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Theorem ballotlemi1 26907
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 9407 . . . . . . 7  |-  0  e.  RR
2 1re 9406 . . . . . . 7  |-  1  e.  RR
31, 2resubcli 9692 . . . . . 6  |-  ( 0  -  1 )  e.  RR
4 0lt1 9883 . . . . . . 7  |-  0  <  1
5 ltsub23 9840 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  e.  RR )  ->  (
( 0  -  1 )  <  0  <->  (
0  -  0 )  <  1 ) )
61, 2, 1, 5mp3an 1314 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0  -  0 )  <  1 )
7 0m0e0 10452 . . . . . . . . 9  |-  ( 0  -  0 )  =  0
87breq1i 4320 . . . . . . . 8  |-  ( ( 0  -  0 )  <  1  <->  0  <  1 )
96, 8bitr2i 250 . . . . . . 7  |-  ( 0  <  1  <->  ( 0  -  1 )  <  0 )
104, 9mpbi 208 . . . . . 6  |-  ( 0  -  1 )  <  0
113, 10gtneii 9507 . . . . 5  |-  0  =/=  ( 0  -  1 )
12 eqcom 2445 . . . . . 6  |-  ( 0  =  ( 0  -  1 )  <->  ( 0  -  1 )  =  0 )
1312necon3abii 2632 . . . . 5  |-  ( 0  =/=  ( 0  -  1 )  <->  -.  (
0  -  1 )  =  0 )
1411, 13mpbi 208 . . . 4  |-  -.  (
0  -  1 )  =  0
15 ballotth.m . . . . . . . . 9  |-  M  e.  NN
16 ballotth.n . . . . . . . . 9  |-  N  e.  NN
17 ballotth.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
18 ballotth.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
19 ballotth.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
20 eldifi 3499 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
21 1nn 10354 . . . . . . . . . 10  |-  1  e.  NN
2221a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
2315, 16, 17, 18, 19, 20, 22ballotlemfp1 26896 . . . . . . . 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 ) ) ) )
2423simpld 459 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
2524imp 429 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
26 1m1e0 10411 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2726fveq2i 5715 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
2827oveq1i 6122 . . . . . . 7  |-  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 )
2928a1i 11 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 ) )
3015, 16, 17, 18, 19ballotlemfval0 26900 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
3120, 30syl 16 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
3231adantr 465 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  0 )  =  0 )
3332oveq1d 6127 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  0 )  -  1 )  =  ( 0  -  1 ) )
3425, 29, 333eqtrrd 2480 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( 0  -  1 )  =  ( ( F `  C ) `  1
) )
3534eqeq1d 2451 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
0  -  1 )  =  0  <->  ( ( F `  C ) `  1 )  =  0 ) )
3614, 35mtbii 302 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
( F `  C
) `  1 )  =  0 )
37 ballotth.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
38 ballotth.mgtn . . . . . . 7  |-  N  < 
M
39 ballotth.i . . . . . . 7  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
4015, 16, 17, 18, 19, 37, 38, 39ballotlemiex 26906 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
4140simprd 463 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
4241ad2antrr 725 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  (
I `  C )
)  =  0 )
43 fveq2 5712 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
4443eqeq1d 2451 . . . . 5  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
4544adantl 466 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( ( F `
 C ) `  ( I `  C
) )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
4642, 45mpbid 210 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  1
)  =  0 )
4736, 46mtand 659 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
I `  C )  =  1 )
4847neneqad 2705 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 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   {crab 2740    \ cdif 3346    i^i cin 3348   ~Pcpw 3881   class class class wbr 4313    e. cmpt 4371   `'ccnv 4860   ` cfv 5439  (class class class)co 6112   supcsup 7711   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    < clt 9439    - cmin 9616    / cdiv 10014   NNcn 10343   ZZcz 10667   ...cfz 11458   #chash 12124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-hash 12125
This theorem is referenced by:  ballotlemic  26911
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