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Theorem ballotlem4 28701
Description: If the first pick is a vote for B, A is not ahead throughout the count (Contributed by Thierry Arnoux, 25-Nov-2016.)
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 ) }
Assertion
Ref Expression
ballotlem4  |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  -.  C  e.  E
) )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O, c    F, c, i    C, i
Allowed substitution hints:    C( x, c)    P( x, i, c)    E( x, i, c)    F( x)    M( x)    N( x)    O( x)

Proof of Theorem ballotlem4
StepHypRef Expression
1 ballotth.m . . . . . . . 8  |-  M  e.  NN
2 ballotth.n . . . . . . . 8  |-  N  e.  NN
3 nnaddcl 10553 . . . . . . . 8  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
41, 2, 3mp2an 670 . . . . . . 7  |-  ( M  +  N )  e.  NN
5 elnnuz 11118 . . . . . . 7  |-  ( ( M  +  N )  e.  NN  <->  ( M  +  N )  e.  (
ZZ>= `  1 ) )
64, 5mpbi 208 . . . . . 6  |-  ( M  +  N )  e.  ( ZZ>= `  1 )
7 eluzfz1 11696 . . . . . 6  |-  ( ( M  +  N )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... ( M  +  N )
) )
86, 7ax-mp 5 . . . . 5  |-  1  e.  ( 1 ... ( M  +  N )
)
9 0le1 10072 . . . . . . . . . 10  |-  0  <_  1
10 0re 9585 . . . . . . . . . . 11  |-  0  e.  RR
11 1re 9584 . . . . . . . . . . 11  |-  1  e.  RR
1210, 11lenlti 9693 . . . . . . . . . 10  |-  ( 0  <_  1  <->  -.  1  <  0 )
139, 12mpbi 208 . . . . . . . . 9  |-  -.  1  <  0
14 ltsub13 10029 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  0  e.  RR  /\  1  e.  RR )  ->  (
0  <  ( 0  -  1 )  <->  1  <  ( 0  -  0 ) ) )
1510, 10, 11, 14mp3an 1322 . . . . . . . . . 10  |-  ( 0  <  ( 0  -  1 )  <->  1  <  ( 0  -  0 ) )
16 0m0e0 10641 . . . . . . . . . . 11  |-  ( 0  -  0 )  =  0
1716breq2i 4447 . . . . . . . . . 10  |-  ( 1  <  ( 0  -  0 )  <->  1  <  0 )
1815, 17bitri 249 . . . . . . . . 9  |-  ( 0  <  ( 0  -  1 )  <->  1  <  0 )
1913, 18mtbir 297 . . . . . . . 8  |-  -.  0  <  ( 0  -  1 )
20 1m1e0 10600 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
2120fveq2i 5851 . . . . . . . . . . 11  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
22 ballotth.o . . . . . . . . . . . 12  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
23 ballotth.p . . . . . . . . . . . 12  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
24 ballotth.f . . . . . . . . . . . 12  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
251, 2, 22, 23, 24ballotlemfval0 28698 . . . . . . . . . . 11  |-  ( C  e.  O  ->  (
( F `  C
) `  0 )  =  0 )
2621, 25syl5eq 2507 . . . . . . . . . 10  |-  ( C  e.  O  ->  (
( F `  C
) `  ( 1  -  1 ) )  =  0 )
2726oveq1d 6285 . . . . . . . . 9  |-  ( C  e.  O  ->  (
( ( F `  C ) `  (
1  -  1 ) )  -  1 )  =  ( 0  -  1 ) )
2827breq2d 4451 . . . . . . . 8  |-  ( C  e.  O  ->  (
0  <  ( (
( F `  C
) `  ( 1  -  1 ) )  -  1 )  <->  0  <  ( 0  -  1 ) ) )
2919, 28mtbiri 301 . . . . . . 7  |-  ( C  e.  O  ->  -.  0  <  ( ( ( F `  C ) `
 ( 1  -  1 ) )  - 
1 ) )
3029adantr 463 . . . . . 6  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  0  <  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
31 simpl 455 . . . . . . . . . . 11  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  C  e.  O
)
32 1nn 10542 . . . . . . . . . . . 12  |-  1  e.  NN
3332a1i 11 . . . . . . . . . . 11  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  1  e.  NN )
341, 2, 22, 23, 24, 31, 33ballotlemfp1 28694 . . . . . . . . . 10  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( -.  1  e.  C  -> 
( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) )  /\  ( 1  e.  C  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  +  1 ) ) ) )
3534simpld 457 . . . . . . . . 9  |-  ( ( C  e.  O  /\  1  e.  ( 1 ... ( M  +  N ) ) )  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `
 1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) ) )
368, 35mpan2 669 . . . . . . . 8  |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  ( ( F `  C ) `  1
)  =  ( ( ( F `  C
) `  ( 1  -  1 ) )  -  1 ) ) )
3736imp 427 . . . . . . 7  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  ( ( F `  C ) `  1 )  =  ( ( ( F `
 C ) `  ( 1  -  1 ) )  -  1 ) )
3837breq2d 4451 . . . . . 6  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  ( 0  <  ( ( F `
 C ) ` 
1 )  <->  0  <  ( ( ( F `  C ) `  (
1  -  1 ) )  -  1 ) ) )
3930, 38mtbird 299 . . . . 5  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  0  <  ( ( F `  C ) `  1
) )
40 fveq2 5848 . . . . . . . 8  |-  ( i  =  1  ->  (
( F `  C
) `  i )  =  ( ( F `
 C ) ` 
1 ) )
4140breq2d 4451 . . . . . . 7  |-  ( i  =  1  ->  (
0  <  ( ( F `  C ) `  i )  <->  0  <  ( ( F `  C
) `  1 )
) )
4241notbid 292 . . . . . 6  |-  ( i  =  1  ->  ( -.  0  <  ( ( F `  C ) `
 i )  <->  -.  0  <  ( ( F `  C ) `  1
) ) )
4342rspcev 3207 . . . . 5  |-  ( ( 1  e.  ( 1 ... ( M  +  N ) )  /\  -.  0  <  ( ( F `  C ) `
 1 ) )  ->  E. i  e.  ( 1 ... ( M  +  N ) )  -.  0  <  (
( F `  C
) `  i )
)
448, 39, 43sylancr 661 . . . 4  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  E. i  e.  ( 1 ... ( M  +  N )
)  -.  0  < 
( ( F `  C ) `  i
) )
45 rexnal 2902 . . . 4  |-  ( E. i  e.  ( 1 ... ( M  +  N ) )  -.  0  <  ( ( F `  C ) `
 i )  <->  -.  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
)
4644, 45sylib 196 . . 3  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
)
47 ballotth.e . . . . 5  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
481, 2, 22, 23, 24, 47ballotleme 28699 . . . 4  |-  ( C  e.  E  <->  ( C  e.  O  /\  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
) )
4948simprbi 462 . . 3  |-  ( C  e.  E  ->  A. i  e.  ( 1 ... ( M  +  N )
) 0  <  (
( F `  C
) `  i )
)
5046, 49nsyl 121 . 2  |-  ( ( C  e.  O  /\  -.  1  e.  C
)  ->  -.  C  e.  E )
5150ex 432 1  |-  ( C  e.  O  ->  ( -.  1  e.  C  ->  -.  C  e.  E
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    \ cdif 3458    i^i cin 3460   ~Pcpw 3999   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388
This theorem is referenced by:  ballotth  28740
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