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Theorem ballotlemiex 28313
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 28312 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 } ,  RR ,  `'  <  ) )
10 gtso 9669 . . . . 5  |-  `'  <  Or  RR
1110a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  `'  <  Or  RR )
12 fzfi 12061 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  e. 
Fin
13 ssrab2 3570 . . . . . 6  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  ( 1 ... ( M  +  N )
)
14 ssfi 7742 . . . . . 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 )
1512, 13, 14mp2an 672 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin
1615a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  e.  Fin )
171, 2, 3, 4, 5, 6, 7ballotlem5 28311 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  E. k  e.  ( 1 ... ( M  +  N )
) ( ( F `
 C ) `  k )  =  0 )
18 rabn0 3791 . . . . 5  |-  ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `  C ) `  k
)  =  0 }  =/=  (/)  <->  E. k  e.  ( 1 ... ( M  +  N ) ) ( ( F `  C ) `  k
)  =  0 )
1917, 18sylibr 212 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  =/=  (/) )
20 fzssuz 11733 . . . . . . . 8  |-  ( 1 ... ( M  +  N ) )  C_  ( ZZ>= `  1 )
21 uzssz 11109 . . . . . . . 8  |-  ( ZZ>= ` 
1 )  C_  ZZ
2220, 21sstri 3498 . . . . . . 7  |-  ( 1 ... ( M  +  N ) )  C_  ZZ
23 zssre 10877 . . . . . . 7  |-  ZZ  C_  RR
2422, 23sstri 3498 . . . . . 6  |-  ( 1 ... ( M  +  N ) )  C_  RR
2513, 24sstri 3498 . . . . 5  |-  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR
2625a1i 11 . . . 4  |-  ( C  e.  ( O  \  E )  ->  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  C
) `  k )  =  0 }  C_  RR )
27 fisupcl 7930 . . . 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 } )
2811, 16, 19, 26, 27syl13anc 1231 . . 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 } )
299, 28eqeltrd 2531 . 2  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 C ) `  k )  =  0 } )
30 fveq2 5856 . . . 4  |-  ( k  =  ( I `  C )  ->  (
( F `  C
) `  k )  =  ( ( F `
 C ) `  ( I `  C
) ) )
3130eqeq1d 2445 . . 3  |-  ( k  =  ( I `  C )  ->  (
( ( F `  C ) `  k
)  =  0  <->  (
( F `  C
) `  ( I `  C ) )  =  0 ) )
3231elrab 3243 . 2  |-  ( ( I `  C )  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  C ) `
 k )  =  0 }  <->  ( (
I `  C )  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  C ) `  ( I `  C
) )  =  0 ) )
3329, 32sylib 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 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3770   ~Pcpw 3997   class class class wbr 4437    |-> cmpt 4495    Or wor 4789   `'ccnv 4988   ` cfv 5578  (class class class)co 6281   Fincfn 7518   supcsup 7902   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    < clt 9631    - cmin 9810    / cdiv 10212   NNcn 10542   ZZcz 10870   ZZ>=cuz 11090   ...cfz 11681   #chash 12384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-hash 12385
This theorem is referenced by:  ballotlemi1  28314  ballotlemii  28315  ballotlemimin  28317  ballotlemic  28318  ballotlem1c  28319  ballotlemsgt1  28322  ballotlemsdom  28323  ballotlemsel1i  28324  ballotlemsf1o  28325  ballotlemsi  28326  ballotlemsima  28327  ballotlemrv2  28333  ballotlemfrc  28338  ballotlemfrci  28339  ballotlemfrceq  28340  ballotlemfrcn0  28341  ballotlemrc  28342  ballotlemirc  28343  ballotlem1ri  28346
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