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Theorem ballotlem4 23057
Description: If the first pick is a vote for B, A is not ahead throughout the count (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
) ) ) ) )
ballotth.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
Assertion
Ref Expression
ballotlem4  |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  -.  C  e.  E
) )
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)    E( x, i, c)    F( x)    M( x)    N( x)    O( x)

Proof of Theorem ballotlem4
StepHypRef Expression
1 0le1 9297 . . . . . . . . . 10  |-  0  <_  1
2 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
3 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
42, 3lenlti 8938 . . . . . . . . . 10  |-  ( 0  <_  1  <->  -.  1  <  0 )
51, 4mpbi 199 . . . . . . . . 9  |-  -.  1  <  0
6 ltsub13 9255 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  0  e.  RR  /\  1  e.  RR )  ->  (
0  <  ( 0  -  1 )  <->  1  <  ( 0  -  0 ) ) )
72, 2, 3, 6mp3an 1277 . . . . . . . . . . 11  |-  ( 0  <  ( 0  -  1 )  <->  1  <  ( 0  -  0 ) )
8 0cn 8831 . . . . . . . . . . . . 13  |-  0  e.  CC
98subidi 9117 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
109breq2i 4031 . . . . . . . . . . 11  |-  ( 1  <  ( 0  -  0 )  <->  1  <  0 )
117, 10bitri 240 . . . . . . . . . 10  |-  ( 0  <  ( 0  -  1 )  <->  1  <  0 )
1211notbii 287 . . . . . . . . 9  |-  ( -.  0  <  ( 0  -  1 )  <->  -.  1  <  0 )
135, 12mpbir 200 . . . . . . . 8  |-  -.  0  <  ( 0  -  1 )
14 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
1514subidi 9117 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
1615fveq2i 5528 . . . . . . . . . . 11  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
17 ballotth.m . . . . . . . . . . . 12  |-  M  e.  NN
18 ballotth.n . . . . . . . . . . . 12  |-  N  e.  NN
19 ballotth.o . . . . . . . . . . . 12  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
20 ballotth.p . . . . . . . . . . . 12  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
21 ballotth.f . . . . . . . . . . . 12  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
2217, 18, 19, 20, 21ballotlemfval0 23054 . . . . . . . . . . 11  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2316, 22syl5eq 2327 . . . . . . . . . 10  |-  ( C  e.  O  ->  (
( F `  C
) `  ( 1  -  1 ) )  =  0 )
2423oveq1d 5873 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( ( F `  C ) `  (
1  -  1 ) )  -  1 )  =  ( 0  -  1 ) )
2524breq2d 4035 . . . . . . . 8  |-  ( C  e.  O  ->  (
0  <  ( (
( F `  C
) `  ( 1  -  1 ) )  -  1 )  <->  0  <  ( 0  -  1 ) ) )
2613, 25mtbiri 294 . . . . . . 7  |-  ( C  e.  O  ->  -.  0  <  ( ( ( F `  C ) `
 ( 1  -  1 ) )  - 
1 ) )
2726adantr 451 . . . . . 6  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  0  <  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
28 nnaddcl 9768 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
2917, 18, 28mp2an 653 . . . . . . . . . . . 12  |-  ( M  +  N )  e.  NN
30 elnnuz 10264 . . . . . . . . . . . 12  |-  ( ( M  +  N )  e.  NN  <->  ( M  +  N )  e.  (
ZZ>= `  1 ) )
3129, 30mpbi 199 . . . . . . . . . . 11  |-  ( M  +  N )  e.  ( ZZ>= `  1 )
32 eluzfz1 10803 . . . . . . . . . . 11  |-  ( ( M  +  N )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( M  +  N )
) )
3331, 32ax-mp 8 . . . . . . . . . 10  |-  1  e.  ( 1 ... ( M  +  N )
)
3433jctr 526 . . . . . . . . 9  |-  ( C  e.  O  ->  ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) ) )
35 simpl 443 . . . . . . . . . . 11  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  C  e.  O
)
36 1nn 9757 . . . . . . . . . . . 12  |-  1  e.  NN
3736a1i 10 . . . . . . . . . . 11  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  1  e.  NN )
3817, 18, 19, 20, 21, 35, 37ballotlemfp1 23050 . . . . . . . . . 10  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( -.  1  e.  C  -> 
( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) )  /\  ( 1  e.  C  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  +  1 ) ) ) )
3938simpld 445 . . . . . . . . 9  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `
 1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) ) )
4034, 39syl 15 . . . . . . . 8  |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
4140imp 418 . . . . . . 7  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
4241breq2d 4035 . . . . . 6  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  ( 0  <  ( ( F `
 C ) ` 
1 )  <->  0  <  ( ( ( F `  C ) `  (
1  -  1 ) )  -  1 ) ) )
4327, 42mtbird 292 . . . . 5  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  0  <  ( ( F `  C ) `  1
) )
44 fveq2 5525 . . . . . . . . 9  |-  ( i  =  1  ->  (
( F `  C
) `  i )  =  ( ( F `
 C ) ` 
1 ) )
4544breq2d 4035 . . . . . . . 8  |-  ( i  =  1  ->  (
0  <  ( ( F `  C ) `  i )  <->  0  <  ( ( F `  C
) `  1 )
) )
4645notbid 285 . . . . . . 7  |-  ( i  =  1  ->  ( -.  0  <  ( ( F `  C ) `
 i )  <->  -.  0  <  ( ( F `  C ) `  1
) ) )
4746rspcev 2884 . . . . . 6  |-  ( ( 1  e.  ( 1 ... ( M  +  N ) )  /\  -.  0  <  ( ( F `  C ) `
 1 ) )  ->  E. i  e.  ( 1 ... ( M  +  N ) )  -.  0  <  (
( F `  C
) `  i )
)
4833, 47mpan 651 . . . . 5  |-  ( -.  0  <  ( ( F `  C ) `
 1 )  ->  E. i  e.  (
1 ... ( M  +  N ) )  -.  0  <  ( ( F `  C ) `
 i ) )
4943, 48syl 15 . . . 4  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  E. i  e.  ( 1 ... ( M  +  N )
)  -.  0  < 
( ( F `  C ) `  i
) )
50 rexnal 2554 . . . 4  |-  ( E. i  e.  ( 1 ... ( M  +  N ) )  -.  0  <  ( ( F `  C ) `
 i )  <->  -.  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
)
5149, 50sylib 188 . . 3  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
)
52 ballotth.e . . . . . 6  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
5317, 18, 19, 20, 21, 52ballotleme 23055 . . . . 5  |-  ( C  e.  E  <->  ( C  e.  O  /\  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
) )
5453simprbi 450 . . . 4  |-  ( C  e.  E  ->  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
)
5554con3i 127 . . 3  |-  ( -. 
A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  C ) `
 i )  ->  -.  C  e.  E
)
5651, 55syl 15 . 2  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  C  e.  E )
5756ex 423 1  |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  -.  C  e.  E
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    i^i cin 3151   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   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:  ballotth  23096
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-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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338
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