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Theorem ballotlemfval0 28307
Description:  ( F `  C ) always starts counting at 0 . (Contributed by Thierry Arnoux, 25-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
) ) ) ) )
Assertion
Ref Expression
ballotlemfval0  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O, c    F, c, i    C, i
Allowed substitution hints:    C( x, c)    P( x, i, c)    F( x)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemfval0
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 id 22 . . 3  |-  ( C  e.  O  ->  C  e.  O )
7 0zd 10882 . . 3  |-  ( C  e.  O  ->  0  e.  ZZ )
81, 2, 3, 4, 5, 6, 7ballotlemfval 28301 . 2  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  ( ( # `  ( ( 1 ... 0 )  i^i  C
) )  -  ( # `
 ( ( 1 ... 0 )  \  C ) ) ) )
9 fz10 11715 . . . . . . . 8  |-  ( 1 ... 0 )  =  (/)
109ineq1i 3681 . . . . . . 7  |-  ( ( 1 ... 0 )  i^i  C )  =  ( (/)  i^i  C )
11 incom 3676 . . . . . . 7  |-  ( C  i^i  (/) )  =  (
(/)  i^i  C )
12 in0 3797 . . . . . . 7  |-  ( C  i^i  (/) )  =  (/)
1310, 11, 123eqtr2i 2478 . . . . . 6  |-  ( ( 1 ... 0 )  i^i  C )  =  (/)
1413fveq2i 5859 . . . . 5  |-  ( # `  ( ( 1 ... 0 )  i^i  C
) )  =  (
# `  (/) )
15 hash0 12416 . . . . 5  |-  ( # `  (/) )  =  0
1614, 15eqtri 2472 . . . 4  |-  ( # `  ( ( 1 ... 0 )  i^i  C
) )  =  0
179difeq1i 3603 . . . . . . 7  |-  ( ( 1 ... 0 ) 
\  C )  =  ( (/)  \  C )
18 0dif 3885 . . . . . . 7  |-  ( (/)  \  C )  =  (/)
1917, 18eqtri 2472 . . . . . 6  |-  ( ( 1 ... 0 ) 
\  C )  =  (/)
2019fveq2i 5859 . . . . 5  |-  ( # `  ( ( 1 ... 0 )  \  C
) )  =  (
# `  (/) )
2120, 15eqtri 2472 . . . 4  |-  ( # `  ( ( 1 ... 0 )  \  C
) )  =  0
2216, 21oveq12i 6293 . . 3  |-  ( (
# `  ( (
1 ... 0 )  i^i 
C ) )  -  ( # `  ( ( 1 ... 0 ) 
\  C ) ) )  =  ( 0  -  0 )
23 0m0e0 10651 . . 3  |-  ( 0  -  0 )  =  0
2422, 23eqtri 2472 . 2  |-  ( (
# `  ( (
1 ... 0 )  i^i 
C ) )  -  ( # `  ( ( 1 ... 0 ) 
\  C ) ) )  =  0
258, 24syl6eq 2500 1  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   {crab 2797    \ cdif 3458    i^i cin 3460   (/)c0 3770   ~Pcpw 3997    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   0cc0 9495   1c1 9496    + caddc 9498    - cmin 9810    / cdiv 10212   NNcn 10542   ZZcz 10870   ...cfz 11681   #chash 12384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-hash 12385
This theorem is referenced by:  ballotlem4  28310  ballotlemi1  28314  ballotlemii  28315  ballotlemic  28318  ballotlem1c  28319
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