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Theorem pmapat 29952
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 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmapat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 29479 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 pmapat.m . . . 4  |-  M  =  ( pmap `  K
)
61, 4, 2, 5pmapval 29946 . . 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 29550 . . . . 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 29501 . . . 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 2779 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  { q  e.  A  |  q ( le
`  K ) P }  =  { q  e.  A  |  q  =  P } )
15 rabsn 3697 . . 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 2319 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 1623    e. wcel 1684   {crab 2547   {csn 3640   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   pmapcpmap 29686
This theorem is referenced by:  elpmapat  29953  2polatN  30121  paddatclN  30138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-poset 14080  df-plt 14092  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-pmap 29693
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