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Theorem ballotlemiminOLD 29449
Description:  ( I `  C ) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.) Obsolete version of ballotlemimin 29411 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
ballotlemiminOLD  |-  ( C  e.  ( O  \  E )  ->  -.  E. k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) ( ( F `  C
) `  k )  =  0 )
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 ballotlemiminOLD
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfzle2 11829 . . . . . 6  |-  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  ->  k  <_  ( ( I `  C )  -  1 ) )
21adantl 473 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  k  <_  (
( I `  C
)  -  1 ) )
3 elfzelz 11826 . . . . . 6  |-  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  ->  k  e.  ZZ )
4 ballotthOLD.m . . . . . . . . . 10  |-  M  e.  NN
5 ballotthOLD.n . . . . . . . . . 10  |-  N  e.  NN
6 ballotthOLD.o . . . . . . . . . 10  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
7 ballotthOLD.p . . . . . . . . . 10  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
8 ballotthOLD.f . . . . . . . . . 10  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
9 ballotthOLD.e . . . . . . . . . 10  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
10 ballotthOLD.mgtn . . . . . . . . . 10  |-  N  < 
M
11 ballotthOLD.i . . . . . . . . . 10  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
124, 5, 6, 7, 8, 9, 10, 11ballotlemiexOLD 29445 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1312simpld 466 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
14 elfznn 11854 . . . . . . . 8  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  NN )
1513, 14syl 17 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  NN )
1615nnzd 11062 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
17 zltlem1 11013 . . . . . 6  |-  ( ( k  e.  ZZ  /\  ( I `  C
)  e.  ZZ )  ->  ( k  < 
( I `  C
)  <->  k  <_  (
( I `  C
)  -  1 ) ) )
183, 16, 17syl2anr 486 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  ( k  < 
( I `  C
)  <->  k  <_  (
( I `  C
)  -  1 ) ) )
192, 18mpbird 240 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  k  <  (
I `  C )
)
2019adantr 472 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  k  e.  ( 1 ... ( ( I `  C )  -  1 ) ) )  /\  ( ( F `  C ) `
 k )  =  0 )  ->  k  <  ( I `  C
) )
21 1zzd 10992 . . . . . . . . . . . . 13  |-  ( C  e.  ( O  \  E )  ->  1  e.  ZZ )
2216, 21zsubcld 11068 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  e.  ZZ )
2322zred 11063 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  e.  RR )
24 nnaddcl 10653 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
254, 5, 24mp2an 686 . . . . . . . . . . . . 13  |-  ( M  +  N )  e.  NN
2625a1i 11 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  NN )
2726nnred 10646 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  RR )
28 elfzle2 11829 . . . . . . . . . . . . 13  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  <_  ( M  +  N
) )
2913, 28syl 17 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  <_  ( M  +  N
) )
3026nnzd 11062 . . . . . . . . . . . . 13  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  ZZ )
31 zlem1lt 11012 . . . . . . . . . . . . 13  |-  ( ( ( I `  C
)  e.  ZZ  /\  ( M  +  N
)  e.  ZZ )  ->  ( ( I `
 C )  <_ 
( M  +  N
)  <->  ( ( I `
 C )  - 
1 )  <  ( M  +  N )
) )
3216, 30, 31syl2anc 673 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  <_  ( M  +  N )  <->  ( (
I `  C )  -  1 )  < 
( M  +  N
) ) )
3329, 32mpbid 215 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  <  ( M  +  N ) )
3423, 27, 33ltled 9800 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  <_  ( M  +  N ) )
35 eluz 11196 . . . . . . . . . . 11  |-  ( ( ( ( I `  C )  -  1 )  e.  ZZ  /\  ( M  +  N
)  e.  ZZ )  ->  ( ( M  +  N )  e.  ( ZZ>= `  ( (
I `  C )  -  1 ) )  <-> 
( ( I `  C )  -  1 )  <_  ( M  +  N ) ) )
3622, 30, 35syl2anc 673 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( M  +  N
)  e.  ( ZZ>= `  ( ( I `  C )  -  1 ) )  <->  ( (
I `  C )  -  1 )  <_ 
( M  +  N
) ) )
3734, 36mpbird 240 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  ( ZZ>= `  ( (
I `  C )  -  1 ) ) )
38 fzss2 11864 . . . . . . . . 9  |-  ( ( M  +  N )  e.  ( ZZ>= `  (
( I `  C
)  -  1 ) )  ->  ( 1 ... ( ( I `
 C )  - 
1 ) )  C_  ( 1 ... ( M  +  N )
) )
3937, 38syl 17 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
1 ... ( ( I `
 C )  - 
1 ) )  C_  ( 1 ... ( M  +  N )
) )
4039sseld 3417 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) )  -> 
k  e.  ( 1 ... ( M  +  N ) ) ) )
41 rabid 2953 . . . . . . . 8  |-  ( k  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 }  <->  ( k  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  C ) `  k )  =  0 ) )
424, 5, 6, 7, 8, 9, 10, 11ballotlemsupOLD 29448 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) )
43 gtso 9733 . . . . . . . . . . . 12  |-  `'  <  Or  RR
4443a1i 11 . . . . . . . . . . 11  |-  ( E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) )  ->  `'  <  Or  RR )
45 id 22 . . . . . . . . . . 11  |-  ( E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) )  ->  E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) ) )
4644, 45supub 7991 . . . . . . . . . 10  |-  ( E. z  e.  RR  ( A. w  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  -.  z `'  <  w  /\  A. w  e.  RR  (
w `'  <  z  ->  E. y  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } w `'  <  y
) )  ->  (
k  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  ->  -. 
sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
4742, 46syl 17 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  ->  -. 
sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
484, 5, 6, 7, 8, 9, 10, 11ballotlemiOLD 29444 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
4948breq1d 4405 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
) `'  <  k  <->  sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
5049notbid 301 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  ( -.  ( I `  C
) `'  <  k  <->  -. 
sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
5147, 50sylibrd 242 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  ->  -.  ( I `  C
) `'  <  k
) )
5241, 51syl5bir 226 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( k  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `
 C ) `  k )  =  0 )  ->  -.  (
I `  C ) `'  <  k ) )
5340, 52syland 489 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 )  ->  -.  (
I `  C ) `'  <  k ) )
5453imp 436 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 ) )  ->  -.  ( I `  C
) `'  <  k
)
55 ltrel 9714 . . . . . 6  |-  Rel  <
5655relbrcnv 5216 . . . . 5  |-  ( ( I `  C ) `'  <  k  <->  k  <  ( I `  C ) )
5754, 56sylnib 311 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 ) )  ->  -.  k  <  ( I `  C ) )
5857anassrs 660 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  k  e.  ( 1 ... ( ( I `  C )  -  1 ) ) )  /\  ( ( F `  C ) `
 k )  =  0 )  ->  -.  k  <  ( I `  C ) )
5920, 58pm2.65da 586 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  -.  ( ( F `  C ) `  k )  =  0 )
6059nrexdv 2842 1  |-  ( C  e.  ( O  \  E )  ->  -.  E. k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) ( ( F `  C
) `  k )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760    \ cdif 3387    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   class class class wbr 4395    |-> cmpt 4454    Or wor 4759   `'ccnv 4838   ` cfv 5589  (class class class)co 6308   supcsup 7972   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554
This theorem is referenced by:  ballotlemicOLD  29450  ballotlem1cOLD  29451
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