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Theorem pmapssat 39742
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 2735 . . 3 (le‘𝐾) = (le‘𝐾)
3 pmapssat.a . . 3 𝐴 = (Atoms‘𝐾)
4 pmapssat.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 39740 . 2 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑝𝐴𝑝(le‘𝐾)𝑋})
6 ssrab2 4090 . 2 {𝑝𝐴𝑝(le‘𝐾)𝑋} ⊆ 𝐴
75, 6eqsstrdi 4050 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  wss 3963   class class class wbr 5148  cfv 6563  Basecbs 17245  lecple 17305  Atomscatm 39245  pmapcpmap 39480
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-pmap 39487
This theorem is referenced by:  pmapssbaN  39743  pmapglb2N  39754  pmapglb2xN  39755  pmapjoin  39835  pmapjat1  39836  pmapjat2  39837  pmapjlln1  39838  hlmod1i  39839  polpmapN  39895  2pmaplubN  39909  pmapj2N  39912  pmapocjN  39913  polatN  39914  pmapsubclN  39929  ispsubcl2N  39930  pl42lem2N  39963  pl42lem3N  39964
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