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Theorem ballotlemroOLD 29405
Description: Range of  R is included in  O. (Contributed by Thierry Arnoux, 17-Apr-2017.) Obsolete version of ballotlemro 29367 as of 6-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ballotthOLD.m  |-  M  e.  NN
ballotthOLD.n  |-  N  e.  NN
ballotthOLD.o  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
ballotthOLD.p  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
ballotthOLD.f  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
ballotthOLD.e  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
ballotthOLD.mgtn  |-  N  < 
M
ballotthOLD.i  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
ballotthOLD.s  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
ballotthOLD.r  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
Assertion
Ref Expression
ballotlemroOLD  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
Distinct variable groups:    M, c    N, c    O, c    i, M   
i, N    i, O    k, M    k, N    k, O    C, i, k    i,
c, F, k    i, E, k    k, I, c    E, c    i, I, c    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)    M( x)    N( x)    O( x)

Proof of Theorem ballotlemroOLD
StepHypRef Expression
1 ballotthOLD.m . . . 4  |-  M  e.  NN
2 ballotthOLD.n . . . 4  |-  N  e.  NN
3 ballotthOLD.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 ballotthOLD.p . . . 4  |-  P  =  ( x  e.  ~P O  |->  ( ( # `  x )  /  ( # `
 O ) ) )
5 ballotthOLD.f . . . 4  |-  F  =  ( c  e.  O  |->  ( i  e.  ZZ  |->  ( ( # `  (
( 1 ... i
)  i^i  c )
)  -  ( # `  ( ( 1 ... i )  \  c
) ) ) ) )
6 ballotthOLD.e . . . 4  |-  E  =  { c  e.  O  |  A. i  e.  ( 1 ... ( M  +  N ) ) 0  <  ( ( F `  c ) `
 i ) }
7 ballotthOLD.mgtn . . . 4  |-  N  < 
M
8 ballotthOLD.i . . . 4  |-  I  =  ( c  e.  ( O  \  E ) 
|->  sup ( { k  e.  ( 1 ... ( M  +  N
) )  |  ( ( F `  c
) `  k )  =  0 } ,  RR ,  `'  <  ) )
9 ballotthOLD.s . . . 4  |-  S  =  ( c  e.  ( O  \  E ) 
|->  ( i  e.  ( 1 ... ( M  +  N ) ) 
|->  if ( i  <_ 
( I `  c
) ,  ( ( ( I `  c
)  +  1 )  -  i ) ,  i ) ) )
10 ballotthOLD.r . . . 4  |-  R  =  ( c  e.  ( O  \  E ) 
|->  ( ( S `  c ) " c
) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10ballotlemrvalOLD 29400 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
12 imassrn 5182 . . . 4  |-  ( ( S `  C )
" C )  C_  ran  ( S `  C
)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1oOLD 29396 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1413simpld 461 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
15 f1ofo 5826 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-onto-> ( 1 ... ( M  +  N )
) )
16 forn 5801 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -onto-> ( 1 ... ( M  +  N ) )  ->  ran  ( S `  C
)  =  ( 1 ... ( M  +  N ) ) )
1714, 15, 163syl 18 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ran  ( S `  C )  =  ( 1 ... ( M  +  N
) ) )
1812, 17syl5sseq 3482 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
C_  ( 1 ... ( M  +  N
) ) )
1911, 18eqsstrd 3468 . 2  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  C_  ( 1 ... ( M  +  N )
) )
20 f1of1 5818 . . . . . . 7  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
2114, 20syl 17 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-> ( 1 ... ( M  +  N ) ) )
22 eldifi 3557 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
231, 2, 3ballotlemelo 29332 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
2422, 23sylib 200 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( C  C_  ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
2524simpld 461 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
26 id 22 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  e.  ( O  \  E
) )
27 f1imaeng 7634 . . . . . 6  |-  ( ( ( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-> ( 1 ... ( M  +  N ) )  /\  C  C_  (
1 ... ( M  +  N ) )  /\  C  e.  ( O  \  E ) )  -> 
( ( S `  C ) " C
)  ~~  C )
2821, 25, 26, 27syl3anc 1269 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
~~  C )
2911, 28eqbrtrd 4426 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  ~~  C )
30 hasheni 12538 . . . 4  |-  ( ( R `  C ) 
~~  C  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3129, 30syl 17 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3224simprd 465 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 C )  =  M )
3331, 32eqtrd 2487 . 2  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  M )
341, 2, 3ballotlemelo 29332 . 2  |-  ( ( R `  C )  e.  O  <->  ( ( R `  C )  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  ( R `  C )
)  =  M ) )
3519, 33, 34sylanbrc 671 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   {crab 2743    \ cdif 3403    i^i cin 3405    C_ wss 3406   ifcif 3883   ~Pcpw 3953   class class class wbr 4405    |-> cmpt 4464   `'ccnv 4836   ran crn 4838   "cima 4840   -1-1->wf1 5582   -onto->wfo 5583   -1-1-onto->wf1o 5584   ` cfv 5585  (class class class)co 6295    ~~ cen 7571   supcsup 7959   RRcr 9543   0cc0 9544   1c1 9545    + caddc 9547    < clt 9680    <_ cle 9681    - cmin 9865    / cdiv 10276   NNcn 10616   ZZcz 10944   ...cfz 11791   #chash 12522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-hash 12523
This theorem is referenced by:  ballotlemfrcOLD  29409  ballotlemfrceqOLD  29411  ballotlemfrcn0OLD  29412  ballotlemrcOLD  29413
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