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Theorem ballotlemi1 28419
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 0re 9599 . . . . . . 7  |-  0  e.  RR
2 1re 9598 . . . . . . 7  |-  1  e.  RR
31, 2resubcli 9886 . . . . . 6  |-  ( 0  -  1 )  e.  RR
4 0lt1 10082 . . . . . . 7  |-  0  <  1
5 ltsub23 10039 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  e.  RR )  ->  (
( 0  -  1 )  <  0  <->  (
0  -  0 )  <  1 ) )
61, 2, 1, 5mp3an 1325 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0  -  0 )  <  1 )
7 0m0e0 10652 . . . . . . . . 9  |-  ( 0  -  0 )  =  0
87breq1i 4444 . . . . . . . 8  |-  ( ( 0  -  0 )  <  1  <->  0  <  1 )
96, 8bitr2i 250 . . . . . . 7  |-  ( 0  <  1  <->  ( 0  -  1 )  <  0 )
104, 9mpbi 208 . . . . . 6  |-  ( 0  -  1 )  <  0
113, 10gtneii 9699 . . . . 5  |-  0  =/=  ( 0  -  1 )
1211nesymi 2716 . . . 4  |-  -.  (
0  -  1 )  =  0
13 ballotth.m . . . . . . . . 9  |-  M  e.  NN
14 ballotth.n . . . . . . . . 9  |-  N  e.  NN
15 ballotth.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
16 ballotth.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
17 ballotth.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
18 eldifi 3611 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
19 1nn 10554 . . . . . . . . . 10  |-  1  e.  NN
2019a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
2113, 14, 15, 16, 17, 18, 20ballotlemfp1 28408 . . . . . . . 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 459 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
2322imp 429 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
24 1m1e0 10611 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2524fveq2i 5859 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
2625oveq1i 6291 . . . . . . 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 28412 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2918, 28syl 16 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
3029adantr 465 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  0 )  =  0 )
3130oveq1d 6296 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  0 )  -  1 )  =  ( 0  -  1 ) )
3223, 27, 313eqtrrd 2489 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( 0  -  1 )  =  ( ( F `  C ) `  1
) )
3332eqeq1d 2445 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
0  -  1 )  =  0  <->  ( ( F `  C ) `  1 )  =  0 ) )
3412, 33mtbii 302 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
( F `  C
) `  1 )  =  0 )
35 ballotth.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
36 ballotth.mgtn . . . . . . 7  |-  N  < 
M
37 ballotth.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, 37ballotlemiex 28418 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
3938simprd 463 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
4039ad2antrr 725 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  (
I `  C )
)  =  0 )
41 fveq2 5856 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
4241eqeq1d 2445 . . . . 5  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
4342adantl 466 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( ( F `
 C ) `  ( I `  C
) )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
4440, 43mpbid 210 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  1
)  =  0 )
4534, 44mtand 659 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
I `  C )  =  1 )
4645neqned 2646 1  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( I `  C )  =/=  1
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797    \ cdif 3458    i^i cin 3460   ~Pcpw 3997   class class class wbr 4437    |-> cmpt 4495   `'ccnv 4988   ` cfv 5578  (class class class)co 6281   supcsup 7902   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    < clt 9631    - cmin 9810    / cdiv 10213   NNcn 10543   ZZcz 10871   ...cfz 11683   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-hash 12388
This theorem is referenced by:  ballotlemic  28423
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