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Theorem pmapssat 35772
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 2797 . . 3 (le‘𝐾) = (le‘𝐾)
3 pmapssat.a . . 3 𝐴 = (Atoms‘𝐾)
4 pmapssat.m . . 3 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapval 35770 . 2 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑝𝐴𝑝(le‘𝐾)𝑋})
6 ssrab2 3881 . 2 {𝑝𝐴𝑝(le‘𝐾)𝑋} ⊆ 𝐴
75, 6syl6eqss 3849 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {crab 3091  wss 3767   class class class wbr 4841  cfv 6099  Basecbs 16181  lecple 16271  Atomscatm 35276  pmapcpmap 35510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-pmap 35517
This theorem is referenced by:  pmapssbaN  35773  pmapglb2N  35784  pmapglb2xN  35785  pmapjoin  35865  pmapjat1  35866  pmapjat2  35867  pmapjlln1  35868  hlmod1i  35869  polpmapN  35925  2pmaplubN  35939  pmapj2N  35942  pmapocjN  35943  polatN  35944  pmapsubclN  35959  ispsubcl2N  35960  pl42lem2N  35993  pl42lem3N  35994
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