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Theorem ballotlemimin 28084
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 11686 . . . . . 6  |-  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  ->  k  <_  ( ( I `  C )  -  1 ) )
21adantl 466 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  k  <_  (
( I `  C
)  -  1 ) )
3 elfzelz 11684 . . . . . 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 28080 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1312simpld 459 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
14 elfznn 11710 . . . . . . . 8  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  NN )
1513, 14syl 16 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  NN )
1615nnzd 10961 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
17 zltlem1 10911 . . . . . 6  |-  ( ( k  e.  ZZ  /\  ( I `  C
)  e.  ZZ )  ->  ( k  < 
( I `  C
)  <->  k  <_  (
( I `  C
)  -  1 ) ) )
183, 16, 17syl2anr 478 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  ( k  < 
( I `  C
)  <->  k  <_  (
( I `  C
)  -  1 ) ) )
192, 18mpbird 232 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  k  <  (
I `  C )
)
2019adantr 465 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  k  e.  ( 1 ... ( ( I `  C )  -  1 ) ) )  /\  ( ( F `  C ) `
 k )  =  0 )  ->  k  <  ( I `  C
) )
21 1z 10890 . . . . . . . . . . . . . 14  |-  1  e.  ZZ
2221a1i 11 . . . . . . . . . . . . 13  |-  ( C  e.  ( O  \  E )  ->  1  e.  ZZ )
2316, 22zsubcld 10967 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  e.  ZZ )
2423zred 10962 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  e.  RR )
25 nnaddcl 10554 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN  /\  N  e.  NN )  ->  ( M  +  N
)  e.  NN )
264, 5, 25mp2an 672 . . . . . . . . . . . . 13  |-  ( M  +  N )  e.  NN
2726a1i 11 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  NN )
2827nnred 10547 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  RR )
29 elfzle2 11686 . . . . . . . . . . . . 13  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  <_  ( M  +  N
) )
3013, 29syl 16 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  <_  ( M  +  N
) )
3127nnzd 10961 . . . . . . . . . . . . 13  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  ZZ )
32 zlem1lt 10910 . . . . . . . . . . . . 13  |-  ( ( ( I `  C
)  e.  ZZ  /\  ( M  +  N
)  e.  ZZ )  ->  ( ( I `
 C )  <_ 
( M  +  N
)  <->  ( ( I `
 C )  - 
1 )  <  ( M  +  N )
) )
3316, 31, 32syl2anc 661 . . . . . . . . . . . 12  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  <_  ( M  +  N )  <->  ( (
I `  C )  -  1 )  < 
( M  +  N
) ) )
3430, 33mpbid 210 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  <  ( M  +  N ) )
3524, 28, 34ltled 9728 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  -  1 )  <_  ( M  +  N ) )
36 eluz 11091 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( M  +  N
)  e.  ( ZZ>= `  ( ( I `  C )  -  1 ) )  <->  ( (
I `  C )  -  1 )  <_ 
( M  +  N
) ) )
3835, 37mpbird 232 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  ( M  +  N )  e.  ( ZZ>= `  ( (
I `  C )  -  1 ) ) )
39 fzss2 11719 . . . . . . . . 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 3503 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) )  -> 
k  e.  ( 1 ... ( M  +  N ) ) ) )
42 rabid 3038 . . . . . . . 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 28083 . . . . . . . . . 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 9661 . . . . . . . . . . . . 13  |-  <  Or  RR
45 cnvso 5544 . . . . . . . . . . . . 13  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
4644, 45mpbi 208 . . . . . . . . . . . 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 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
) ) )
4947, 48supub 7915 . . . . . . . . . 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 28079 . . . . . . . . . . 11  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
5251breq1d 4457 . . . . . . . . . 10  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
) `'  <  k  <->  sup ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
5352notbid 294 . . . . . . . . 9  |-  ( C  e.  ( O  \  E )  ->  ( -.  ( I `  C
) `'  <  k  <->  -. 
sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) `'  <  k ) )
5450, 53sylibrd 234 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
k  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  ->  -.  ( I `  C
) `'  <  k
) )
5542, 54syl5bir 218 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
( k  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `
 C ) `  k )  =  0 )  ->  -.  (
I `  C ) `'  <  k ) )
5641, 55syland 481 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 )  ->  -.  (
I `  C ) `'  <  k ) )
5756imp 429 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 ) )  ->  -.  ( I `  C
) `'  <  k
)
58 ltrel 9645 . . . . . 6  |-  Rel  <
5958relbrcnv 5375 . . . . 5  |-  ( ( I `  C ) `'  <  k  <->  k  <  ( I `  C ) )
6057, 59sylnib 304 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  ( k  e.  ( 1 ... ( ( I `  C )  -  1 ) )  /\  ( ( F `
 C ) `  k )  =  0 ) )  ->  -.  k  <  ( I `  C ) )
6160anassrs 648 . . 3  |-  ( ( ( C  e.  ( O  \  E )  /\  k  e.  ( 1 ... ( ( I `  C )  -  1 ) ) )  /\  ( ( F `  C ) `
 k )  =  0 )  ->  -.  k  <  ( I `  C ) )
6220, 61pm2.65da 576 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  k  e.  ( 1 ... ( ( I `
 C )  - 
1 ) ) )  ->  -.  ( ( F `  C ) `  k )  =  0 )
6362nrexdv 2920 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505    Or wor 4799   `'ccnv 4998   ` cfv 5586  (class class class)co 6282   supcsup 7896   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   NNcn 10532   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   #chash 12369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370
This theorem is referenced by:  ballotlemic  28085  ballotlem1c  28086
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