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

Theorem ballotlemiex 23076
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 23075 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
10 ltso 8919 . . . . . 6  |-  <  Or  RR
11 cnvso 5230 . . . . . 6  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1210, 11mpbi 199 . . . . 5  |-  `'  <  Or  RR
1312a1i 10 . . . 4  |-  ( C  e.  ( O  \  E )  ->  `'  <  Or  RR )
14 fzfi 11050 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
15 ssrab2 3271 . . . . . 6  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  ( 1 ... ( M  +  N )
)
16 ssfi 7099 . . . . . 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 653 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin
1817a1i 10 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin )
191, 2, 3, 4, 5, 6, 7ballotlem5 23074 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  E. k  e.  ( 1 ... ( M  +  N )
) ( ( F `
 C ) `  k )  =  0 )
20 rabn0 3487 . . . . 5  |-  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  <->  E. k  e.  ( 1 ... ( M  +  N ) ) ( ( F `  C ) `  k
)  =  0 )
2119, 20sylibr 203 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  =/=  (/) )
22 fzssuz 10848 . . . . . . . 8  |-  ( 1 ... ( M  +  N ) )  C_  ( ZZ>= `  1 )
23 uzssz 10263 . . . . . . . 8  |-  ( ZZ>= ` 
1 )  C_  ZZ
2422, 23sstri 3201 . . . . . . 7  |-  ( 1 ... ( M  +  N ) )  C_  ZZ
25 zssre 10047 . . . . . . 7  |-  ZZ  C_  RR
2624, 25sstri 3201 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  C_  RR
2715, 26sstri 3201 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR
2827a1i 10 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR )
29 fisupcl 7234 . . . 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 1184 . . 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 2370 . 2  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
32 fveq2 5541 . . . 4  |-  ( k  =  ( I `  C )  ->  (
( F `  C
) `  k )  =  ( ( F `
 C ) `  ( I `  C
) ) )
3332eqeq1d 2304 . . 3  |-  ( k  =  ( I `  C )  ->  (
( ( F `  C ) `  k
)  =  0  <->  (
( F `  C
) `  ( I `  C ) )  =  0 ) )
3433elrab 2936 . 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 188 1  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   class class class wbr 4039    e. cmpt 4093    Or wor 4329   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   Fincfn 6879   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   #chash 11353
This theorem is referenced by:  ballotlemi1  23077  ballotlemii  23078  ballotlemimin  23080  ballotlemic  23081  ballotlem1c  23082  ballotlemsgt1  23085  ballotlemsdom  23086  ballotlemsel1i  23087  ballotlemsf1o  23088  ballotlemsi  23089  ballotlemsima  23090  ballotlemrv2  23096  ballotlemfrc  23101  ballotlemfrci  23102  ballotlemfrceq  23103  ballotlemfrcn0  23104  ballotlemrc  23105  ballotlemirc  23106  ballotlem1ri  23109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
  Copyright terms: Public domain W3C validator