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Theorem ballotlemii 29283
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 ) 
|-> inf ( { 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 11025 . . . . . 6  |-  1  =  ( 0  +  1 )
2 ax-1ne0 9554 . . . . . 6  |-  1  =/=  0
31, 2eqnetrri 2667 . . . . 5  |-  ( 0  +  1 )  =/=  0
43neii 2598 . . . 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 3525 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
11 1nn 10566 . . . . . . . . . 10  |-  1  e.  NN
1211a1i 11 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  1  e.  NN )
135, 6, 7, 8, 9, 10, 12ballotlemfp1 29271 . . . . . . . 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 10624 . . . . . . . . 9  |-  ( 1  -  1 )  =  0
1716fveq2i 5823 . . . . . . . 8  |-  ( ( F `  C ) `
 ( 1  -  1 ) )  =  ( ( F `  C ) `  0
)
1817oveq1i 6254 . . . . . . 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 29275 . . . . . . . . 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 6259 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( ( ( F `
 C ) ` 
0 )  +  1 )  =  ( 0  +  1 ) )
2415, 19, 233eqtrrd 2462 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  1  e.  C )  ->  ( 0  +  1 )  =  ( ( F `  C ) `
 1 ) )
2524eqeq1d 2425 . . . 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 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 ) 
|-> inf ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  c ) `
 k )  =  0 } ,  RR ,  <  ) )
305, 6, 7, 8, 9, 27, 28, 29ballotlemiex 29281 . . . . . 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 5820 . . . . . 6  |-  ( ( I `  C )  =  1  ->  (
( F `  C
) `  ( I `  C ) )  =  ( ( F `  C ) `  1
) )
3433eqeq1d 2425 . . . . 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 2603 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 2594   A.wral 2709   {crab 2713    \ cdif 3371    i^i cin 3373   ~Pcpw 3919   class class class wbr 4361    |-> cmpt 4420   ` cfv 5539  (class class class)co 6244  infcinf 7903   RRcr 9484   0cc0 9485   1c1 9486    + caddc 9488    < clt 9621    - cmin 9806    / cdiv 10215   NNcn 10555   ZZcz 10883   ...cfz 11730   #chash 12460
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 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562
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 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-sup 7904  df-inf 7905  df-card 8320  df-cda 8544  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-nn 10556  df-2 10614  df-n0 10816  df-z 10884  df-uz 11106  df-fz 11731  df-hash 12461
This theorem is referenced by:  ballotlem1c  29287
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