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Theorem ballotlemirc 29216
Description: Applying  R does not change first ties. (Contributed by Thierry Arnoux, 19-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 ) ) )
ballotth.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
Assertion
Ref Expression
ballotlemirc  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
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    S, k, i, c    R, i, k    x, k, C    x, F    x, M    x, N
Allowed substitution hints:    C( c)    P( x, i, k, c)    R( x, c)    S( x)    E( x)    I( x)    O( x)

Proof of Theorem ballotlemirc
Dummy variables  y 
v  u are mutually distinct and distinct from all other variables.
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 ,  `'  <  ) )
9 ballotth.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
10 ballotth.r . . . 4  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrc 29215 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E
) )
121, 2, 3, 4, 5, 6, 7, 8ballotlemi 29185 . . 3  |-  ( ( R `  C )  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  ( R `  C )
) `  k )  =  0 } ,  RR ,  `'  <  ) )
1311, 12syl 17 . 2  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  ( R `  C )
) `  k )  =  0 } ,  RR ,  `'  <  ) )
14 gtso 9704 . . . 4  |-  `'  <  Or  RR
1514a1i 11 . . 3  |-  ( C  e.  ( O  \  E )  ->  `'  <  Or  RR )
161, 2, 3, 4, 5, 6, 7, 8ballotlemiex 29186 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1716simpld 460 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
18 elfzelz 11787 . . . . 5  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ZZ )
1917, 18syl 17 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
2019zred 11029 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  RR )
21 eqid 2420 . . . . 5  |-  ( u  e.  Fin ,  v  e.  Fin  |->  ( (
# `  ( v  i^i  u ) )  -  ( # `  ( v 
\  u ) ) ) )  =  ( u  e.  Fin , 
v  e.  Fin  |->  ( ( # `  (
v  i^i  u )
)  -  ( # `  ( v  \  u
) ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21ballotlemfrci 29212 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
23 fveq2 5872 . . . . . 6  |-  ( k  =  ( I `  C )  ->  (
( F `  ( R `  C )
) `  k )  =  ( ( F `
 ( R `  C ) ) `  ( I `  C
) ) )
2423eqeq1d 2422 . . . . 5  |-  ( k  =  ( I `  C )  ->  (
( ( F `  ( R `  C ) ) `  k )  =  0  <->  ( ( F `  ( R `  C ) ) `  ( I `  C
) )  =  0 ) )
2524elrab 3226 . . . 4  |-  ( ( I `  C )  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  <->  ( (
I `  C )  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  ( R `  C ) ) `  ( I `  C
) )  =  0 ) )
2617, 22, 25sylanbrc 668 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } )
27 elrabi 3223 . . . . 5  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  ->  y  e.  ( 1 ... ( M  +  N )
) )
2827anim2i 571 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 } )  -> 
( C  e.  ( O  \  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) ) )
2917adantr 466 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
30 vex 3081 . . . . . . . . . 10  |-  y  e. 
_V
31 brcnvg 5026 . . . . . . . . . 10  |-  ( ( ( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  y  e.  _V )  ->  ( ( I `  C ) `'  <  y  <-> 
y  <  ( I `  C ) ) )
3229, 30, 31sylancl 666 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( I `
 C ) `'  <  y  <->  y  <  ( I `  C ) ) )
3332biimpa 486 . . . . . . . 8  |-  ( ( ( C  e.  ( O  \  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  /\  ( I `
 C ) `'  <  y )  -> 
y  <  ( I `  C ) )
341, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemfrcn0 29214 . . . . . . . . . . 11  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  -> 
( ( F `  ( R `  C ) ) `  y )  =/=  0 )
3534neneqd 2623 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  ->  -.  ( ( F `  ( R `  C ) ) `  y )  =  0 )
36 fveq2 5872 . . . . . . . . . . . . 13  |-  ( k  =  y  ->  (
( F `  ( R `  C )
) `  k )  =  ( ( F `
 ( R `  C ) ) `  y ) )
3736eqeq1d 2422 . . . . . . . . . . . 12  |-  ( k  =  y  ->  (
( ( F `  ( R `  C ) ) `  k )  =  0  <->  ( ( F `  ( R `  C ) ) `  y )  =  0 ) )
3837elrab 3226 . . . . . . . . . . 11  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  <->  ( y  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  ( R `  C ) ) `  y )  =  0 ) )
3938simprbi 465 . . . . . . . . . 10  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  ->  (
( F `  ( R `  C )
) `  y )  =  0 )
4035, 39nsyl 124 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  ->  -.  y  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  ( R `  C )
) `  k )  =  0 } )
41403expa 1205 . . . . . . . 8  |-  ( ( ( C  e.  ( O  \  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  /\  y  < 
( I `  C
) )  ->  -.  y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 } )
4233, 41syldan 472 . . . . . . 7  |-  ( ( ( C  e.  ( O  \  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  /\  ( I `
 C ) `'  <  y )  ->  -.  y  e.  { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  ( R `  C )
) `  k )  =  0 } )
4342ex 435 . . . . . 6  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( I `
 C ) `'  <  y  ->  -.  y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 } ) )
4443con2d 118 . . . . 5  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( y  e. 
{ k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 }  ->  -.  (
I `  C ) `'  <  y ) )
4544imp 430 . . . 4  |-  ( ( ( C  e.  ( O  \  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  /\  y  e. 
{ k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } )  ->  -.  ( I `  C
) `'  <  y
)
4628, 45sylancom 671 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 } )  ->  -.  ( I `  C
) `'  <  y
)
4715, 20, 26, 46supmax 7978 . 2  |-  ( C  e.  ( O  \  E )  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } ,  RR ,  `'  <  )  =  ( I `  C ) )
4813, 47eqtrd 2461 1  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   {crab 2777   _Vcvv 3078    \ cdif 3430    i^i cin 3432   ifcif 3906   ~Pcpw 3976   class class class wbr 4417    |-> cmpt 4475    Or wor 4765   `'ccnv 4844   "cima 4848   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7568   supcsup 7951   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    < clt 9664    <_ cle 9665    - cmin 9849    / cdiv 10258   NNcn 10598   ZZcz 10926   ...cfz 11771   #chash 12501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-hash 12502
This theorem is referenced by:  ballotlemrinv0  29217
  Copyright terms: Public domain W3C validator