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Theorem ballotlemi1OLD 29373
Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.) Obsolete version of ballotlemi1 29335 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ballotthOLD.m  |-  M  e.  NN
ballotthOLD.n  |-  N  e.  NN
ballotthOLD.o  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
ballotthOLD.p  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
ballotthOLD.f  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
ballotthOLD.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
ballotthOLD.mgtn  |-  N  < 
M
ballotthOLD.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
Assertion
Ref Expression
ballotlemi1OLD  |-  ( ( 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    C, i, k    i,
c, F, k    i, E, 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 ballotlemi1OLD
StepHypRef Expression
1 0re 9643 . . . . . . 7  |-  0  e.  RR
2 1re 9642 . . . . . . 7  |-  1  e.  RR
31, 2resubcli 9936 . . . . . 6  |-  ( 0  -  1 )  e.  RR
4 0lt1 10136 . . . . . . 7  |-  0  <  1
5 ltsub23 10094 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  e.  RR )  ->  (
( 0  -  1 )  <  0  <->  (
0  -  0 )  <  1 ) )
61, 2, 1, 5mp3an 1364 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0  -  0 )  <  1 )
7 0m0e0 10719 . . . . . . . . 9  |-  ( 0  -  0 )  =  0
87breq1i 4409 . . . . . . . 8  |-  ( ( 0  -  0 )  <  1  <->  0  <  1 )
96, 8bitr2i 254 . . . . . . 7  |-  ( 0  <  1  <->  ( 0  -  1 )  <  0 )
104, 9mpbi 212 . . . . . 6  |-  ( 0  -  1 )  <  0
113, 10gtneii 9746 . . . . 5  |-  0  =/=  ( 0  -  1 )
1211nesymi 2681 . . . 4  |-  -.  (
0  -  1 )  =  0
13 ballotthOLD.m . . . . . . . . 9  |-  M  e.  NN
14 ballotthOLD.n . . . . . . . . 9  |-  N  e.  NN
15 ballotthOLD.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
16 ballotthOLD.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
17 ballotthOLD.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
18 eldifi 3555 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
19 1nn 10620 . . . . . . . . . 10  |-  1  e.  NN
2019a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
2113, 14, 15, 16, 17, 18, 20ballotlemfp1 29324 . . . . . . . 8  |-  ( 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 ) ) ) )
2221simpld 461 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
2322imp 431 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
24 1m1e0 10678 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2524fveq2i 5868 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
2625oveq1i 6300 . . . . . . 7  |-  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 )
2726a1i 11 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 ) )
2813, 14, 15, 16, 17ballotlemfval0 29328 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2918, 28syl 17 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
3029adantr 467 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  0 )  =  0 )
3130oveq1d 6305 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  0 )  -  1 )  =  ( 0  -  1 ) )
3223, 27, 313eqtrrd 2490 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( 0  -  1 )  =  ( ( F `  C ) `  1
) )
3332eqeq1d 2453 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
0  -  1 )  =  0  <->  ( ( F `  C ) `  1 )  =  0 ) )
3412, 33mtbii 304 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
( F `  C
) `  1 )  =  0 )
35 ballotthOLD.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
36 ballotthOLD.mgtn . . . . . . 7  |-  N  < 
M
37 ballotthOLD.i . . . . . . 7  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
3813, 14, 15, 16, 17, 35, 36, 37ballotlemiexOLD 29372 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
3938simprd 465 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
4039ad2antrr 732 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  (
I `  C )
)  =  0 )
41 fveq2 5865 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
4241eqeq1d 2453 . . . . 5  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
4342adantl 468 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( ( F `
 C ) `  ( I `  C
) )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
4440, 43mpbid 214 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  1
)  =  0 )
4534, 44mtand 665 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
I `  C )  =  1 )
4645neqned 2631 1  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( I `  C )  =/=  1
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   {crab 2741    \ cdif 3401    i^i cin 3403   ~Pcpw 3951   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   ` cfv 5582  (class class class)co 6290   supcsup 7954   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    - cmin 9860    / cdiv 10269   NNcn 10609   ZZcz 10937   ...cfz 11784   #chash 12515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-hash 12516
This theorem is referenced by:  ballotlemicOLD  29377
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