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Theorem ballotlemro 26910
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 26905 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  =  ( ( S `
 C ) " C ) )
12 imassrn 5185 . . . 4  |-  ( ( S `  C )
" C )  C_  ran  ( S `  C
)
131, 2, 3, 4, 5, 6, 7, 8, 9ballotlemsf1o 26901 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) : ( 1 ... ( M  +  N ) ) -1-1-onto-> ( 1 ... ( M  +  N ) )  /\  `' ( S `  C )  =  ( S `  C ) ) )
1413simpld 459 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
) )
15 f1ofo 5653 . . . . 5  |-  ( ( S `  C ) : ( 1 ... ( M  +  N
) ) -1-1-onto-> ( 1 ... ( M  +  N )
)  ->  ( S `  C ) : ( 1 ... ( M  +  N ) )
-onto-> ( 1 ... ( M  +  N )
) )
16 forn 5628 . . . . 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 3409 . . 3  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
C_  ( 1 ... ( M  +  N
) ) )
1911, 18eqsstrd 3395 . 2  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  C_  ( 1 ... ( M  +  N )
) )
20 f1of1 5645 . . . . . . 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 3483 . . . . . . . 8  |-  ( C  e.  ( O  \  E )  ->  C  e.  O )
231, 2, 3ballotlemelo 26875 . . . . . . . 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 459 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  C_  ( 1 ... ( M  +  N )
) )
26 id 22 . . . . . 6  |-  ( C  e.  ( O  \  E )  ->  C  e.  ( O  \  E
) )
27 f1imaeng 7374 . . . . . 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 1218 . . . . 5  |-  ( C  e.  ( O  \  E )  ->  (
( S `  C
) " C ) 
~~  C )
2911, 28eqbrtrd 4317 . . . 4  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  ~~  C )
30 hasheni 12124 . . . 4  |-  ( ( R `  C ) 
~~  C  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3129, 30syl 16 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  ( # `  C
) )
3224simprd 463 . . 3  |-  ( C  e.  ( O  \  E )  ->  ( # `
 C )  =  M )
3331, 32eqtrd 2475 . 2  |-  ( C  e.  ( O  \  E )  ->  ( # `
 ( R `  C ) )  =  M )
341, 2, 3ballotlemelo 26875 . 2  |-  ( ( R `  C )  e.  O  <->  ( ( R `  C )  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  ( R `  C )
)  =  M ) )
3519, 33, 34sylanbrc 664 1  |-  ( C  e.  ( O  \  E )  ->  ( R `  C )  e.  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   {crab 2724    \ cdif 3330    i^i cin 3332    C_ wss 3333   ifcif 3796   ~Pcpw 3865   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   ran crn 4846   "cima 4848   -1-1->wf1 5420   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    ~~ cen 7312   supcsup 7695   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    < clt 9423    <_ cle 9424    - cmin 9600    / cdiv 9998   NNcn 10327   ZZcz 10651   ...cfz 11442   #chash 12108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-hash 12109
This theorem is referenced by:  ballotlemfrc  26914  ballotlemfrceq  26916  ballotlemfrcn0  26917  ballotlemrc  26918
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