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Theorem ballotlemfmpn 27923
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 22 . . 3  |-  ( C  e.  O  ->  C  e.  O )
7 nnaddcl 10547 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
81, 2, 7mp2an 672 . . . . 5  |-  ( M  +  N )  e.  NN
98nnzi 10877 . . . 4  |-  ( M  +  N )  e.  ZZ
109a1i 11 . . 3  |-  ( C  e.  O  ->  ( M  +  N )  e.  ZZ )
111, 2, 3, 4, 5, 6, 10ballotlemfval 27918 . 2  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( ( # `  (
( 1 ... ( M  +  N )
)  i^i  C )
)  -  ( # `  ( ( 1 ... ( M  +  N
) )  \  C
) ) ) )
12 ssrab2 3578 . . . . . . . . 9  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
~P ( 1 ... ( M  +  N
) )
133, 12eqsstri 3527 . . . . . . . 8  |-  O  C_  ~P ( 1 ... ( M  +  N )
)
1413sseli 3493 . . . . . . 7  |-  ( C  e.  O  ->  C  e.  ~P ( 1 ... ( M  +  N
) ) )
1514elpwid 4013 . . . . . 6  |-  ( C  e.  O  ->  C  C_  ( 1 ... ( M  +  N )
) )
16 dfss1 3696 . . . . . 6  |-  ( C 
C_  ( 1 ... ( M  +  N
) )  <->  ( (
1 ... ( M  +  N ) )  i^i 
C )  =  C )
1715, 16sylib 196 . . . . 5  |-  ( C  e.  O  ->  (
( 1 ... ( M  +  N )
)  i^i  C )  =  C )
1817fveq2d 5861 . . . 4  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  i^i 
C ) )  =  ( # `  C
) )
19 rabssab 3580 . . . . . . 7  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
{ c  |  (
# `  c )  =  M }
2019sseli 3493 . . . . . 6  |-  ( C  e.  { c  e. 
~P ( 1 ... ( M  +  N
) )  |  (
# `  c )  =  M }  ->  C  e.  { c  |  (
# `  c )  =  M } )
2120, 3eleq2s 2568 . . . . 5  |-  ( C  e.  O  ->  C  e.  { c  |  (
# `  c )  =  M } )
22 fveq2 5857 . . . . . . 7  |-  ( b  =  C  ->  ( # `
 b )  =  ( # `  C
) )
2322eqeq1d 2462 . . . . . 6  |-  ( b  =  C  ->  (
( # `  b )  =  M  <->  ( # `  C
)  =  M ) )
24 fveq2 5857 . . . . . . . 8  |-  ( c  =  b  ->  ( # `
 c )  =  ( # `  b
) )
2524eqeq1d 2462 . . . . . . 7  |-  ( c  =  b  ->  (
( # `  c )  =  M  <->  ( # `  b
)  =  M ) )
2625cbvabv 2603 . . . . . 6  |-  { c  |  ( # `  c
)  =  M }  =  { b  |  (
# `  b )  =  M }
2723, 26elab2g 3245 . . . . 5  |-  ( C  e.  O  ->  ( C  e.  { c  |  ( # `  c
)  =  M }  <->  (
# `  C )  =  M ) )
2821, 27mpbid 210 . . . 4  |-  ( C  e.  O  ->  ( # `
 C )  =  M )
2918, 28eqtrd 2501 . . 3  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  i^i 
C ) )  =  M )
30 fzfi 12038 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
31 hashssdif 12427 . . . . 5  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  C  C_  ( 1 ... ( M  +  N
) ) )  -> 
( # `  ( ( 1 ... ( M  +  N ) ) 
\  C ) )  =  ( ( # `  ( 1 ... ( M  +  N )
) )  -  ( # `
 C ) ) )
3230, 15, 31sylancr 663 . . . 4  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  \  C ) )  =  ( ( # `  (
1 ... ( M  +  N ) ) )  -  ( # `  C
) ) )
338nnnn0i 10792 . . . . . 6  |-  ( M  +  N )  e. 
NN0
34 hashfz1 12374 . . . . . 6  |-  ( ( M  +  N )  e.  NN0  ->  ( # `  ( 1 ... ( M  +  N )
) )  =  ( M  +  N ) )
3533, 34mp1i 12 . . . . 5  |-  ( C  e.  O  ->  ( # `
 ( 1 ... ( M  +  N
) ) )  =  ( M  +  N
) )
3635, 28oveq12d 6293 . . . 4  |-  ( C  e.  O  ->  (
( # `  ( 1 ... ( M  +  N ) ) )  -  ( # `  C
) )  =  ( ( M  +  N
)  -  M ) )
371nncni 10535 . . . . . 6  |-  M  e.  CC
382nncni 10535 . . . . . 6  |-  N  e.  CC
39 pncan2 9816 . . . . . 6  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  +  N )  -  M
)  =  N )
4037, 38, 39mp2an 672 . . . . 5  |-  ( ( M  +  N )  -  M )  =  N
4140a1i 11 . . . 4  |-  ( C  e.  O  ->  (
( M  +  N
)  -  M )  =  N )
4232, 36, 413eqtrd 2505 . . 3  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  \  C ) )  =  N )
4329, 42oveq12d 6293 . 2  |-  ( C  e.  O  ->  (
( # `  ( ( 1 ... ( M  +  N ) )  i^i  C ) )  -  ( # `  (
( 1 ... ( M  +  N )
)  \  C )
) )  =  ( M  -  N ) )
4411, 43eqtrd 2501 1  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( M  -  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   {cab 2445   {crab 2811    \ cdif 3466    i^i cin 3468    C_ wss 3469   ~Pcpw 4003    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Fincfn 7506   CCcc 9479   1c1 9482    + caddc 9484    - cmin 9794    / cdiv 10195   NNcn 10525   NN0cn0 10784   ZZcz 10853   ...cfz 11661   #chash 12360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-hash 12361
This theorem is referenced by:  ballotlem5  27928
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