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Theorem pmapssat 36889
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 2821 . . 3 (le‘𝐾) = (le‘𝐾)
3 pmapssat.a . . 3 𝐴 = (Atoms‘𝐾)
4 pmapssat.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 36887 . 2 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑝𝐴𝑝(le‘𝐾)𝑋})
6 ssrab2 4055 . 2 {𝑝𝐴𝑝(le‘𝐾)𝑋} ⊆ 𝐴
75, 6eqsstrdi 4020 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  {crab 3142  wss 3935   class class class wbr 5058  cfv 6349  Basecbs 16477  lecple 16566  Atomscatm 36393  pmapcpmap 36627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-pmap 36634
This theorem is referenced by:  pmapssbaN  36890  pmapglb2N  36901  pmapglb2xN  36902  pmapjoin  36982  pmapjat1  36983  pmapjat2  36984  pmapjlln1  36985  hlmod1i  36986  polpmapN  37042  2pmaplubN  37056  pmapj2N  37059  pmapocjN  37060  polatN  37061  pmapsubclN  37076  ispsubcl2N  37077  pl42lem2N  37110  pl42lem3N  37111
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