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Theorem ballotlemfval0 28074
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 10872 . . 3  |-  ( C  e.  O  ->  0  e.  ZZ )
81, 2, 3, 4, 5, 6, 7ballotlemfval 28068 . 2  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  ( ( # `  ( ( 1 ... 0 )  i^i  C
) )  -  ( # `
 ( ( 1 ... 0 )  \  C ) ) ) )
9 fz10 11702 . . . . . . . 8  |-  ( 1 ... 0 )  =  (/)
109ineq1i 3696 . . . . . . 7  |-  ( ( 1 ... 0 )  i^i  C )  =  ( (/)  i^i  C )
11 incom 3691 . . . . . . 7  |-  ( C  i^i  (/) )  =  (
(/)  i^i  C )
12 in0 3811 . . . . . . 7  |-  ( C  i^i  (/) )  =  (/)
1310, 11, 123eqtr2i 2502 . . . . . 6  |-  ( ( 1 ... 0 )  i^i  C )  =  (/)
1413fveq2i 5867 . . . . 5  |-  ( # `  ( ( 1 ... 0 )  i^i  C
) )  =  (
# `  (/) )
15 hash0 12401 . . . . 5  |-  ( # `  (/) )  =  0
1614, 15eqtri 2496 . . . 4  |-  ( # `  ( ( 1 ... 0 )  i^i  C
) )  =  0
179difeq1i 3618 . . . . . . 7  |-  ( ( 1 ... 0 ) 
\  C )  =  ( (/)  \  C )
18 0dif 3898 . . . . . . 7  |-  ( (/)  \  C )  =  (/)
1917, 18eqtri 2496 . . . . . 6  |-  ( ( 1 ... 0 ) 
\  C )  =  (/)
2019fveq2i 5867 . . . . 5  |-  ( # `  ( ( 1 ... 0 )  \  C
) )  =  (
# `  (/) )
2120, 15eqtri 2496 . . . 4  |-  ( # `  ( ( 1 ... 0 )  \  C
) )  =  0
2216, 21oveq12i 6294 . . 3  |-  ( (
# `  ( (
1 ... 0 )  i^i 
C ) )  -  ( # `  ( ( 1 ... 0 ) 
\  C ) ) )  =  ( 0  -  0 )
23 0m0e0 10641 . . 3  |-  ( 0  -  0 )  =  0
2422, 23eqtri 2496 . 2  |-  ( (
# `  ( (
1 ... 0 )  i^i 
C ) )  -  ( # `  ( ( 1 ... 0 ) 
\  C ) ) )  =  0
258, 24syl6eq 2524 1  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818    \ cdif 3473    i^i cin 3475   (/)c0 3785   ~Pcpw 4010    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801    / cdiv 10202   NNcn 10532   ZZcz 10860   ...cfz 11668   #chash 12369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370
This theorem is referenced by:  ballotlem4  28077  ballotlemi1  28081  ballotlemii  28082  ballotlemic  28085  ballotlem1c  28086
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