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Theorem ballotlemsgt1 27057
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 11573 . . . . 5  |-  ( J  e.  ( 1 ... ( M  +  N
) )  ->  J  e.  ZZ )
213ad2ant2 1010 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  e.  ZZ )
32zred 10861 . . 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 27048 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1312simpld 459 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
14 elfzelz 11573 . . . . . . 7  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ZZ )
1513, 14syl 16 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
16153ad2ant1 1009 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( I `  C
)  e.  ZZ )
1716zred 10861 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( I `  C
)  e.  RR )
18 1re 9499 . . . . 5  |-  1  e.  RR
1918a1i 11 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  e.  RR )
2017, 19readdcld 9527 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( I `  C )  +  1 )  e.  RR )
21 simp3 990 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  <  ( I `  C ) )
2216zcnd 10862 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( I `  C
)  e.  CC )
23 ax-1cn 9454 . . . . . 6  |-  1  e.  CC
2423a1i 11 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  e.  CC )
2522, 24pncand 9834 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( ( I `
 C )  +  1 )  -  1 )  =  ( I `
 C ) )
2621, 25breqtrrd 4429 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  <  ( ( ( I `  C )  +  1 )  - 
1 ) )
273, 20, 19, 26ltsub13d 10059 . 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 27056 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( S `
 C ) `  J )  =  if ( J  <_  (
I `  C ) ,  ( ( ( I `  C )  +  1 )  -  J ) ,  J
) )
30293adant3 1008 . . 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 9636 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  ->  J  <_  ( I `  C ) )
32 iftrue 3908 . . . 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 2495 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
( ( S `  C ) `  J
)  =  ( ( ( I `  C
)  +  1 )  -  J ) )
3527, 34breqtrrd 4429 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) )  /\  J  <  ( I `  C ) )  -> 
1  <  ( ( S `  C ) `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803    \ cdif 3436    i^i cin 3438   ifcif 3902   ~Pcpw 3971   class class class wbr 4403    |-> cmpt 4461   `'ccnv 4950   ` cfv 5529  (class class class)co 6203   supcsup 7804   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    < clt 9532    <_ cle 9533    - cmin 9709    / cdiv 10107   NNcn 10436   ZZcz 10760   ...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:  ballotlemfrcn0  27076
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