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Theorem ballotlemfrci 24746
Description: Reverse counting preserves a tie at the first tie. (Contributed by Thierry Arnoux, 21-Apr-2017.)
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
) ) ) ) )
ballotth.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
ballotth.mgtn  |-  N  < 
M
ballotth.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
ballotth.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
ballotth.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
ballotlemg  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
Assertion
Ref Expression
ballotlemfrci  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O    k, M    k, N    k, O    i, c, F, k    C, i, k    i, E, k    C, k    k, I, c    E, c    i, I, c    S, k, i, c    R, i    v, u, C   
u, I, v    u, R, v    u, S, v
Allowed substitution hints:    C( x, c)    P( x, v, u, i, k, c)    R( x, k, c)    S( x)    E( x, v, u)    .^ ( x, v, u, i, k, c)    F( x, v, u)    I( x)    M( x, v, u)    N( x, v, u)    O( x, v, u)

Proof of Theorem ballotlemfrci
StepHypRef Expression
1 ballotth.m . . . . . . 7  |-  M  e.  NN
2 ballotth.n . . . . . . 7  |-  N  e.  NN
3 ballotth.o . . . . . . 7  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotth.p . . . . . . 7  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotth.f . . . . . . 7  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 ballotth.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
7 ballotth.mgtn . . . . . . 7  |-  N  < 
M
8 ballotth.i . . . . . . 7  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
91, 2, 3, 4, 5, 6, 7, 8ballotlemiex 24720 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
109simpld 446 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
11 elfzuz 11019 . . . . 5  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ( ZZ>= `  1 )
)
12 eluzfz2 11029 . . . . 5  |-  ( ( I `  C )  e.  ( ZZ>= `  1
)  ->  ( I `  C )  e.  ( 1 ... ( I `
 C ) ) )
1310, 11, 123syl 19 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... (
I `  C )
) )
14 ballotth.s . . . . 5  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
15 ballotth.r . . . . 5  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
16 ballotlemg . . . . 5  |-  .^  =  ( u  e.  Fin ,  v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
171, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16ballotlemfrc 24745 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  ( I `  C
)  e.  ( 1 ... ( I `  C ) ) )  ->  ( ( F `
 ( R `  C ) ) `  ( I `  C
) )  =  ( C  .^  ( (
( S `  C
) `  ( I `  C ) ) ... ( I `  C
) ) ) )
1813, 17mpdan 650 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  ( C  .^  (
( ( S `  C ) `  (
I `  C )
) ... ( I `  C ) ) ) )
191, 2, 3, 4, 5, 6, 7, 8, 14ballotlemsi 24733 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) `  ( I `  C ) )  =  1 )
2019oveq1d 6063 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
( ( S `  C ) `  (
I `  C )
) ... ( I `  C ) )  =  ( 1 ... (
I `  C )
) )
2120oveq2d 6064 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( C  .^  ( ( ( S `  C ) `
 ( I `  C ) ) ... ( I `  C
) ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
2218, 21eqtrd 2444 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
23 1nn0 10201 . . . . . 6  |-  1  e.  NN0
24 nn0uz 10484 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
2523, 24eleqtri 2484 . . . . 5  |-  1  e.  ( ZZ>= `  0 )
26 fzss1 11055 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( M  +  N ) )  C_  ( 0 ... ( M  +  N )
) )
2725, 26ax-mp 8 . . . 4  |-  ( 1 ... ( M  +  N ) )  C_  ( 0 ... ( M  +  N )
)
2827, 10sseldi 3314 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 0 ... ( M  +  N )
) )
291, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16ballotlemfg 24744 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  ( I `  C
)  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  ( I `  C
) )  =  ( C  .^  ( 1 ... ( I `  C ) ) ) )
3028, 29mpdan 650 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
319simprd 450 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
3222, 30, 313eqtr2d 2450 1  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678    \ cdif 3285    i^i cin 3287    C_ wss 3288   ifcif 3707   ~Pcpw 3767   class class class wbr 4180    e. cmpt 4234   `'ccnv 4844   "cima 4848   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   Fincfn 7076   supcsup 7411   RRcr 8953   0cc0 8954   1c1 8955    + caddc 8957    < clt 9084    <_ cle 9085    - cmin 9255    / cdiv 9641   NNcn 9964   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007   #chash 11581
This theorem is referenced by:  ballotlemrc  24749  ballotlemirc  24750
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-hash 11582
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