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Theorem ballotlemfmpn 23053
Description:  ( F `  C ) finishes counting at  ( M  -  N ). (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
ballotlemfmpn  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( M  -  N
) )
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 ballotlemfmpn
Dummy variable  b is distinct from all other variables.
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 nnaddcl 9768 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
81, 2, 7mp2an 653 . . . . . 6  |-  ( M  +  N )  e.  NN
98nnnn0i 9973 . . . . 5  |-  ( M  +  N )  e. 
NN0
109nn0zi 10048 . . . 4  |-  ( M  +  N )  e.  ZZ
1110a1i 10 . . 3  |-  ( C  e.  O  ->  ( M  +  N )  e.  ZZ )
121, 2, 3, 4, 5, 6, 11ballotlemfval 23048 . 2  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( ( # `  (
( 1 ... ( M  +  N )
)  i^i  C )
)  -  ( # `  ( ( 1 ... ( M  +  N
) )  \  C
) ) ) )
13 ssrab2 3258 . . . . . . . . 9  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
~P ( 1 ... ( M  +  N
) )
143, 13eqsstri 3208 . . . . . . . 8  |-  O  C_  ~P ( 1 ... ( M  +  N )
)
1514sseli 3176 . . . . . . 7  |-  ( C  e.  O  ->  C  e.  ~P ( 1 ... ( M  +  N
) ) )
16 elpwg 3632 . . . . . . 7  |-  ( C  e.  O  ->  ( C  e.  ~P (
1 ... ( M  +  N ) )  <->  C  C_  (
1 ... ( M  +  N ) ) ) )
1715, 16mpbid 201 . . . . . 6  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
18 df-ss 3166 . . . . . . 7  |-  ( C 
C_  ( 1 ... ( M  +  N
) )  <->  ( C  i^i  ( 1 ... ( M  +  N )
) )  =  C )
19 incom 3361 . . . . . . . 8  |-  ( C  i^i  ( 1 ... ( M  +  N
) ) )  =  ( ( 1 ... ( M  +  N
) )  i^i  C
)
2019eqeq1i 2290 . . . . . . 7  |-  ( ( C  i^i  ( 1 ... ( M  +  N ) ) )  =  C  <->  ( (
1 ... ( M  +  N ) )  i^i 
C )  =  C )
2118, 20bitri 240 . . . . . 6  |-  ( C 
C_  ( 1 ... ( M  +  N
) )  <->  ( (
1 ... ( M  +  N ) )  i^i 
C )  =  C )
2217, 21sylib 188 . . . . 5  |-  ( C  e.  O  ->  (
( 1 ... ( M  +  N )
)  i^i  C )  =  C )
2322fveq2d 5529 . . . 4  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  i^i 
C ) )  =  ( # `  C
) )
243eleq2i 2347 . . . . . 6  |-  ( C  e.  O  <->  C  e.  { c  e.  ~P (
1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }
)
25 ssv 3198 . . . . . . . . 9  |-  ~P (
1 ... ( M  +  N ) )  C_  _V
26 rabss2 3256 . . . . . . . . 9  |-  ( ~P ( 1 ... ( M  +  N )
)  C_  _V  ->  { c  e.  ~P (
1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
{ c  e.  _V  |  ( # `  c
)  =  M }
)
2725, 26ax-mp 8 . . . . . . . 8  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
{ c  e.  _V  |  ( # `  c
)  =  M }
28 rabab 2805 . . . . . . . 8  |-  { c  e.  _V  |  (
# `  c )  =  M }  =  {
c  |  ( # `  c )  =  M }
2927, 28sseqtri 3210 . . . . . . 7  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
{ c  |  (
# `  c )  =  M }
3029sseli 3176 . . . . . 6  |-  ( C  e.  { c  e. 
