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Theorem ballotlemro 26835
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 ) 
|->  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
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 ) 
|->  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, 10ballotlemrval 26830 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
12 imassrn 5177 . . . 4  |-  ( ( S `  C )
" C )  C_  ran  ( S `  C
)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 26826 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1413simpld 456 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
15 f1ofo 5645 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-onto-> ( 1 ... ( M  +  N )
) )
16 forn 5620 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -onto-> ( 1 ... ( M  +  N ) )  ->  ran  ( S `  C
)  =  ( 1 ... ( M  +  N ) ) )
1714, 15, 163syl 20 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ran  ( S `  C )  =  ( 1 ... ( M  +  N
) ) )
1812, 17syl5sseq 3401 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
C_  ( 1 ... ( M  +  N
) ) )
1911, 18eqsstrd 3387 . 2  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  C_  ( 1 ... ( M  +  N )
) )
20 f1of1 5637 . . . . . . 7  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-1-1-> ( 1 ... ( M  +  N )
) )
2114, 20syl 16 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-> ( 1 ... ( M  +  N ) ) )
22 eldifi 3475 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
231, 2, 3ballotlemelo 26800 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
2422, 23sylib 196 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( C  C_  ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
2524simpld 456 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
26 id 22 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  e.  ( O  \  E
) )
27 f1imaeng 7365 . . . . . 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 1213 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
~~  C )
2911, 28eqbrtrd 4309 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  ~~  C )
30 hasheni 12115 . . . 4  |-  ( ( R `  C ) 
~~  C  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3129, 30syl 16 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3224simprd 460 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 C )  =  M )
3331, 32eqtrd 2473 . 2  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  M )
341, 2, 3ballotlemelo 26800 . 2  |-  ( ( R `  C )  e.  O  <->  ( ( R `  C )  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  ( R `  C )
)  =  M ) )
3519, 33, 34sylanbrc 659 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   {crab 2717    \ cdif 3322    i^i cin 3324    C_ wss 3325   ifcif 3788   ~Pcpw 3857   class class class wbr 4289    e. cmpt 4347   `'ccnv 4835   ran crn 4837   "cima 4839   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    ~~ cen 7303   supcsup 7686   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    <_ cle 9415    - cmin 9591    / cdiv 9989   NNcn 10318   ZZcz 10642   ...cfz 11433   #chash 12099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-hash 12100
This theorem is referenced by:  ballotlemfrc  26839  ballotlemfrceq  26841  ballotlemfrcn0  26842  ballotlemrc  26843
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