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Theorem ballotlemfg 24775
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 3461 . . . 4  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
76adantr 452 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  C  e.  O
)
8 elfzelz 11051 . . . 4  |-  ( J  e.  ( 0 ... ( M  +  N
) )  ->  J  e.  ZZ )
98adantl 453 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  J  e.  ZZ )
101, 2, 3, 4, 5, 7, 9ballotlemfval 24739 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  J )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
11 fzfi 11303 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
121, 2, 3ballotlemelo 24737 . . . . . 6  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
1312simplbi 447 . . . . 5  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
14 ssfi 7321 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  ->  C  e.  Fin )
1511, 13, 14sylancr 645 . . . 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 11304 . . 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 24773 . . 3  |-  ( ( C  e.  Fin  /\  ( 1 ... J
)  e.  Fin )  ->  ( C  .^  (
1 ... J ) )  =  ( ( # `  ( ( 1 ... J )  i^i  C
) )  -  ( # `
 ( ( 1 ... J )  \  C ) ) ) )
2516, 17, 24syl2anc 643 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( C  .^  ( 1 ... J
) )  =  ( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) ) )
2610, 25eqtr4d 2470 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  J )  =  ( C  .^  ( 1 ... J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309    i^i cin 3311    C_ wss 3312   ifcif 3731   ~Pcpw 3791   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   "cima 4873   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   Fincfn 7101   supcsup 7437   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   ZZcz 10274   ...cfz 11035   #chash 11610
This theorem is referenced by:  ballotlemfrci  24777  ballotlemfrceq  24778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036
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