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

Theorem ballotlemrv 29425
Description: Value of  R evaluated at  J. (Contributed by Thierry Arnoux, 17-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 ) 
|-> inf ( { 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
ballotlemrv  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( J  e.  ( R `  C
)  <->  if ( J  <_ 
( I `  C
) ,  ( ( ( I `  C
)  +  1 )  -  J ) ,  J )  e.  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    k, J    S, k, i, c
Allowed substitution hints:    C( x, c)    P( x, i, k, c)    R( x, i, k, c)    S( x)    E( x)    F( x)    I( x)    J( x, i, c)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemrv
StepHypRef Expression
1 simpl 464 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  C  e.  ( O  \  E ) )
2 ballotth.m . . . . . 6  |-  M  e.  NN
3 ballotth.n . . . . . 6  |-  N  e.  NN
4 ballotth.o . . . . . 6  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
5 ballotth.p . . . . . 6  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
6 ballotth.f . . . . . 6  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
7 ballotth.e . . . . . 6  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
8 ballotth.mgtn . . . . . 6  |-  N  < 
M
9 ballotth.i . . . . . 6  |-  I  =  ( c  e.  ( O  \  E ) 
|-> inf ( { k  e.  ( 1 ... ( M  +  N )
)  |  ( ( F `  c ) `
 k )  =  0 } ,  RR ,  <  ) )
10 ballotth.s . . . . . 6  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
112, 3, 4, 5, 6, 7, 8, 9, 10ballotlemsf1o 29419 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1211simpld 466 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
13 f1ofun 5830 . . . 4  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  Fun  ( S `
 C ) )
141, 12, 133syl 18 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  Fun  ( S `  C ) )
15 simpr 468 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  J  e.  ( 1 ... ( M  +  N ) ) )
16 f1odm 5832 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  dom  ( S `
 C )  =  ( 1 ... ( M  +  N )
) )
171, 12, 163syl 18 . . . 4  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  dom  ( S `  C )  =  ( 1 ... ( M  +  N ) ) )
1815, 17eleqtrrd 2552 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  J  e.  dom  ( S `  C ) )
19 fvimacnv 6012 . . 3  |-  ( ( Fun  ( S `  C )  /\  J  e.  dom  ( S `  C ) )  -> 
( ( ( S `
 C ) `  J )  e.  C  <->  J  e.  ( `' ( S `  C )
" C ) ) )
2014, 18, 19syl2anc 673 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( ( S `  C ) `
 J )  e.  C  <->  J  e.  ( `' ( S `  C ) " C
) ) )
212, 3, 4, 5, 6, 7, 8, 9, 10ballotlemsv 29415 . . 3  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( S `
 C ) `  J )  =  if ( J  <_  (
I `  C ) ,  ( ( ( I `  C )  +  1 )  -  J ) ,  J
) )
2221eleq1d 2533 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( ( ( S `  C ) `
 J )  e.  C  <->  if ( J  <_ 
( I `  C
) ,  ( ( ( I `  C
)  +  1 )  -  J ) ,  J )  e.  C
) )
2311simprd 470 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  `' ( S `  C )  =  ( S `  C ) )
2423imaeq1d 5173 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( `' ( S `  C ) " C
)  =  ( ( S `  C )
" C ) )
25 ballotth.r . . . . . 6  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
262, 3, 4, 5, 6, 7, 8, 9, 10, 25ballotlemrval 29423 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
2724, 26eqtr4d 2508 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( `' ( S `  C ) " C
)  =  ( R `
 C ) )
2827eleq2d 2534 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( J  e.  ( `' ( S `  C )
" C )  <->  J  e.  ( R `  C ) ) )
291, 28syl 17 . 2  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( J  e.  ( `' ( S `
 C ) " C )  <->  J  e.  ( R `  C ) ) )
3020, 22, 293bitr3rd 292 1  |-  ( ( C  e.  ( O 
\  E )  /\  J  e.  ( 1 ... ( M  +  N ) ) )  ->  ( J  e.  ( R `  C
)  <->  if ( J  <_ 
( I `  C
) ,  ( ( ( I `  C
)  +  1 )  -  J ) ,  J )  e.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    \ cdif 3387    i^i cin 3389   ifcif 3872   ~Pcpw 3942   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   "cima 4842   Fun wfun 5583   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308  infcinf 7973   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   ZZcz 10961   ...cfz 11810   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-hash 12554
This theorem is referenced by:  ballotlemrv1  29426  ballotlemrv2  29427
  Copyright terms: Public domain W3C validator