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Theorem psubssat 39736
Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atpsub.a 𝐴 = (Atoms‘𝐾)
atpsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
psubssat ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)

Proof of Theorem psubssat
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . 3 (le‘𝐾) = (le‘𝐾)
2 eqid 2734 . . 3 (join‘𝐾) = (join‘𝐾)
3 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
4 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4ispsubsp 39727 . 2 (𝐾𝐵 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
65simprbda 498 1 ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  wss 3962   class class class wbr 5147  cfv 6562  (class class class)co 7430  lecple 17304  joincjn 18368  Atomscatm 39244  PSubSpcpsubsp 39478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-psubsp 39485
This theorem is referenced by:  psubatN  39737  paddidm  39823  paddclN  39824  paddss  39827  pmodlem1  39828  pmod1i  39830  pmodl42N  39833  elpcliN  39875  pclidN  39878  pclbtwnN  39879  pclunN  39880  pclun2N  39881  pclfinN  39882  polssatN  39890  psubclsubN  39922
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