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Theorem ballotlem4 23073
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 9313 . . . . . . . . . 10  |-  0  <_  1
2 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
3 1re 8853 . . . . . . . . . . 11  |-  1  e.  RR
42, 3lenlti 8954 . . . . . . . . . 10  |-  ( 0  <_  1  <->  -.  1  <  0 )
51, 4mpbi 199 . . . . . . . . 9  |-  -.  1  <  0
6 ltsub13 9271 . . . . . . . . . . . 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 8847 . . . . . . . . . . . . 13  |-  0  e.  CC
98subidi 9133 . . . . . . . . . . . 12  |-  ( 0  -  0 )  =  0
109breq2i 4047 . . . . . . . . . . 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 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
1514subidi 9133 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
1615fveq2i 5544 . . . . . . . . . . 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 23070 . . . . . . . . . . 11  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2316, 22syl5eq 2340 . . . . . . . . . 10  |-  ( C  e.  O  ->  (
( F `  C
) `  ( 1  -  1 ) )  =  0 )
2423oveq1d 5889 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( ( F `  C ) `  (
1  -  1 ) )  -  1 )  =  ( 0  -  1 ) )
2524breq2d 4051 . . . . . . . 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 9784 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
2917, 18, 28mp2an 653 . . . . . . . . . . . 12  |-  ( M  +  N )  e.  NN
30 elnnuz 10280 . . . . . . . . . . . 12  |-  ( ( M  +  N )  e.  NN  <->  ( M  +  N )  e.  (
ZZ>= `  1 ) )
3129, 30mpbi 199 . . . . . . . . . . 11  |-  ( M  +  N )  e.  ( ZZ>= `  1 )
32 eluzfz1 10819 . . . . . . . . . . 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 9773 . . . . . . . . . . . 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 23066 . . . . . . . . . 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 4051 . . . . . 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 5541 . . . . . . . . 9  |-  ( i  =  1  ->  (
( F `  C
) `  i )  =  ( ( F `
 C ) ` 
1 ) )
4544breq2d 4051 . . . . . . . 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 2897 . . . . . 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 2567 . . . 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 23071 . . . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    i^i cin 3164   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotth  23112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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