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Theorem ballotlemfrci 28730
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 28704 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
109simpld 457 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
11 elfzuz 11687 . . . . 5  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ( ZZ>= `  1 )
)
12 eluzfz2 11697 . . . . 5  |-  ( ( I `  C )  e.  ( ZZ>= `  1
)  ->  ( I `  C )  e.  ( 1 ... ( I `
 C ) ) )
1310, 11, 123syl 20 . . . 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 28729 . . . 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 666 . . 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 28717 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) `  ( I `  C ) )  =  1 )
2019oveq1d 6285 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
( ( S `  C ) `  (
I `  C )
) ... ( I `  C ) )  =  ( 1 ... (
I `  C )
) )
2120oveq2d 6286 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( C  .^  ( ( ( S `  C ) `
 ( I `  C ) ) ... ( I `  C
) ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
2218, 21eqtrd 2495 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
23 1eluzge0 11125 . . . . 5  |-  1  e.  ( ZZ>= `  0 )
24 fzss1 11726 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( M  +  N ) )  C_  ( 0 ... ( M  +  N )
) )
2523, 24ax-mp 5 . . . 4  |-  ( 1 ... ( M  +  N ) )  C_  ( 0 ... ( M  +  N )
)
2625, 10sseldi 3487 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 0 ... ( M  +  N )
) )
271, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16ballotlemfg 28728 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  ( I `  C
)  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  ( I `  C
) )  =  ( C  .^  ( 1 ... ( I `  C ) ) ) )
2826, 27mpdan 666 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
299simprd 461 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
3022, 28, 293eqtr2d 2501 1  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    \ cdif 3458    i^i cin 3460    C_ wss 3461   ifcif 3929   ~Pcpw 3999   class class class wbr 4439    |-> cmpt 4497   `'ccnv 4987   "cima 4991   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Fincfn 7509   supcsup 7892   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-hash 12388
This theorem is referenced by:  ballotlemrc  28733  ballotlemirc  28734
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