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Theorem ballotlemirc 28221
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 28220 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  ( O  \  E
) )
121, 2, 3, 4, 5, 6, 7, 8ballotlemi 28190 . . 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 9666 . . . . 5  |-  <  Or  RR
15 cnvso 5546 . . . . 5  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
1614, 15mpbi 208 . . . 4  |-  `'  <  Or  RR
1716a1i 11 . . 3  |-  ( C  e.  ( O  \  E )  ->  `'  <  Or  RR )
181, 2, 3, 4, 5, 6, 7, 8ballotlemiex 28191 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  ( ( F `  C ) `  (
I `  C )
)  =  0 ) )
1918simpld 459 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ( 1 ... ( M  +  N )
) )
20 elfzelz 11689 . . . . 5  |-  ( ( I `  C )  e.  ( 1 ... ( M  +  N
) )  ->  (
I `  C )  e.  ZZ )
2119, 20syl 16 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  ZZ )
2221zred 10967 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  RR )
23 eqid 2467 . . . . 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 28217 . . . 4  |-  ( C  e.  ( O  \  E )  ->  (
( F `  ( R `  C )
) `  ( I `  C ) )  =  0 )
25 fveq2 5866 . . . . . 6  |-  ( k  =  ( I `  C )  ->  (
( F `  ( R `  C )
) `  k )  =  ( ( F `
 ( R `  C ) ) `  ( I `  C
) ) )
2625eqeq1d 2469 . . . . 5  |-  ( k  =  ( I `  C )  ->  (
( ( F `  ( R `  C ) ) `  k )  =  0  <->  ( ( F `  ( R `  C ) ) `  ( I `  C
) )  =  0 ) )
2726elrab 3261 . . . 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 664 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
I `  C )  e.  { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } )
29 elrabi 3258 . . . . 5  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  ->  y  e.  ( 1 ... ( M  +  N )
) )
3029anim2i 569 . . . 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 465 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( I `  C )  e.  ( 1 ... ( M  +  N ) ) )
32 vex 3116 . . . . . . . . . 10  |-  y  e. 
_V
33 brcnvg 5183 . . . . . . . . . 10  |-  ( ( ( I `  C
)  e.  ( 1 ... ( M  +  N ) )  /\  y  e.  _V )  ->  ( ( I `  C ) `'  <  y  <-> 
y  <  ( I `  C ) ) )
3431, 32, 33sylancl 662 . . . . . . . . 9  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( I `
 C ) `'  <  y  <->  y  <  ( I `  C ) ) )
3534biimpa 484 . . . . . . . 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 28219 . . . . . . . . . . 11  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  -> 
( ( F `  ( R `  C ) ) `  y )  =/=  0 )
3736neneqd 2669 . . . . . . . . . 10  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  ( 1 ... ( M  +  N ) )  /\  y  <  ( I `  C ) )  ->  -.  ( ( F `  ( R `  C ) ) `  y )  =  0 )
38 fveq2 5866 . . . . . . . . . . . . 13  |-  ( k  =  y  ->  (
( F `  ( R `  C )
) `  k )  =  ( ( F `
 ( R `  C ) ) `  y ) )
3938eqeq1d 2469 . . . . . . . . . . . 12  |-  ( k  =  y  ->  (
( ( F `  ( R `  C ) ) `  k )  =  0  <->  ( ( F `  ( R `  C ) ) `  y )  =  0 ) )
4039elrab 3261 . . . . . . . . . . 11  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  <->  ( y  e.  ( 1 ... ( M  +  N )
)  /\  ( ( F `  ( R `  C ) ) `  y )  =  0 ) )
4140simprbi 464 . . . . . . . . . 10  |-  ( y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 }  ->  (
( F `  ( R `  C )
) `  y )  =  0 )
4237, 41nsyl 121 . . . . . . . . 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 1196 . . . . . . . 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 470 . . . . . . 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 434 . . . . . 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 115 . . . . 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 429 . . . 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 667 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  y  e.  { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  ( R `
 C ) ) `
 k )  =  0 } )  ->  -.  ( I `  C
) `'  <  y
)
4917, 22, 28, 48supmax 7926 . 2  |-  ( C  e.  ( O  \  E )  ->  sup ( { k  e.  ( 1 ... ( M  +  N ) )  |  ( ( F `
 ( R `  C ) ) `  k )  =  0 } ,  RR ,  `'  <  )  =  ( I `  C ) )
5013, 49eqtrd 2508 1  |-  ( C  e.  ( O  \  E )  ->  (
I `  ( R `  C ) )  =  ( I `  C
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    i^i cin 3475   ifcif 3939   ~Pcpw 4010   class class class wbr 4447    |-> cmpt 4505    Or wor 4799   `'ccnv 4998   "cima 5002   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   Fincfn 7517   supcsup 7901   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    < clt 9629    <_ cle 9630    - cmin 9806    / cdiv 10207   NNcn 10537   ZZcz 10865   ...cfz 11673   #chash 12374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-hash 12375
This theorem is referenced by:  ballotlemrinv0  28222
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