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Theorem ballotlemirc 24746
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 24745 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E
) )
121, 2, 3, 4, 5, 6, 7, 8ballotlemi 24715 . . 3  |-  ( ( R `  C )  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  ( R `  C )
) `  k )  =  0 } ,  RR ,  `'  <  ) )
1311, 12syl 16 . 2  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  ( R `  C )
) `  k )  =  0 } ,  RR ,  `'  <  ) )
14 ltso 9116 . . . . 5  |-  <  Or  RR
15 cnvso 5374 . . . . 5  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1614, 15mpbi 200 . . . 4  |-  `'  <  Or  RR
1716a1i 11 . . 3  |-  ( C  e.  ( O  \  E )  ->  `'  <  Or  RR )
181, 2, 3, 4, 5, 6, 7, 8ballotlemiex 24716 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1918simpld 446 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
20 elfzelz 11019 . . . . 5  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ZZ )
2119, 20syl 16 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
2221zred 10335 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  RR )
23 eqid 2408 . . . . 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
) ) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 23ballotlemfrci 24742 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
25 fveq2 5691 . . . . . 6  |-  ( k  =  ( I `  C )  ->  (
( F `  ( R `  C )
) `  k )  =  ( ( F `
 ( R `  C ) ) `  ( I `  C
) ) )
2625eqeq1d 2416 . . . . 5  |-  ( k  =  ( I `  C )  ->  (
( ( F `  ( R `  C ) ) `  k )  =  0  <->  ( ( F `  ( R `  C ) ) `  ( I `  C
) )  =  0 ) )
2726elrab 3056 . . . 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 ) )
2819, 24, 27sylanbrc 646 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } )
29 elrabi 3054 . . . . 5  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  ->  y  e.  ( 1 ... ( M  +  N )
) )
3029anim2i 553 . . . 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 ) ) ) )
3119adantr 452 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
32 vex 2923 . . . . . . . . . 10  |-  y  e. 
_V
33 brcnvg 5016 . . . . . . . . . 10  |-  ( ( ( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  y  e.  _V )  ->  ( ( I `  C ) `'  <  y  <-> 
y  <  ( I `  C ) ) )
3431, 32, 33sylancl 644 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( I `
 C ) `'  <  y  <->  y  <  ( I `  C ) ) )
3534biimpa 471 . . . . . . . 8  |-  ( ( ( C  e.  ( O  \  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  /\  ( I `
 C ) `'  <  y )  -> 
y  <  ( I `  C ) )
361, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemfrcn0 24744 . . . . . . . . . . 11  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  -> 
( ( F `  ( R `  C ) ) `  y )  =/=  0 )
3736neneqd 2587 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  ->  -.  ( ( F `  ( R `  C ) ) `  y )  =  0 )
38 fveq2 5691 . . . . . . . . . . . . 13  |-  ( k  =  y  ->  (
( F `  ( R `  C )
) `  k )  =  ( ( F `
 ( R `  C ) ) `  y ) )
3938eqeq1d 2416 . . . . . . . . . . . 12  |-  ( k  =  y  ->  (
( ( F `  ( R `  C ) ) `  k )  =  0  <->  ( ( F `  ( R `  C ) ) `  y )  =  0 ) )
4039elrab 3056 . . . . . . . . . . 11  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  <->  ( y  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  ( R `  C ) ) `  y )  =  0 ) )
4140simprbi 451 . . . . . . . . . 10  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  ->  (
( F `  ( R `  C )
) `  y )  =  0 )
4237, 41nsyl 115 . . . . . . . . 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 } )
43423expa 1153 . . . . . . . 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 } )
4435, 43syldan 457 . . . . . . 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 } )
4544ex 424 . . . . . 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 } ) )
4645con2d 109 . . . . 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 ) )
4746imp 419 . . . 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
)
4830, 47sylancom 649 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 } )  ->  -.  ( I `  C
) `'  <  y
)
4917, 22, 28, 48supmax 7430 . 2  |-  ( C  e.  ( O  \  E )  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } ,  RR ,  `'  <  )  =  ( I `  C ) )
5013, 49eqtrd 2440 1  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   {crab 2674   _Vcvv 2920    \ cdif 3281    i^i cin 3283   ifcif 3703   ~Pcpw 3763   class class class wbr 4176    e. cmpt 4230    Or wor 4466   `'ccnv 4840   "cima 4844   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   Fincfn 7072   supcsup 7407   RRcr 8949   0cc0 8950   1c1 8951    + caddc 8953    < clt 9080    <_ cle 9081    - cmin 9251    / cdiv 9637   NNcn 9960   ZZcz 10242   ...cfz 11003   #chash 11577
This theorem is referenced by:  ballotlemrinv0  24747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-hash 11578
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