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Theorem ballotlemii 28948
Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-Apr-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
ballotlemii  |-  ( ( 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 ballotlemii
StepHypRef Expression
1 1e0p1 11047 . . . . . 6  |-  1  =  ( 0  +  1 )
2 ax-1ne0 9591 . . . . . 6  |-  1  =/=  0
31, 2eqnetrri 2700 . . . . 5  |-  ( 0  +  1 )  =/=  0
43neii 2602 . . . 4  |-  -.  (
0  +  1 )  =  0
5 ballotth.m . . . . . . . . 9  |-  M  e.  NN
6 ballotth.n . . . . . . . . 9  |-  N  e.  NN
7 ballotth.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
8 ballotth.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
9 ballotth.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
10 eldifi 3565 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
11 1nn 10587 . . . . . . . . . 10  |-  1  e.  NN
1211a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
135, 6, 7, 8, 9, 10, 12ballotlemfp1 28936 . . . . . . . 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 ) ) ) )
1413simprd 461 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
1  e.  C  -> 
( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  +  1 ) ) )
1514imp 427 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  +  1 ) )
16 1m1e0 10645 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
1716fveq2i 5852 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
1817oveq1i 6288 . . . . . . 7  |-  ( ( ( F `  C
) `  ( 1  -  1 ) )  +  1 )  =  ( ( ( F `
 C ) ` 
0 )  +  1 )
1918a1i 11 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( ( F `
 C ) `  ( 1  -  1 ) )  +  1 )  =  ( ( ( F `  C
) `  0 )  +  1 ) )
205, 6, 7, 8, 9ballotlemfval0 28940 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2110, 20syl 17 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
2221adantr 463 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( F `  C ) `  0
)  =  0 )
2322oveq1d 6293 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( ( F `
 C ) ` 
0 )  +  1 )  =  ( 0  +  1 ) )
2415, 19, 233eqtrrd 2448 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( 0  +  1 )  =  ( ( F `  C ) `
 1 ) )
2524eqeq1d 2404 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( 0  +  1 )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
264, 25mtbii 300 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  -.  ( ( F `
 C ) ` 
1 )  =  0 )
27 ballotth.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
28 ballotth.mgtn . . . . . . 7  |-  N  < 
M
29 ballotth.i . . . . . . 7  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
305, 6, 7, 8, 9, 27, 28, 29ballotlemiex 28946 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
3130simprd 461 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
3231ad2antrr 724 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  1  e.  C
)  /\  ( I `  C )  =  1 )  ->  ( ( F `  C ) `  ( I `  C
) )  =  0 )
33 fveq2 5849 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
3433eqeq1d 2404 . . . . 5  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
3534adantl 464 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  1  e.  C
)  /\  ( I `  C )  =  1 )  ->  ( (
( F `  C
) `  ( I `  C ) )  =  0  <->  ( ( F `
 C ) ` 
1 )  =  0 ) )
3632, 35mpbid 210 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  1  e.  C
)  /\  ( I `  C )  =  1 )  ->  ( ( F `  C ) `  1 )  =  0 )
3726, 36mtand 657 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  -.  ( I `  C )  =  1 )
3837neqned 2606 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 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   {crab 2758    \ cdif 3411    i^i cin 3413   ~Pcpw 3955   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4822   ` cfv 5569  (class class class)co 6278   supcsup 7934   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    < clt 9658    - cmin 9841    / cdiv 10247   NNcn 10576   ZZcz 10905   ...cfz 11726   #chash 12452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-hash 12453
This theorem is referenced by:  ballotlem1c  28952
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