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Theorem ballotlem1ri 23093
Description: When the vote on the first tie is for A, the first vote is also for A on the reverse counting. (Contributed by Thierry Arnoux, 18-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
) )
Assertion
Ref Expression
ballotlem1ri  |-  ( C  e.  ( O  \  E )  ->  (
1  e.  ( R `
 C )  <->  ( I `  C )  e.  C
) )
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, k    x, c   
x, C    x, F    x, M    x, N
Allowed substitution hints:    C( c)    P( x, i, k, c)    R( x, c)    S( x)    E( x)    I( x)    O( x)

Proof of Theorem ballotlem1ri
StepHypRef Expression
1 ballotth.m . . . . . 6  |-  M  e.  NN
2 ballotth.n . . . . . 6  |-  N  e.  NN
3 nnaddcl 9768 . . . . . 6  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
41, 2, 3mp2an 653 . . . . 5  |-  ( M  +  N )  e.  NN
5 nnuz 10263 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
64, 5eleqtri 2355 . . . 4  |-  ( M  +  N )  e.  ( ZZ>= `  1 )
7 eluzfz1 10803 . . . 4  |-  ( ( M  +  N )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( M  +  N )
) )
86, 7mp1i 11 . . 3  |-  ( C  e.  ( O  \  E )  ->  1  e.  ( 1 ... ( M  +  N )
) )
9 ballotth.o . . . . . 6  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
10 ballotth.p . . . . . 6  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
11 ballotth.f . . . . . 6  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
12 ballotth.e . . . . . 6  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
13 ballotth.mgtn . . . . . 6  |-  N  < 
M
14 ballotth.i . . . . . 6  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
151, 2, 9, 10, 11, 12, 13, 14ballotlemiex 23060 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1615simpld 445 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
17 elfzle1 10799 . . . 4  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  1  <_  ( I `  C
) )
1816, 17syl 15 . . 3  |-  ( C  e.  ( O  \  E )  ->  1  <_  ( I `  C
) )
19 ballotth.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
20 ballotth.r . . . 4  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
211, 2, 9, 10, 11, 12, 13, 14, 19, 20ballotlemrv1 23079 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  ( 1 ... ( M  +  N ) )  /\  1  <_  ( I `  C ) )  -> 
( 1  e.  ( R `  C )  <-> 
( ( ( I `
 C )  +  1 )  -  1 )  e.  C ) )
228, 18, 21mpd3an23 1279 . 2  |-  ( C  e.  ( O  \  E )  ->  (
1  e.  ( R `
 C )  <->  ( (
( I `  C
)  +  1 )  -  1 )  e.  C ) )
23 elfzelz 10798 . . . . . 6  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ZZ )
2416, 23syl 15 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
2524zcnd 10118 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  CC )
26 ax-1cn 8795 . . . . 5  |-  1  e.  CC
2726a1i 10 . . . 4  |-  ( C  e.  ( O  \  E )  ->  1  e.  CC )
2825, 27pncand 9158 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( ( I `  C )  +  1 )  -  1 )  =  ( I `  C ) )
2928eleq1d 2349 . 2  |-  ( C  e.  ( O  \  E )  ->  (
( ( ( I `
 C )  +  1 )  -  1 )  e.  C  <->  ( I `  C )  e.  C
) )
3022, 29bitrd 244 1  |-  ( C  e.  ( O  \  E )  ->  (
1  e.  ( R `
 C )  <->  ( I `  C )  e.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151   ifcif 3565   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlem7  23094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-hash 11338
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