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Theorem ballotlemfg 26756
Description: Express the value of  ( F `
 C ) in terms of  .^. (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
ballotlemfg  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  J )  =  ( C  .^  ( 1 ... J ) ) )
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 ballotlemfg
StepHypRef Expression
1 ballotth.m . . 3  |-  M  e.  NN
2 ballotth.n . . 3  |-  N  e.  NN
3 ballotth.o . . 3  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotth.p . . 3  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotth.f . . 3  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 eldifi 3466 . . . 4  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
76adantr 462 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  C  e.  O
)
8 elfzelz 11440 . . . 4  |-  ( J  e.  ( 0 ... ( M  +  N
) )  ->  J  e.  ZZ )
98adantl 463 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  J  e.  ZZ )
101, 2, 3, 4, 5, 7, 9ballotlemfval 26720 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  J )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
11 fzfi 11778 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
121, 2, 3ballotlemelo 26718 . . . . . 6  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
1312simplbi 457 . . . . 5  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
14 ssfi 7521 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  ->  C  e.  Fin )
1511, 13, 14sylancr 656 . . . 4  |-  ( C  e.  O  ->  C  e.  Fin )
167, 15syl 16 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  C  e.  Fin )
17 fzfid 11779 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( 1 ... J )  e.  Fin )
18 ballotth.e . . . 4  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
19 ballotth.mgtn . . . 4  |-  N  < 
M
20 ballotth.i . . . 4  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
21 ballotth.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
22 ballotth.r . . . 4  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
23 ballotlemg . . . 4  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
241, 2, 3, 4, 5, 18, 19, 20, 21, 22, 23ballotlemgval 26754 . . 3  |-  ( ( C  e.  Fin  /\  ( 1 ... J
)  e.  Fin )  ->  ( C  .^  (
1 ... J ) )  =  ( ( # `  ( ( 1 ... J )  i^i  C
) )  -  ( # `
 ( ( 1 ... J )  \  C ) ) ) )
2516, 17, 24syl2anc 654 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( C  .^  ( 1 ... J
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
2610, 25eqtr4d 2468 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  J )  =  ( C  .^  ( 1 ... J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   {crab 2709    \ cdif 3313    i^i cin 3315    C_ wss 3316   ifcif 3779   ~Pcpw 3848   class class class wbr 4280    e. cmpt 4338   `'ccnv 4826   "cima 4830   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   Fincfn 7298   supcsup 7678   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   ZZcz 10634   ...cfz 11424   #chash 12087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425
This theorem is referenced by:  ballotlemfrci  26758  ballotlemfrceq  26759
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