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Theorem ballotlemfelz 26825
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 26824 . 2  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
9 fzfi 11786 . . . . . 6  |-  ( 1 ... J )  e. 
Fin
10 inss1 3565 . . . . . 6  |-  ( ( 1 ... J )  i^i  C )  C_  ( 1 ... J
)
11 ssfi 7525 . . . . . 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 12118 . . . . 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 10663 . . 3  |-  ( # `  ( ( 1 ... J )  i^i  C
) )  e.  ZZ
16 difss 3478 . . . . . 6  |-  ( ( 1 ... J ) 
\  C )  C_  ( 1 ... J
)
17 ssfi 7525 . . . . . 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 12118 . . . . 5  |-  ( ( ( 1 ... J
)  \  C )  e.  Fin  ->  ( # `  (
( 1 ... J
)  \  C )
)  e.  NN0 )
2018, 19ax-mp 5 . . . 4  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  NN0
2120nn0zi 10663 . . 3  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  ZZ
22 zsubcl 10679 . . 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 2526 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2714    \ cdif 3320    i^i cin 3322    C_ wss 3323   ~Pcpw 3855    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Fincfn 7302   1c1 9275    + caddc 9277    - cmin 9587    / cdiv 9985   NNcn 10314   NN0cn0 10571   ZZcz 10638   ...cfz 11429   #chash 12095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096
This theorem is referenced by:  ballotlemfc0  26827  ballotlemfcc  26828  ballotlemodife  26832  ballotlemic  26841  ballotlem1c  26842  ballotlemfrceq  26863  ballotlemfrcn0  26864
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