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Theorem ballotlemi1 23061
Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-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 ,  `'  <  ) )
Assertion
Ref Expression
ballotlemi1  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( I `  C )  =/=  1
)
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   
k, c, E    i, I
Allowed substitution hints:    C( x, c)    P( x, i, k, c)    E( x)    F( x)    I( x, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemi1
StepHypRef Expression
1 notnot 282 . . . 4  |-  ( ( I `  C )  =  1  <->  -.  -.  (
I `  C )  =  1 )
2 ballotth.m . . . . . . . 8  |-  M  e.  NN
3 ballotth.n . . . . . . . 8  |-  N  e.  NN
4 ballotth.o . . . . . . . 8  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
5 ballotth.p . . . . . . . 8  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
6 ballotth.f . . . . . . . 8  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
7 ballotth.e . . . . . . . 8  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
8 ballotth.mgtn . . . . . . . 8  |-  N  < 
M
9 ballotth.i . . . . . . . 8  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
102, 3, 4, 5, 6, 7, 8, 9ballotlemiex 23060 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1110simprd 449 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
1211ad2antrr 706 . . . . 5  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  (
I `  C )
)  =  0 )
13 fveq2 5525 . . . . . . 7  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
1413eqeq1d 2291 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
1514adantl 452 . . . . 5  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( ( F `
 C ) `  ( I `  C
) )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
1612, 15mpbid 201 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  1
)  =  0 )
171, 16sylan2br 462 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  -.  -.  ( I `  C
)  =  1 )  ->  ( ( F `
 C ) ` 
1 )  =  0 )
18 0lt1 9296 . . . . . . . . 9  |-  0  <  1
19 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
20 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
21 ltsub23 9254 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  e.  RR )  ->  (
( 0  -  1 )  <  0  <->  (
0  -  0 )  <  1 ) )
2219, 20, 19, 21mp3an 1277 . . . . . . . . . 10  |-  ( ( 0  -  1 )  <  0  <->  ( 0  -  0 )  <  1 )
23 0cn 8831 . . . . . . . . . . . 12  |-  0  e.  CC
2423subidi 9117 . . . . . . . . . . 11  |-  ( 0  -  0 )  =  0
2524breq1i 4030 . . . . . . . . . 10  |-  ( ( 0  -  0 )  <  1  <->  0  <  1 )
2622, 25bitr2i 241 . . . . . . . . 9  |-  ( 0  <  1  <->  ( 0  -  1 )  <  0 )
2718, 26mpbi 199 . . . . . . . 8  |-  ( 0  -  1 )  <  0
2819, 20resubcli 9109 . . . . . . . . 9  |-  ( 0  -  1 )  e.  RR
2928, 19ltnei 8943 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  ->  0  =/=  ( 0  -  1 ) )
3027, 29ax-mp 8 . . . . . . 7  |-  0  =/=  ( 0  -  1 )
31 eqcom 2285 . . . . . . . 8  |-  ( 0  =  ( 0  -  1 )  <->  ( 0  -  1 )  =  0 )
3231necon3abii 2476 . . . . . . 7  |-  ( 0  =/=  ( 0  -  1 )  <->  -.  (
0  -  1 )  =  0 )
3330, 32mpbi 199 . . . . . 6  |-  -.  (
0  -  1 )  =  0
34 eldifi 3298 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
35 1nn 9757 . . . . . . . . . . . . 13  |-  1  e.  NN
3635a1i 10 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
372, 3, 4, 5, 6, 34, 36ballotlemfp1 23050 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( -.  1  e.  C  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )  /\  (
1  e.  C  -> 
( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  +  1 ) ) ) )
3837simpld 445 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
3938imp 418 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
40 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
4140subidi 9117 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
4241fveq2i 5528 . . . . . . . . . . 11  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
4342oveq1i 5868 . . . . . . . . . 10  |-  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 )
4443a1i 10 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 ) )
452, 3, 4, 5, 6ballotlemfval0 23054 . . . . . . . . . . . 12  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
4634, 45syl 15 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
4746adantr 451 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  0 )  =  0 )
4847oveq1d 5873 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  0 )  -  1 )  =  ( 0  -  1 ) )
4939, 44, 483eqtrrd 2320 . . . . . . . 8  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( 0  -  1 )  =  ( ( F `  C ) `  1
) )
5049eqeq1d 2291 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
0  -  1 )  =  0  <->  ( ( F `  C ) `  1 )  =  0 ) )
5150notbid 285 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( -.  ( 0  -  1 )  =  0  <->  -.  ( ( F `  C ) `  1
)  =  0 ) )
5233, 51mpbii 202 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
( F `  C
) `  1 )  =  0 )
5352adantr 451 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  ->  -.  ( ( F `  C ) `  1
)  =  0 )
541, 53sylan2br 462 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  -.  -.  ( I `  C
)  =  1 )  ->  -.  ( ( F `  C ) `  1 )  =  0 )
5517, 54condan 769 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
I `  C )  =  1 )
56 df-ne 2448 . 2  |-  ( ( I `  C )  =/=  1  <->  -.  (
I `  C )  =  1 )
5755, 56sylibr 203 1  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( I `  C )  =/=  1
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    \ cdif 3149    i^i cin 3151   ~Pcpw 3625   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   ZZcz 10024   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemic  23065
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-fz 10783  df-hash 11338
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