Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemiex Structured version   Unicode version

Theorem ballotlemiex 27048
Description: Properties of  ( I `
 C ). (Contributed by Thierry Arnoux, 12-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
ballotlemiex  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  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
Allowed substitution hints:    C( x, c)    P( x, i, k, c)    E( x)    F( x)    I( x, i, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemiex
StepHypRef Expression
1 ballotth.m . . . 4  |-  M  e.  NN
2 ballotth.n . . . 4  |-  N  e.  NN
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotth.p . . . 4  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotth.f . . . 4  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 ballotth.e . . . 4  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
7 ballotth.mgtn . . . 4  |-  N  < 
M
8 ballotth.i . . . 4  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
91, 2, 3, 4, 5, 6, 7, 8ballotlemi 27047 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
10 ltso 9569 . . . . . 6  |-  <  Or  RR
11 cnvso 5487 . . . . . 6  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1210, 11mpbi 208 . . . . 5  |-  `'  <  Or  RR
1312a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  `'  <  Or  RR )
14 fzfi 11914 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
15 ssrab2 3548 . . . . . 6  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  ( 1 ... ( M  +  N )
)
16 ssfi 7647 . . . . . 6  |-  ( ( ( 1 ... ( M  +  N )
)  e.  Fin  /\  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } 
C_  ( 1 ... ( M  +  N
) ) )  ->  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  e.  Fin )
1714, 15, 16mp2an 672 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin
1817a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin )
191, 2, 3, 4, 5, 6, 7ballotlem5 27046 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  E. k  e.  ( 1 ... ( M  +  N )
) ( ( F `
 C ) `  k )  =  0 )
20 rabn0 3768 . . . . 5  |-  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  <->  E. k  e.  ( 1 ... ( M  +  N ) ) ( ( F `  C ) `  k
)  =  0 )
2119, 20sylibr 212 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  =/=  (/) )
22 fzssuz 11619 . . . . . . . 8  |-  ( 1 ... ( M  +  N ) )  C_  ( ZZ>= `  1 )
23 uzssz 10994 . . . . . . . 8  |-  ( ZZ>= ` 
1 )  C_  ZZ
2422, 23sstri 3476 . . . . . . 7  |-  ( 1 ... ( M  +  N ) )  C_  ZZ
25 zssre 10767 . . . . . . 7  |-  ZZ  C_  RR
2624, 25sstri 3476 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  C_  RR
2715, 26sstri 3476 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR
2827a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR )
29 fisupcl 7831 . . . 4  |-  ( ( `'  <  Or  RR  /\  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 }  e.  Fin  /\  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  /\  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR ) )  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } )
3013, 18, 21, 28, 29syl13anc 1221 . . 3  |-  ( C  e.  ( O  \  E )  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } ,  RR ,  `'  <  )  e.  {
k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 } )
319, 30eqeltrd 2542 . 2  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
32 fveq2 5802 . . . 4  |-  ( k  =  ( I `  C )  ->  (
( F `  C
) `  k )  =  ( ( F `
 C ) `  ( I `  C
) ) )
3332eqeq1d 2456 . . 3  |-  ( k  =  ( I `  C )  ->  (
( ( F `  C ) `  k
)  =  0  <->  (
( F `  C
) `  ( I `  C ) )  =  0 ) )
3433elrab 3224 . 2  |-  ( ( I `  C )  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 }  <->  ( (
I `  C )  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  C ) `  ( I `  C
) )  =  0 ) )
3531, 34sylib 196 1  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    \ cdif 3436    i^i cin 3438    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   class class class wbr 4403    |-> cmpt 4461    Or wor 4751   `'ccnv 4950   ` cfv 5529  (class class class)co 6203   Fincfn 7423   supcsup 7804   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    < clt 9532    - cmin 9709    / cdiv 10107   NNcn 10436   ZZcz 10760   ZZ>=cuz 10975   ...cfz 11557   #chash 12223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-hash 12224
This theorem is referenced by:  ballotlemi1  27049  ballotlemii  27050  ballotlemimin  27052  ballotlemic  27053  ballotlem1c  27054  ballotlemsgt1  27057  ballotlemsdom  27058  ballotlemsel1i  27059  ballotlemsf1o  27060  ballotlemsi  27061  ballotlemsima  27062  ballotlemrv2  27068  ballotlemfrc  27073  ballotlemfrci  27074  ballotlemfrceq  27075  ballotlemfrcn0  27076  ballotlemrc  27077  ballotlemirc  27078  ballotlem1ri  27081
  Copyright terms: Public domain W3C validator