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Theorem pmapat 29877
Description: The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
Hypotheses
Ref Expression
pmapat.a  |-  A  =  ( Atoms `  K )
pmapat.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapat  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( M `  P
)  =  { P } )

Proof of Theorem pmapat
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmapat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 29404 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4 eqid 2387 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 pmapat.m . . . 4  |-  M  =  ( pmap `  K
)
61, 4, 2, 5pmapval 29871 . . 3  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K ) )  -> 
( M `  P
)  =  { q  e.  A  |  q ( le `  K
) P } )
73, 6sylan2 461 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( M `  P
)  =  { q  e.  A  |  q ( le `  K
) P } )
8 hlatl 29475 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
98ad2antrr 707 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  K  e.  AtLat
)
10 simpr 448 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  q  e.  A )
11 simplr 732 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  P  e.  A )
124, 2atcmp 29426 . . . 4  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  (
q ( le `  K ) P  <->  q  =  P ) )
139, 10, 11, 12syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  ( q
( le `  K
) P  <->  q  =  P ) )
1413rabbidva 2890 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  { q  e.  A  |  q ( le
`  K ) P }  =  { q  e.  A  |  q  =  P } )
15 rabsn 3816 . . 3  |-  ( P  e.  A  ->  { q  e.  A  |  q  =  P }  =  { P } )
1615adantl 453 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  { q  e.  A  |  q  =  P }  =  { P } )
177, 14, 163eqtrd 2423 1  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( M `  P
)  =  { P } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653   {csn 3757   class class class wbr 4153   ` cfv 5394   Basecbs 13396   lecple 13463   Atomscatm 29378   AtLatcal 29379   HLchlt 29465   pmapcpmap 29611
This theorem is referenced by:  elpmapat  29878  2polatN  30046  paddatclN  30063
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-poset 14330  df-plt 14342  df-lat 14402  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-pmap 29618
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