~P ( 1 ... ( M  +  N
) )  |  (
# `  c )  =  M }  ->  C  e.  { c  |  (
# `  c )  =  M } )
3124, 30sylbi 187 . . . . 5  |-  ( C  e.  O  ->  C  e.  { c  |  (
# `  c )  =  M } )
32 fveq2 5525 . . . . . . . . 9  |-  ( c  =  b  ->  ( # `
 c )  =  ( # `  b
) )
3332eqeq1d 2291 . . . . . . . 8  |-  ( c  =  b  ->  (
( # `  c )  =  M  <->  ( # `  b
)  =  M ) )
3433cbvabv 2402 . . . . . . 7  |-  { c  |  ( # `  c
)  =  M }  =  { b  |  (
# `  b )  =  M }
3534eleq2i 2347 . . . . . 6  |-  ( C  e.  { c  |  ( # `  c
)  =  M }  <->  C  e.  { b  |  ( # `  b
)  =  M }
)
36 fveq2 5525 . . . . . . . 8  |-  ( b  =  C  ->  ( # `
 b )  =  ( # `  C
) )
3736eqeq1d 2291 . . . . . . 7  |-  ( b  =  C  ->  (
( # `  b )  =  M  <->  ( # `  C
)  =  M ) )
3837elabg 2915 . . . . . 6  |-  ( C  e.  O  ->  ( C  e.  { b  |  ( # `  b
)  =  M }  <->  (
# `  C )  =  M ) )
3935, 38syl5bb 248 . . . . 5  |-  ( C  e.  O  ->  ( C  e.  { c  |  ( # `  c
)  =  M }  <->  (
# `  C )  =  M ) )
4031, 39mpbid 201 . . . 4  |-  ( C  e.  O  ->  ( # `
 C )  =  M )
4123, 40eqtrd 2315 . . 3  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  i^i 
C ) )  =  M )
42 fzfi 11034 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
4317, 42jctil 523 . . . . 5  |-  ( C  e.  O  ->  (
( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) ) )
44 hashssdif 11374 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  -> 
( # `  ( ( 1 ... ( M  +  N ) ) 
\  C ) )  =  ( ( # `  ( 1 ... ( M  +  N )
) )  -  ( # `
 C ) ) )
4543, 44syl 15 . . . 4  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  \  C ) )  =  ( ( # `  (
1 ... ( M  +  N ) ) )  -  ( # `  C
) ) )
46 hashfz1 11345 . . . . . . . 8  |-  ( ( M  +  N )  e.  NN0  ->  ( # `  ( 1 ... ( M  +  N )
) )  =  ( M  +  N ) )
479, 46ax-mp 8 . . . . . . 7  |-  ( # `  ( 1 ... ( M  +  N )
) )  =  ( M  +  N )
4847a1i 10 . . . . . 6  |-  ( C  e.  O  ->  ( # `
 ( 1 ... ( M  +  N
) ) )  =  ( M  +  N
) )
4948, 40oveq12d 5876 . . . . 5  |-  ( C  e.  O  ->  (
( # `  ( 1 ... ( M  +  N ) ) )  -  ( # `  C
) )  =  ( ( M  +  N
)  -  M ) )
501nncni 9756 . . . . . . 7  |-  M  e.  CC
512nncni 9756 . . . . . . 7  |-  N  e.  CC
52 pncan2 9058 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
5350, 51, 52mp2an 653 . . . . . 6  |-  ( ( M  +  N )  -  M )  =  N
5453a1i 10 . . . . 5  |-  ( C  e.  O  ->  (
( M  +  N
)  -  M )  =  N )
5549, 54eqtrd 2315 . . . 4  |-  ( C  e.  O  ->  (
( # `  ( 1 ... ( M  +  N ) ) )  -  ( # `  C
) )  =  N )
5645, 55eqtrd 2315 . . 3  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  \  C ) )  =  N )
5741, 56oveq12d 5876 . 2  |-  ( C  e.  O  ->  (
( # `  ( ( 1 ... ( M  +  N ) )  i^i  C ) )  -  ( # `  (
( 1 ... ( M  +  N )
)  \  C )
) )  =  ( M  -  N ) )
5812, 57eqtrd 2315 1  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( M  -  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   1c1 8738    + caddc 8740    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlem5  23058
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-rmo 2551  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-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794  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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