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Theorem ballotlemro 29355
Description: Range of  R is included in  O. (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
ballotlemro  |-  ( 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    i, c, F, k    C, i, k    i, E, k    C, 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 ballotlemro
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 ) 
|-> inf ( { 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, 10ballotlemrval 29350 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
12 imassrn 5179 . . . 4  |-  ( ( S `  C )
" C )  C_  ran  ( S `  C
)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 29346 . . . . . 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 5821 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-onto-> ( 1 ... ( M  +  N )
) )
16 forn 5796 . . . . 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 3480 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
C_  ( 1 ... ( M  +  N
) ) )
1911, 18eqsstrd 3466 . 2  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  C_  ( 1 ... ( M  +  N )
) )
20 f1of1 5813 . . . . . . 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 3555 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
231, 2, 3ballotlemelo 29320 . . . . . . . 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 7629 . . . . . 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 1268 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
~~  C )
2911, 28eqbrtrd 4423 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  ~~  C )
30 hasheni 12531 . . . 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 2485 . 2  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  M )
341, 2, 3ballotlemelo 29320 . 2  |-  ( ( R `  C )  e.  O  <->  ( ( R `  C )  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  ( R `  C )
)  =  M ) )
3519, 33, 34sylanbrc 670 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   {crab 2741    \ cdif 3401    i^i cin 3403    C_ wss 3404   ifcif 3881   ~Pcpw 3951   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   ran crn 4835   "cima 4837   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290    ~~ cen 7566  infcinf 7955   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   ZZcz 10937   ...cfz 11784   #chash 12515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-hash 12516
This theorem is referenced by:  ballotlemfrc  29359  ballotlemfrceq  29361  ballotlemfrcn0  29362  ballotlemrc  29363
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