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Theorem ballotlemiiOLD 29382
Description: The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-Apr-2017.) Obsolete version of ballotlemii 29344 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
ballotlemiiOLD  |-  ( ( 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 ballotlemiiOLD
StepHypRef Expression
1 1e0p1 11086 . . . . . 6  |-  1  =  ( 0  +  1 )
2 ax-1ne0 9615 . . . . . 6  |-  1  =/=  0
31, 2eqnetrri 2717 . . . . 5  |-  ( 0  +  1 )  =/=  0
43neii 2618 . . . 4  |-  -.  (
0  +  1 )  =  0
5 ballotthOLD.m . . . . . . . . 9  |-  M  e.  NN
6 ballotthOLD.n . . . . . . . . 9  |-  N  e.  NN
7 ballotthOLD.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
8 ballotthOLD.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
9 ballotthOLD.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
10 eldifi 3587 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
11 1nn 10627 . . . . . . . . . 10  |-  1  e.  NN
1211a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
135, 6, 7, 8, 9, 10, 12ballotlemfp1 29332 . . . . . . . 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 464 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
1  e.  C  -> 
( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  +  1 ) ) )
1514imp 430 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  +  1 ) )
16 1m1e0 10685 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
1716fveq2i 5884 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
1817oveq1i 6315 . . . . . . 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 29336 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2110, 20syl 17 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  0 )  =  0 )
2221adantr 466 . . . . . . 7  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( F `  C ) `  0
)  =  0 )
2322oveq1d 6320 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( ( F `
 C ) ` 
0 )  +  1 )  =  ( 0  +  1 ) )
2415, 19, 233eqtrrd 2468 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( 0  +  1 )  =  ( ( F `  C ) `
 1 ) )
2524eqeq1d 2424 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( 0  +  1 )  =  0  <-> 
( ( F `  C ) `  1
)  =  0 ) )
264, 25mtbii 303 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  -.  ( ( F `
 C ) ` 
1 )  =  0 )
27 ballotthOLD.e . . . . . . 7  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
28 ballotthOLD.mgtn . . . . . . 7  |-  N  < 
M
29 ballotthOLD.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, 29ballotlemiexOLD 29380 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
3130simprd 464 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( F `  C
) `  ( I `  C ) )  =  0 )
3231ad2antrr 730 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  1  e.  C
)  /\  ( I `  C )  =  1 )  ->  ( ( F `  C ) `  ( I `  C
) )  =  0 )
33 fveq2 5881 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
3433eqeq1d 2424 . . . . 5  |-  ( ( I `  C )  =  1  ->  (
( ( F `  C ) `  (
I `  C )
)  =  0  <->  (
( F `  C
) `  1 )  =  0 ) )
3534adantl 467 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  1  e.  C
)  /\  ( I `  C )  =  1 )  ->  ( (
( F `  C
) `  ( I `  C ) )  =  0  <->  ( ( F `
 C ) ` 
1 )  =  0 ) )
3632, 35mpbid 213 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  1  e.  C
)  /\  ( I `  C )  =  1 )  ->  ( ( F `  C ) `  1 )  =  0 )
3726, 36mtand 663 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  -.  ( I `  C )  =  1 )
3837neqned 2623 1  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( I `  C
)  =/=  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   {crab 2775    \ cdif 3433    i^i cin 3435   ~Pcpw 3981   class class class wbr 4423    |-> cmpt 4482   `'ccnv 4852   ` cfv 5601  (class class class)co 6305   supcsup 7963   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549    < clt 9682    - cmin 9867    / cdiv 10276   NNcn 10616   ZZcz 10944   ...cfz 11791   #chash 12521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12522
This theorem is referenced by:  ballotlem1cOLD  29386
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