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Theorem ballotlemfval0 23054
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 19 . . 3  |-  ( C  e.  O  ->  C  e.  O )
7 0z 10035 . . . 4  |-  0  e.  ZZ
87a1i 10 . . 3  |-  ( C  e.  O  ->  0  e.  ZZ )
91, 2, 3, 4, 5, 6, 8ballotlemfval 23048 . 2  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  ( ( # `  ( ( 1 ... 0 )  i^i  C
) )  -  ( # `
 ( ( 1 ... 0 )  \  C ) ) ) )
10 fz10 10814 . . . . . . . 8  |-  ( 1 ... 0 )  =  (/)
1110ineq1i 3366 . . . . . . 7  |-  ( ( 1 ... 0 )  i^i  C )  =  ( (/)  i^i  C )
12 incom 3361 . . . . . . . 8  |-  ( C  i^i  (/) )  =  (
(/)  i^i  C )
13 in0 3480 . . . . . . . 8  |-  ( C  i^i  (/) )  =  (/)
1412, 13eqtr3i 2305 . . . . . . 7  |-  ( (/)  i^i 
C )  =  (/)
1511, 14eqtri 2303 . . . . . 6  |-  ( ( 1 ... 0 )  i^i  C )  =  (/)
1615fveq2i 5528 . . . . 5  |-  ( # `  ( ( 1 ... 0 )  i^i  C
) )  =  (
# `  (/) )
17 hash0 11355 . . . . 5  |-  ( # `  (/) )  =  0
1816, 17eqtri 2303 . . . 4  |-  ( # `  ( ( 1 ... 0 )  i^i  C
) )  =  0
1910difeq1i 3290 . . . . . . 7  |-  ( ( 1 ... 0 ) 
\  C )  =  ( (/)  \  C )
20 0dif 3525 . . . . . . 7  |-  ( (/)  \  C )  =  (/)
2119, 20eqtri 2303 . . . . . 6  |-  ( ( 1 ... 0 ) 
\  C )  =  (/)
2221fveq2i 5528 . . . . 5  |-  ( # `  ( ( 1 ... 0 )  \  C
) )  =  (
# `  (/) )
2322, 17eqtri 2303 . . . 4  |-  ( # `  ( ( 1 ... 0 )  \  C
) )  =  0
2418, 23oveq12i 5870 . . 3  |-  ( (
# `  ( (
1 ... 0 )  i^i 
C ) )  -  ( # `  ( ( 1 ... 0 ) 
\  C ) ) )  =  ( 0  -  0 )
25 0cn 8831 . . . 4  |-  0  e.  CC
2625subidi 9117 . . 3  |-  ( 0  -  0 )  =  0
2724, 26eqtri 2303 . 2  |-  ( (
# `  ( (
1 ... 0 )  i^i 
C ) )  -  ( # `  ( ( 1 ... 0 ) 
\  C ) ) )  =  0
289, 27syl6eq 2331 1  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547    \ cdif 3149    i^i cin 3151   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlem4  23057  ballotlemi1  23061  ballotlemii  23062  ballotlemic  23065  ballotlem1c  23066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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