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Theorem ballotlemfelz 28948
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 28947 . 2  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
9 fzfi 12125 . . . . . 6  |-  ( 1 ... J )  e. 
Fin
10 inss1 3661 . . . . . 6  |-  ( ( 1 ... J )  i^i  C )  C_  ( 1 ... J
)
11 ssfi 7777 . . . . . 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 12477 . . . . 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 10932 . . 3  |-  ( # `  ( ( 1 ... J )  i^i  C
) )  e.  ZZ
16 difss 3572 . . . . . 6  |-  ( ( 1 ... J ) 
\  C )  C_  ( 1 ... J
)
17 ssfi 7777 . . . . . 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 12477 . . . . 5  |-  ( ( ( 1 ... J
)  \  C )  e.  Fin  ->  ( # `  (
( 1 ... J
)  \  C )
)  e.  NN0 )
2018, 19ax-mp 5 . . . 4  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  NN0
2120nn0zi 10932 . . 3  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  ZZ
22 zsubcl 10949 . . 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 2500 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   {crab 2760    \ cdif 3413    i^i cin 3415    C_ wss 3416   ~Pcpw 3957    |-> cmpt 4455   ` cfv 5571  (class class class)co 6280   Fincfn 7556   1c1 9525    + caddc 9527    - cmin 9843    / cdiv 10249   NNcn 10578   NN0cn0 10838   ZZcz 10907   ...cfz 11728   #chash 12454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-hash 12455
This theorem is referenced by:  ballotlemfc0  28950  ballotlemfcc  28951  ballotlemodife  28955  ballotlemic  28964  ballotlem1c  28965  ballotlemfrceq  28986  ballotlemfrcn0  28987
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