Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapat Structured version   Unicode version

Theorem pmapat 30487
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 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 pmapat.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 30014 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4 eqid 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 pmapat.m . . . 4  |-  M  =  ( pmap `  K
)
61, 4, 2, 5pmapval 30481 . . 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 30085 . . . . 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 30036 . . . 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 2939 . 2  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  { q  e.  A  |  q ( le
`  K ) P }  =  { q  e.  A  |  q  =  P } )
15 rabsn 3865 . . 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 2471 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 1652    e. wcel 1725   {crab 2701   {csn 3806   class class class wbr 4204   ` cfv 5446   Basecbs 13461   lecple 13528   Atomscatm 29988   AtLatcal 29989   HLchlt 30075   pmapcpmap 30221
This theorem is referenced by:  elpmapat  30488  2polatN  30656  paddatclN  30673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-poset 14395  df-plt 14407  df-lat 14467  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-pmap 30228
  Copyright terms: Public domain W3C validator