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Theorem ballotlemfelz 27010
Description:  ( F `  C ) has values in  ZZ. (Contributed by Thierry Arnoux, 23-Nov-2016.)
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
) ) ) ) )
ballotlemfval.c  |-  ( ph  ->  C  e.  O )
ballotlemfval.j  |-  ( ph  ->  J  e.  ZZ )
Assertion
Ref Expression
ballotlemfelz  |-  ( ph  ->  ( ( F `  C ) `  J
)  e.  ZZ )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O, c    F, c, i    C, i    i, J    ph, i
Allowed substitution hints:    ph( x, c)    C( x, c)    P( x, i, c)    F( x)    J( x, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemfelz
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 ballotlemfval.c . . 3  |-  ( ph  ->  C  e.  O )
7 ballotlemfval.j . . 3  |-  ( ph  ->  J  e.  ZZ )
81, 2, 3, 4, 5, 6, 7ballotlemfval 27009 . 2  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
9 fzfi 11904 . . . . . 6  |-  ( 1 ... J )  e. 
Fin
10 inss1 3671 . . . . . 6  |-  ( ( 1 ... J )  i^i  C )  C_  ( 1 ... J
)
11 ssfi 7637 . . . . . 6  |-  ( ( ( 1 ... J
)  e.  Fin  /\  ( ( 1 ... J )  i^i  C
)  C_  ( 1 ... J ) )  ->  ( ( 1 ... J )  i^i 
C )  e.  Fin )
129, 10, 11mp2an 672 . . . . 5  |-  ( ( 1 ... J )  i^i  C )  e. 
Fin
13 hashcl 12236 . . . . 5  |-  ( ( ( 1 ... J
)  i^i  C )  e.  Fin  ->  ( # `  (
( 1 ... J
)  i^i  C )
)  e.  NN0 )
1412, 13ax-mp 5 . . . 4  |-  ( # `  ( ( 1 ... J )  i^i  C
) )  e.  NN0
1514nn0zi 10775 . . 3  |-  ( # `  ( ( 1 ... J )  i^i  C
) )  e.  ZZ
16 difss 3584 . . . . . 6  |-  ( ( 1 ... J ) 
\  C )  C_  ( 1 ... J
)
17 ssfi 7637 . . . . . 6  |-  ( ( ( 1 ... J
)  e.  Fin  /\  ( ( 1 ... J )  \  C
)  C_  ( 1 ... J ) )  ->  ( ( 1 ... J )  \  C )  e.  Fin )
189, 16, 17mp2an 672 . . . . 5  |-  ( ( 1 ... J ) 
\  C )  e. 
Fin
19 hashcl 12236 . . . . 5  |-  ( ( ( 1 ... J
)  \  C )  e.  Fin  ->  ( # `  (
( 1 ... J
)  \  C )
)  e.  NN0 )
2018, 19ax-mp 5 . . . 4  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  NN0
2120nn0zi 10775 . . 3  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  ZZ
22 zsubcl 10791 . . 3  |-  ( ( ( # `  (
( 1 ... J
)  i^i  C )
)  e.  ZZ  /\  ( # `  ( ( 1 ... J ) 
\  C ) )  e.  ZZ )  -> 
( ( # `  (
( 1 ... J
)  i^i  C )
)  -  ( # `  ( ( 1 ... J )  \  C
) ) )  e.  ZZ )
2315, 21, 22mp2an 672 . 2  |-  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) )  e.  ZZ
248, 23syl6eqel 2547 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {crab 2799    \ cdif 3426    i^i cin 3428    C_ wss 3429   ~Pcpw 3961    |-> cmpt 4451   ` cfv 5519  (class class class)co 6193   Fincfn 7413   1c1 9387    + caddc 9389    - cmin 9699    / cdiv 10097   NNcn 10426   NN0cn0 10683   ZZcz 10750   ...cfz 11547   #chash 12213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-hash 12214
This theorem is referenced by:  ballotlemfc0  27012  ballotlemfcc  27013  ballotlemodife  27017  ballotlemic  27026  ballotlem1c  27027  ballotlemfrceq  27048  ballotlemfrcn0  27049
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