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Theorem ballotlemfmpn 23069
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 9784 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
81, 2, 7mp2an 653 . . . . . 6  |-  ( M  +  N )  e.  NN
98nnnn0i 9989 . . . . 5  |-  ( M  +  N )  e. 
NN0
109nn0zi 10064 . . . 4  |-  ( M  +  N )  e.  ZZ
1110a1i 10 . . 3  |-  ( C  e.  O  ->  ( M  +  N )  e.  ZZ )
121, 2, 3, 4, 5, 6, 11ballotlemfval 23064 . 2  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( ( # `  (
( 1 ... ( M  +  N )
)  i^i  C )
)  -  ( # `  ( ( 1 ... ( M  +  N
) )  \  C
) ) ) )
13 ssrab2 3271 . . . . . . . . 9  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
~P ( 1 ... ( M  +  N
) )
143, 13eqsstri 3221 . . . . . . . 8  |-  O  C_  ~P ( 1 ... ( M  +  N )
)
1514sseli 3189 . . . . . . 7  |-  ( C  e.  O  ->  C  e.  ~P ( 1 ... ( M  +  N
) ) )
16 elpwg 3645 . . . . . . 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 3179 . . . . . . 7  |-  ( C 
C_  ( 1 ... ( M  +  N
) )  <->  ( C  i^i  ( 1 ... ( M  +  N )
) )  =  C )
19 incom 3374 . . . . . . . 8  |-  ( C  i^i  ( 1 ... ( M  +  N
) ) )  =  ( ( 1 ... ( M  +  N
) )  i^i  C
)
2019eqeq1i 2303 . . . . . . 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 5545 . . . 4  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  i^i 
C ) )  =  ( # `  C
) )
243eleq2i 2360 . . . . . 6  |-  ( C  e.  O  <->  C  e.  { c  e.  ~P (
1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }
)
25 ssv 3211 . . . . . . . . 9  |-  ~P (
1 ... ( M  +  N ) )  C_  _V
26 rabss2 3269 . . . . . . . . 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 2818 . . . . . . . 8  |-  { c  e.  _V  |  (
# `  c )  =  M }  =  {
c  |  ( # `  c )  =  M }
2927, 28sseqtri 3223 . . . . . . 7  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  C_ 
{ c  |  (
# `  c )  =  M }
3029sseli 3189 . . . . . 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 5541 . . . . . . . . 9  |-  ( c  =  b  ->  ( # `
 c )  =  ( # `  b
) )
3332eqeq1d 2304 . . . . . . . 8  |-  ( c  =  b  ->  (
( # `  c )  =  M  <->  ( # `  b
)  =  M ) )
3433cbvabv 2415 . . . . . . 7  |-  { c  |  ( # `  c
)  =  M }  =  { b  |  (
# `  b )  =  M }
3534eleq2i 2360 . . . . . 6  |-  ( C  e.  { c  |  ( # `  c
)  =  M }  <->  C  e.  { b  |  ( # `  b
)  =  M }
)
36 fveq2 5541 . . . . . . . 8  |-  ( b  =  C  ->  ( # `
 b )  =  ( # `  C
) )
3736eqeq1d 2304 . . . . . . 7  |-  ( b  =  C  ->  (
( # `  b )  =  M  <->  ( # `  C
)  =  M ) )
3837elabg 2928 . . . . . 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 2328 . . 3  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  i^i 
C ) )  =  M )
42 fzfi 11050 . . . . . 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 11390 . . . . 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 11361 . . . . . . . 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 5892 . . . . 5  |-  ( C  e.  O  ->  (
( # `  ( 1 ... ( M  +  N ) ) )  -  ( # `  C
) )  =  ( ( M  +  N
)  -  M ) )
501nncni 9772 . . . . . . 7  |-  M  e.  CC
512nncni 9772 . . . . . . 7  |-  N  e.  CC
52 pncan2 9074 . . . . . . 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 2328 . . . 4  |-  ( C  e.  O  ->  (
( # `  ( 1 ... ( M  +  N ) ) )  -  ( # `  C
) )  =  N )
5645, 55eqtrd 2328 . . 3  |-  ( C  e.  O  ->  ( # `
 ( ( 1 ... ( M  +  N ) )  \  C ) )  =  N )
5741, 56oveq12d 5892 . 2  |-  ( C  e.  O  ->  (
( # `  ( ( 1 ... ( M  +  N ) )  i^i  C ) )  -  ( # `  (
( 1 ... ( M  +  N )
)  \  C )
) )  =  ( M  -  N ) )
5812, 57eqtrd 2328 1  |-  ( C  e.  O  ->  (
( F `  C
) `  ( M  +  N ) )  =  ( M  -  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   1c1 8754    + caddc 8756    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotlem5  23074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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