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Theorem ballotlemimin 24765
Description:  ( I `  C ) is the first tie. (Contributed by Thierry Arnoux, 1-Dec-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 ) }
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
ballotlemimin  |-  ( 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    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 ballotlemimin
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfzle2 11063 . . . . . 6  |-  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  ->  k  <_  ( ( I `  C )  -  1 ) )
21adantl 454 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  k  <_  (
( I `  C
)  -  1 ) )
3 elfzelz 11061 . . . . . 6  |-  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  ->  k  e.  ZZ )
4 ballotth.m . . . . . . . . . 10  |-  M  e.  NN
5 ballotth.n . . . . . . . . . 10  |-  N  e.  NN
6 ballotth.o . . . . . . . . . 10  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
7 ballotth.p . . . . . . . . . 10  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
8 ballotth.f . . . . . . . . . 10  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
9 ballotth.e . . . . . . . . . 10  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
10 ballotth.mgtn . . . . . . . . . 10  |-  N  < 
M
11 ballotth.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, 11ballotlemiex 24761 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1312simpld 447 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
14 elfznn 11082 . . . . . . . 8  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  NN )
1513, 14syl 16 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  NN )
1615nnzd 10376 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
17 zltlem1 10330 . . . . . 6  |-  ( ( k  e.  ZZ  /\  ( I `  C
)  e.  ZZ )  ->  ( k  < 
( I `  C
)  <->  k  <_  (
( I `  C
)  -  1 ) ) )
183, 16, 17syl2anr 466 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  ( k  < 
( I `  C
)  <->  k  <_  (
( I `  C
)  -  1 ) ) )
192, 18mpbird 225 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  k  <  (
I `  C )
)
2019adantr 453 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  k  e.  ( 1 ... ( ( I `  C )  -  1 ) ) )  /\  ( ( F `  C ) `
 k )  =  0 )  ->  k  <  ( I `  C
) )
21 1z 10313 . . . . . . . . . . . . . 14  |-  1  e.  ZZ
2221a1i 11 . . . . . . . . . . . . 13  |-  ( C  e.  ( O  \  E )  ->  1  e.  ZZ )
2316, 22zsubcld 10382 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  e.  ZZ )
2423zred 10377 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  e.  RR )
25 nnaddcl 10024 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
264, 5, 25mp2an 655 . . . . . . . . . . . . 13  |-  ( M  +  N )  e.  NN
2726a1i 11 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  NN )
2827nnred 10017 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  RR )
29 elfzle2 11063 . . . . . . . . . . . . 13  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  <_  ( M  +  N
) )
3013, 29syl 16 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  <_  ( M  +  N
) )
3127nnzd 10376 . . . . . . . . . . . . 13  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  ZZ )
32 zlem1lt 10329 . . . . . . . . . . . . 13  |-  ( ( ( I `  C
)  e.  ZZ  /\  ( M  +  N
)  e.  ZZ )  ->  ( ( I `
 C )  <_ 
( M  +  N
)  <->  ( ( I `
 C )  - 
1 )  <  ( M  +  N )
) )
3316, 31, 32syl2anc 644 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  <_  ( M  +  N )  <->  ( (
I `  C )  -  1 )  < 
( M  +  N
) ) )
3430, 33mpbid 203 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  <  ( M  +  N ) )
3524, 28, 34ltled 9223 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  <_  ( M  +  N ) )
36 eluz 10501 . . . . . . . . . . 11  |-  ( ( ( ( I `  C )  -  1 )  e.  ZZ  /\  ( M  +  N
)  e.  ZZ )  ->  ( ( M  +  N )  e.  ( ZZ>= `  ( (
I `  C )  -  1 ) )  <-> 
( ( I `  C )  -  1 )  <_  ( M  +  N ) ) )
3723, 31, 36syl2anc 644 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( M  +  N
)  e.  ( ZZ>= `  ( ( I `  C )  -  1 ) )  <->  ( (
I `  C )  -  1 )  <_ 
( M  +  N
) ) )
3835, 37mpbird 225 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  ( ZZ>= `  ( (
I `  C )  -  1 ) ) )
39 fzss2 11094 . . . . . . . . 9  |-  ( ( M  +  N )  e.  ( ZZ>= `  (
( I `  C
)  -  1 ) )  ->  ( 1 ... ( ( I `
 C )  - 
1 ) )  C_  ( 1 ... ( M  +  N )
) )
4038, 39syl 16 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
1 ... ( ( I `
 C )  - 
1 ) )  C_  ( 1 ... ( M  +  N )
) )
4140sseld 3349 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) )  -> 
k  e.  ( 1 ... ( M  +  N ) ) ) )
42 rabid 2886 . . . . . . . 8  |-  ( k  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 }  <->  ( k  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  C ) `  k )  =  0 ) )
434, 5, 6, 7, 8, 9, 10, 11ballotlemsup 24764 . . . . . . . . . 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
) ) )
44 ltso 9158 . . . . . . . . . . . . 13  |-  <  Or  RR
45 cnvso 5413 . . . . . . . . . . . . 13  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
4644, 45mpbi 201 . . . . . . . . . . . 12  |-  `'  <  Or  RR
4746a1i 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 )
48 id 21 . . . . . . . . . . 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
) ) )
4947, 48supub 7466 . . . . . . . . . 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 ) )
5043, 49syl 16 . . . . . . . . 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 ) )
514, 5, 6, 7, 8, 9, 10, 11ballotlemi 24760 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
5251breq1d 4224 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
) `'  <  k  <->  sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
5352notbid 287 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  ( -.  ( I `  C
) `'  <  k  <->  -. 
sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
5450, 53sylibrd 227 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  ->  -.  ( I `  C
) `'  <  k
) )
5542, 54syl5bir 211 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( k  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `
 C ) `  k )  =  0 )  ->  -.  (
I `  C ) `'  <  k ) )
5641, 55syland 469 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 )  ->  -.  (
I `  C ) `'  <  k ) )
5756imp 420 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 ) )  ->  -.  ( I `  C
) `'  <  k
)
58 ltrel 9142 . . . . . 6  |-  Rel  <
5958relbrcnv 5247 . . . . 5  |-  ( ( I `  C ) `'  <  k  <->  k  <  ( I `  C ) )
6057, 59sylnib 297 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 ) )  ->  -.  k  <  ( I `  C ) )
6160anassrs 631 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  k  e.  ( 1 ... ( ( I `  C )  -  1 ) ) )  /\  ( ( F `  C ) `
 k )  =  0 )  ->  -.  k  <  ( I `  C ) )
6220, 61pm2.65da 561 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  -.  ( ( F `  C ) `  k )  =  0 )
6362nrexdv 2811 1  |-  ( C  e.  ( O  \  E )  ->  -.  E. k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) ( ( F `  C
) `  k )  =  0 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   class class class wbr 4214    e. cmpt 4268    Or wor 4504   `'ccnv 4879   ` cfv 5456  (class class class)co 6083   supcsup 7447   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    < clt 9122    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045   #chash 11620
This theorem is referenced by:  ballotlemic  24766  ballotlem1c  24767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-hash 11621
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