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Theorem pmapat 30574
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 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmapat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 30101 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 pmapat.m . . . 4  |-  M  =  ( pmap `  K
)
61, 4, 2, 5pmapval 30568 . . 3  |-  ( ( K  e.  HL  /\  P  e.  ( Base `  K ) )  -> 
( M `  P
)  =  { q  e.  A  |  q ( le `  K
) P } )
73, 6sylan2 460 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( M `  P
)  =  { q  e.  A  |  q ( le `  K
) P } )
8 hlatl 30172 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
98ad2antrr 706 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  K  e.  AtLat
)
10 simpr 447 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  q  e.  A )
11 simplr 731 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  P  e.  A )
124, 2atcmp 30123 . . . 4  |-  ( ( K  e.  AtLat  /\  q  e.  A  /\  P  e.  A )  ->  (
q ( le `  K ) P  <->  q  =  P ) )
139, 10, 11, 12syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A )  /\  q  e.  A
)  ->  ( q
( le `  K
) P  <->  q  =  P ) )
1413rabbidva 2792 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  { q  e.  A  |  q ( le
`  K ) P }  =  { q  e.  A  |  q  =  P } )
15 rabsn 3710 . . 3  |-  ( P  e.  A  ->  { q  e.  A  |  q  =  P }  =  { P } )
1615adantl 452 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  { q  e.  A  |  q  =  P }  =  { P } )
177, 14, 163eqtrd 2332 1  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( M `  P
)  =  { P } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   {csn 3653   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   pmapcpmap 30308
This theorem is referenced by:  elpmapat  30575  2polatN  30743  paddatclN  30760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-poset 14096  df-plt 14108  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-pmap 30315
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