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Theorem ballotlemi1 24541
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 9026 . . . . . . 7  |-  0  e.  RR
2 1re 9025 . . . . . . 7  |-  1  e.  RR
31, 2resubcli 9297 . . . . . 6  |-  ( 0  -  1 )  e.  RR
4 0lt1 9484 . . . . . . 7  |-  0  <  1
5 ltsub23 9442 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  0  e.  RR )  ->  (
( 0  -  1 )  <  0  <->  (
0  -  0 )  <  1 ) )
61, 2, 1, 5mp3an 1279 . . . . . . . 8  |-  ( ( 0  -  1 )  <  0  <->  ( 0  -  0 )  <  1 )
7 0cn 9019 . . . . . . . . . 10  |-  0  e.  CC
87subidi 9305 . . . . . . . . 9  |-  ( 0  -  0 )  =  0
98breq1i 4162 . . . . . . . 8  |-  ( ( 0  -  0 )  <  1  <->  0  <  1 )
106, 9bitr2i 242 . . . . . . 7  |-  ( 0  <  1  <->  ( 0  -  1 )  <  0 )
114, 10mpbi 200 . . . . . 6  |-  ( 0  -  1 )  <  0
123, 11gtneii 9118 . . . . 5  |-  0  =/=  ( 0  -  1 )
13 eqcom 2391 . . . . . 6  |-  ( 0  =  ( 0  -  1 )  <->  ( 0  -  1 )  =  0 )
1413necon3abii 2582 . . . . 5  |-  ( 0  =/=  ( 0  -  1 )  <->  -.  (
0  -  1 )  =  0 )
1512, 14mpbi 200 . . . 4  |-  -.  (
0  -  1 )  =  0
16 ballotth.m . . . . . . . . 9  |-  M  e.  NN
17 ballotth.n . . . . . . . . 9  |-  N  e.  NN
18 ballotth.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
19 ballotth.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
20 ballotth.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
21 eldifi 3414 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
22 1nn 9945 . . . . . . . . . 10  |-  1  e.  NN
2322a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
2416, 17, 18, 19, 20, 21, 23ballotlemfp1 24530 . . . . . . . 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 ) ) ) )
2524simpld 446 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
2625imp 419 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
27 1m1e0 10002 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
2827fveq2i 5673 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
2928oveq1i 6032 . . . . . . 7  |-  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 )
3029a1i 11 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  ( 1  -  1 ) )  -  1 )  =  ( ( ( F `
 C ) ` 
0 )  -  1 ) )
3116, 17, 18, 19, 20ballotlemfval0 24534 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
3221, 31syl 16 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
3332adantr 452 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  0 )  =  0 )
3433oveq1d 6037 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
( F `  C
) `  0 )  -  1 )  =  ( 0  -  1 ) )
3526, 30, 343eqtrrd 2426 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( 0  -  1 )  =  ( ( F `  C ) `  1
) )
3635eqeq1d 2397 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( (
0  -  1 )  =  0  <->  ( ( F `  C ) `  1 )  =  0 ) )
3715, 36mtbii 294 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
( F `  C
) `  1 )  =  0 )
38 ballotth.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
39 ballotth.mgtn . . . . . . 7  |-  N  < 
M
40 ballotth.i . . . . . . 7  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
4116, 17, 18, 19, 20, 38, 39, 40ballotlemiex 24540 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
4241simprd 450 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
4342ad2antrr 707 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  (
I `  C )
)  =  0 )
44 fveq2 5670 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
4544eqeq1d 2397 . . . . 5  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
4645adantl 453 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( ( F `
 C ) `  ( I `  C
) )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
4743, 46mpbid 202 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  -.  1  e.  C )  /\  (
I `  C )  =  1 )  -> 
( ( F `  C ) `  1
)  =  0 )
4837, 47mtand 641 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  -.  (
I `  C )  =  1 )
4948neneqad 2622 1  |-  ( ( C  e.  ( O 
\  E )  /\  -.  1  e.  C
)  ->  ( I `  C )  =/=  1
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   {crab 2655    \ cdif 3262    i^i cin 3264   ~Pcpw 3744   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   ` cfv 5396  (class class class)co 6022   supcsup 7382   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    < clt 9055    - cmin 9225    / cdiv 9611   NNcn 9934   ZZcz 10216   ...cfz 10977   #chash 11547
This theorem is referenced by:  ballotlemic  24545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-hash 11548
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