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Theorem ballotlemfrcOLD 29397
Description: Express the value of  ( F `
 ( R `  C ) ) in terms of the newly defined  .^. (Contributed by Thierry Arnoux, 21-Apr-2017.) Obsolete version of ballotlemfrc 29359 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ballotthOLD.m  |-  M  e.  NN
ballotthOLD.n  |-  N  e.  NN
ballotthOLD.o  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
ballotthOLD.p  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
ballotthOLD.f  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
ballotthOLD.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
ballotthOLD.mgtn  |-  N  < 
M
ballotthOLD.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
ballotthOLD.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
ballotthOLD.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
ballotlemgOLD  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
Assertion
Ref Expression
ballotlemfrcOLD  |-  ( ( 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    C, i, k    i,
c, F, k    i, E, 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 ballotlemfrcOLD
StepHypRef Expression
1 ballotthOLD.m . . . . . . . . 9  |-  M  e.  NN
2 ballotthOLD.n . . . . . . . . 9  |-  N  e.  NN
3 ballotthOLD.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotthOLD.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotthOLD.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 ballotthOLD.e . . . . . . . . 9  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
7 ballotthOLD.mgtn . . . . . . . . 9  |-  N  < 
M
8 ballotthOLD.i . . . . . . . . 9  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
9 ballotthOLD.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, 9ballotlemsf1oOLD 29384 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1110simpld 461 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
12 f1of1 5813 . . . . . . 7  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
1311, 12syl 17 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-> ( 1 ... ( M  +  N ) ) )
1413adantr 467 . . . . 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, 8ballotlemiexOLD 29372 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1615simpld 461 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
1716adantr 467 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
18 elfzuz3 11797 . . . . . . . . 9  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  ( M  +  N )  e.  ( ZZ>= `  ( I `  C ) ) )
1917, 18syl 17 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  ( I `  C ) ) )
20 elfzuz3 11797 . . . . . . . . 9  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  (
I `  C )  e.  ( ZZ>= `  J )
)
2120adantl 468 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( I `  C )  e.  (
ZZ>= `  J ) )
22 uztrn 11175 . . . . . . . 8  |-  ( ( ( M  +  N
)  e.  ( ZZ>= `  ( I `  C
) )  /\  (
I `  C )  e.  ( ZZ>= `  J )
)  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
2319, 21, 22syl2anc 667 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( M  +  N )  e.  (
ZZ>= `  J ) )
24 fzss2 11838 . . . . . . 7  |-  ( ( M  +  N )  e.  ( ZZ>= `  J
)  ->  ( 1 ... J )  C_  ( 1 ... ( M  +  N )
) )
2523, 24syl 17 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( 1 ... J )  C_  (
1 ... ( M  +  N ) ) )
26 ssinss1 3660 . . . . . 6  |-  ( ( 1 ... J ) 
C_  ( 1 ... ( M  +  N
) )  ->  (
( 1 ... J
)  i^i  ( R `  C ) )  C_  ( 1 ... ( M  +  N )
) )
2725, 26syl 17 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  i^i  ( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
28 f1ores 5828 . . . . 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 667 . . . 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 6318 . . . . . 6  |-  ( 1 ... J )  e. 
_V
3130inex1 4544 . . . . 5  |-  ( ( 1 ... J )  i^i  ( R `  C ) )  e. 
_V
3231f1oen 7590 . . . 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 12531 . . . 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 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 ) ) ) ) )
3525ssdifssd 3571 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( 1 ... J )  \ 
( R `  C
) )  C_  (
1 ... ( M  +  N ) ) )
36 f1ores 5828 . . . . 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 667 . . . 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 4551 . . . . . 6  |-  ( ( 1 ... J )  e.  _V  ->  (
( 1 ... J
)  \  ( R `  C ) )  e. 
_V )
3930, 38ax-mp 5 . . . . 5  |-  ( ( 1 ... J ) 
\  ( R `  C ) )  e. 
_V
4039f1oen 7590 . . . 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 12531 . . . 4  |-  ( ( ( 1 ... J
)  \  ( R `  C ) )  ~~  ( ( S `  C ) " (
( 1 ... J
)  \  ( R `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4237, 40, 413syl 18 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( 1 ... J
)  \  ( R `  C ) ) )  =  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) ) )
4334, 42oveq12d 6308 . 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 ballotthOLD.r . . . . 5  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
451, 2, 3, 4, 5, 6, 7, 8, 9, 44ballotlemroOLD 29393 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
4645adantr 467 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( R `  C )  e.  O
)
47 elfzelz 11800 . . . 4  |-  ( J  e.  ( 1 ... ( I `  C
) )  ->  J  e.  ZZ )
4847adantl 468 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  J  e.  ZZ )
491, 2, 3, 4, 5, 46, 48ballotlemfval 29322 . 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 12185 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
51 eldifi 3555 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
521, 2, 3ballotlemelo 29320 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
5352simplbi 462 . . . . . . 7  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
5451, 53syl 17 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
5554adantr 467 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  C_  (
1 ... ( M  +  N ) ) )
56 ssfi 7792 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  ->  C  e.  Fin )
5750, 55, 56sylancr 669 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  C  e.  Fin )
58 fzfid 12186 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  e.  Fin )
59 ballotlemgOLD . . . . 5  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
601, 2, 3, 4, 5, 6, 7, 8, 9, 44, 59ballotlemgvalOLD 29394 . . . 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 667 . . 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 5820 . . . . . . . . 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 466 . . . . . . . 8  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  Fun  `' ( S `  C ) )
64 imain 5659 . . . . . . . 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 18 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J )  i^i  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  i^i  ( ( S `  C ) " ( R `  C )
) ) )
6665adantr 467 . . . . . 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, 9ballotlemsimaOLD 29386 . . . . . . 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, 44ballotlemscrOLD 29389 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( R `
 C ) )  =  C )
6968adantr 467 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( R `  C
) )  =  C )
7067, 69ineq12d 3635 . . . . . 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 2485 . . . . 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 5869 . . . 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 5658 . . . . . . . 8  |-  ( Fun  `' ( S `  C )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7411, 63, 733syl 18 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) )  =  ( ( ( S `  C )
" ( 1 ... J ) )  \ 
( ( S `  C ) " ( R `  C )
) ) )
7574adantr 467 . . . . . 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 3552 . . . . . 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 2485 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( S `
 C ) "
( ( 1 ... J )  \  ( R `  C )
) )  =  ( ( ( ( S `
 C ) `  J ) ... (
I `  C )
)  \  C )
)
7877fveq2d 5869 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( I `  C ) ) )  ->  ( # `  (
( S `  C
) " ( ( 1 ... J ) 
\  ( R `  C ) ) ) )  =  ( # `  ( ( ( ( S `  C ) `
 J ) ... ( I `  C
) )  \  C
) ) )
7972, 78oveq12d 6308 . . 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 2488 . 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 2495 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 371    = wceq 1444    e. wcel 1887   A.wral 2737   {crab 2741   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   ifcif 3881   ~Pcpw 3951   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833    |` cres 4836   "cima 4837   Fun wfun 5576   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292    ~~ cen 7566   Fincfn 7569   supcsup 7954   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   #chash 12515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-hash 12516
This theorem is referenced by:  ballotlemfrciOLD  29398  ballotlemfrceqOLD  29399
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