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Theorem ballotlemro 24552
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 24547 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
12 imassrn 5149 . . . 4  |-  ( ( S `  C )
" C )  C_  ran  ( S `  C
)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 24543 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1413simpld 446 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
15 f1ofo 5614 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-onto-> ( 1 ... ( M  +  N )
) )
16 forn 5589 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -onto-> ( 1 ... ( M  +  N ) )  ->  ran  ( S `  C
)  =  ( 1 ... ( M  +  N ) ) )
1714, 15, 163syl 19 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ran  ( S `  C )  =  ( 1 ... ( M  +  N
) ) )
1812, 17syl5sseq 3332 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
C_  ( 1 ... ( M  +  N
) ) )
1911, 18eqsstrd 3318 . 2  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  C_  ( 1 ... ( M  +  N )
) )
20 f1of1 5606 . . . . . . 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 3405 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
231, 2, 3ballotlemelo 24517 . . . . . . . 8  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
2422, 23sylib 189 . . . . . . 7  |-  ( C  e.  ( O  \  E )  ->  ( C  C_  ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
2524simpld 446 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
26 id 20 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  e.  ( O  \  E
) )
27 f1imaeng 7096 . . . . . 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 1184 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
~~  C )
2911, 28eqbrtrd 4166 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  ~~  C )
30 hasheni 11552 . . . 4  |-  ( ( R `  C ) 
~~  C  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3129, 30syl 16 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3224simprd 450 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 C )  =  M )
3331, 32eqtrd 2412 . 2  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  M )
341, 2, 3ballotlemelo 24517 . 2  |-  ( ( R `  C )  e.  O  <->  ( ( R `  C )  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  ( R `  C )
)  =  M ) )
3519, 33, 34sylanbrc 646 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646    \ cdif 3253    i^i cin 3255    C_ wss 3256   ifcif 3675   ~Pcpw 3735   class class class wbr 4146    e. cmpt 4200   `'ccnv 4810   ran crn 4812   "cima 4814   -1-1->wf1 5384   -onto->wfo 5385   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013    ~~ cen 7035   supcsup 7373   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    < clt 9046    <_ cle 9047    - cmin 9216    / cdiv 9602   NNcn 9925   ZZcz 10207   ...cfz 10968   #chash 11538
This theorem is referenced by:  ballotlemfrc  24556  ballotlemfrceq  24558  ballotlemfrcn0  24559  ballotlemrc  24560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-hash 11539
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