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Theorem pmapssat 34560
Description: The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)
Hypotheses
Ref Expression
pmapssat.b 𝐵 = (Base‘𝐾)
pmapssat.a 𝐴 = (Atoms‘𝐾)
pmapssat.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmapssat ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)

Proof of Theorem pmapssat
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 pmapssat.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
3 pmapssat.a . . 3 𝐴 = (Atoms‘𝐾)
4 pmapssat.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 34558 . 2 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑝𝐴𝑝(le‘𝐾)𝑋})
6 ssrab2 3671 . 2 {𝑝𝐴𝑝(le‘𝐾)𝑋} ⊆ 𝐴
75, 6syl6eqss 3639 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3559   class class class wbr 4618  cfv 5852  Basecbs 15792  lecple 15880  Atomscatm 34065  pmapcpmap 34298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-pmap 34305
This theorem is referenced by:  pmapssbaN  34561  pmapglb2N  34572  pmapglb2xN  34573  pmapjoin  34653  pmapjat1  34654  pmapjat2  34655  pmapjlln1  34656  hlmod1i  34657  polpmapN  34713  2pmaplubN  34727  pmapj2N  34730  pmapocjN  34731  polatN  34732  pmapsubclN  34747  ispsubcl2N  34748  pl42lem2N  34781  pl42lem3N  34782
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