6711, 65, 663syl 18 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) ) )
6867adantr 451 . . . . . 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
) ) ) )
691, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsima 23074 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( 1 ... J
) )  =  ( ( ( S `  C ) `  J
) ... ( I `  C ) ) )
701, 2, 3, 4, 5, 6, 7, 8, 9, 46ballotlemscr 23077 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( R `
 C ) )  =  C )
7170adantr 451 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( R `  C
) )  =  C )
7269, 71ineq12d 3371 . . . . . 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 )
)
7368, 72eqtrd 2315 . . . . 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 )
)
7473fveq2d 5529 . . . 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
) ) )
75 imadif 5327 . . . . . . . 8  |-  ( Fun  `' ( S `  C )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7611, 65, 753syl 18 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7776adantr 451 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  \  ( R `  C )
) )  =  ( ( ( S `  C ) " (
1 ... J ) ) 
\  ( ( S `
 C ) "
( R `  C
) ) ) )
7869, 71difeq12d 3295 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  \  C )
)
7977, 78eqtrd 2315 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  \  ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  \  C )
)
8079fveq2d 5529 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) )  =  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  \  C
) ) )
8174, 80oveq12d 5876 . . 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
) ) ) )
8263, 81eqtr4d 2318 . 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 ) ) ) ) ) )
8345, 51, 823eqtr4d 2325 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   ifcif 3565   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688    |` cres 4691   "cima 4692   Fun wfun 5249   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ~~ cen 6860   Fincfn 6863   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemfrci  23086  ballotlemfrceq  23087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-hash 11338
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