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Theorem ballotlemfelz 24740
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 24739 . 2  |-  ( ph  ->  ( ( F `  C ) `  J
)  =  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) ) )
9 fzfi 11303 . . . . . 6  |-  ( 1 ... J )  e. 
Fin
10 inss1 3553 . . . . . 6  |-  ( ( 1 ... J )  i^i  C )  C_  ( 1 ... J
)
11 ssfi 7321 . . . . . 6  |-  ( ( ( 1 ... J
)  e.  Fin  /\  ( ( 1 ... J )  i^i  C
)  C_  ( 1 ... J ) )  ->  ( ( 1 ... J )  i^i 
C )  e.  Fin )
129, 10, 11mp2an 654 . . . . 5  |-  ( ( 1 ... J )  i^i  C )  e. 
Fin
13 hashcl 11631 . . . . 5  |-  ( ( ( 1 ... J
)  i^i  C )  e.  Fin  ->  ( # `  (
( 1 ... J
)  i^i  C )
)  e.  NN0 )
1412, 13ax-mp 8 . . . 4  |-  ( # `  ( ( 1 ... J )  i^i  C
) )  e.  NN0
1514nn0zi 10298 . . 3  |-  ( # `  ( ( 1 ... J )  i^i  C
) )  e.  ZZ
16 difss 3466 . . . . . 6  |-  ( ( 1 ... J ) 
\  C )  C_  ( 1 ... J
)
17 ssfi 7321 . . . . . 6  |-  ( ( ( 1 ... J
)  e.  Fin  /\  ( ( 1 ... J )  \  C
)  C_  ( 1 ... J ) )  ->  ( ( 1 ... J )  \  C )  e.  Fin )
189, 16, 17mp2an 654 . . . . 5  |-  ( ( 1 ... J ) 
\  C )  e. 
Fin
19 hashcl 11631 . . . . 5  |-  ( ( ( 1 ... J
)  \  C )  e.  Fin  ->  ( # `  (
( 1 ... J
)  \  C )
)  e.  NN0 )
2018, 19ax-mp 8 . . . 4  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  NN0
2120nn0zi 10298 . . 3  |-  ( # `  ( ( 1 ... J )  \  C
) )  e.  ZZ
22 zsubcl 10311 . . 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 654 . 2  |-  ( (
# `  ( (
1 ... J )  i^i 
C ) )  -  ( # `  ( ( 1 ... J ) 
\  C ) ) )  e.  ZZ
248, 23syl6eqel 2523 1  |-  ( ph  ->  ( ( F `  C ) `  J
)  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Fincfn 7101   1c1 8983    + caddc 8985    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ...cfz 11035   #chash 11610
This theorem is referenced by:  ballotlemfc0  24742  ballotlemfcc  24743  ballotlemodife  24747  ballotlemic  24756  ballotlem1c  24757  ballotlemfrceq  24778  ballotlemfrcn0  24779
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-int 4043  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-card 7818  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  df-hash 11611
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