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Theorem ballotlemfrci 27049
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 27023 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
109simpld 459 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
11 elfzuz 11561 . . . . 5  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ( ZZ>= `  1 )
)
12 eluzfz2 11571 . . . . 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 27048 . . . 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 668 . . 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 27036 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) `  ( I `  C ) )  =  1 )
2019oveq1d 6210 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
( ( S `  C ) `  (
I `  C )
) ... ( I `  C ) )  =  ( 1 ... (
I `  C )
) )
2120oveq2d 6211 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( C  .^  ( ( ( S `  C ) `
 ( I `  C ) ) ... ( I `  C
) ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
2218, 21eqtrd 2493 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
23 1nn0 10701 . . . . . 6  |-  1  e.  NN0
24 nn0uz 11001 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
2523, 24eleqtri 2538 . . . . 5  |-  1  e.  ( ZZ>= `  0 )
26 fzss1 11609 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( M  +  N ) )  C_  ( 0 ... ( M  +  N )
) )
2725, 26ax-mp 5 . . . 4  |-  ( 1 ... ( M  +  N ) )  C_  ( 0 ... ( M  +  N )
)
2827, 10sseldi 3457 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 0 ... ( M  +  N )
) )
291, 2, 3, 4, 5, 6, 7, 8, 14, 15, 16ballotlemfg 27047 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  ( I `  C
)  e.  ( 0 ... ( M  +  N ) ) )  ->  ( ( F `
 C ) `  ( I `  C
) )  =  ( C  .^  ( 1 ... ( I `  C ) ) ) )
3028, 29mpdan 668 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  ( C  .^  (
1 ... ( I `  C ) ) ) )
319simprd 463 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
3222, 30, 313eqtr2d 2499 1  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   A.wral 2796   {crab 2800    \ cdif 3428    i^i cin 3430    C_ wss 3431   ifcif 3894   ~Pcpw 3963   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4942   "cima 4946   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   Fincfn 7415   supcsup 7796   RRcr 9387   0cc0 9388   1c1 9389    + caddc 9391    < clt 9524    <_ cle 9525    - cmin 9701    / cdiv 10099   NNcn 10428   NN0cn0 10685   ZZcz 10752   ZZ>=cuz 10967   ...cfz 11549   #chash 12215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-hash 12216
This theorem is referenced by:  ballotlemrc  27052  ballotlemirc  27053
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