Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ballotlemsgt1 Structured version   Unicode version

Theorem ballotlemsgt1 24761
Description:  S maps values less than  (
I `  C ) to values greater than 1. (Contributed by Thierry Arnoux, 28-Apr-2017.)
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 ,  `'  <  ) )
ballotth.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
Assertion
Ref Expression
ballotlemsgt1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  <  ( ( S `  C ) `  J ) )
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, c    E, c    i, I, c
Allowed substitution hints:    C( x, c)    P( x, i, k, c)    S( x, i, k, c)    E( x)    F( x)    I( x)    J( x, i, k, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemsgt1
StepHypRef Expression
1 elfzelz 11052 . . . . 5  |-  ( J  e.  ( 1 ... ( M  +  N
) )  ->  J  e.  ZZ )
213ad2ant2 979 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  e.  ZZ )
32zred 10368 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  e.  RR )
4 ballotth.m . . . . . . . . 9  |-  M  e.  NN
5 ballotth.n . . . . . . . . 9  |-  N  e.  NN
6 ballotth.o . . . . . . . . 9  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
7 ballotth.p . . . . . . . . 9  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
8 ballotth.f . . . . . . . . 9  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
9 ballotth.e . . . . . . . . 9  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
10 ballotth.mgtn . . . . . . . . 9  |-  N  < 
M
11 ballotth.i . . . . . . . . 9  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
124, 5, 6, 7, 8, 9, 10, 11ballotlemiex 24752 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1312simpld 446 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
14 elfzelz 11052 . . . . . . 7  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ZZ )
1513, 14syl 16 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
16153ad2ant1 978 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( I `  C
)  e.  ZZ )
1716zred 10368 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( I `  C
)  e.  RR )
18 1re 9083 . . . . 5  |-  1  e.  RR
1918a1i 11 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  e.  RR )
2017, 19readdcld 9108 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( I `  C )  +  1 )  e.  RR )
21 simp3 959 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  <  ( I `  C ) )
2216zcnd 10369 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( I `  C
)  e.  CC )
23 ax-1cn 9041 . . . . . 6  |-  1  e.  CC
2423a1i 11 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  e.  CC )
2522, 24pncand 9405 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( ( I `
 C )  +  1 )  -  1 )  =  ( I `
 C ) )
2621, 25breqtrrd 4231 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  <  ( ( ( I `  C )  +  1 )  - 
1 ) )
273, 20, 19, 26ltsub13d 9625 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  <  ( (
( I `  C
)  +  1 )  -  J ) )
28 ballotth.s . . . . 5  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
294, 5, 6, 7, 8, 9, 10, 11, 28ballotlemsv 24760 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( S `
 C ) `  J )  =  if ( J  <_  (
I `  C ) ,  ( ( ( I `  C )  +  1 )  -  J ) ,  J
) )
30293adant3 977 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( S `  C ) `  J
)  =  if ( J  <_  ( I `  C ) ,  ( ( ( I `  C )  +  1 )  -  J ) ,  J ) )
313, 17, 21ltled 9214 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  <_  ( I `  C ) )
32 iftrue 3738 . . . 4  |-  ( J  <_  ( I `  C )  ->  if ( J  <_  ( I `
 C ) ,  ( ( ( I `
 C )  +  1 )  -  J
) ,  J )  =  ( ( ( I `  C )  +  1 )  -  J ) )
3331, 32syl 16 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  if ( J  <_  (
I `  C ) ,  ( ( ( I `  C )  +  1 )  -  J ) ,  J
)  =  ( ( ( I `  C
)  +  1 )  -  J ) )
3430, 33eqtrd 2468 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( S `  C ) `  J
)  =  ( ( ( I `  C
)  +  1 )  -  J ) )
3527, 34breqtrrd 4231 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  <  ( ( S `  C ) `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2698   {crab 2702    \ cdif 3310    i^i cin 3312   ifcif 3732   ~Pcpw 3792   class class class wbr 4205    e. cmpt 4259   `'ccnv 4870   ` cfv 5447  (class class class)co 6074   supcsup 7438   CCcc 8981   RRcr 8982   0cc0 8983   1c1 8984    + caddc 8986    < clt 9113    <_ cle 9114    - cmin 9284    / cdiv 9670   NNcn 9993   ZZcz 10275   ...cfz 11036   #chash 11611
This theorem is referenced by:  ballotlemfrcn0  24780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-card 7819  df-cda 8041  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-n0 10215  df-z 10276  df-uz 10482  df-fz 11037  df-hash 11612
  Copyright terms: Public domain W3C validator