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Theorem psubssat 35562
 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 2771 . . 3 (le‘𝐾) = (le‘𝐾)
2 eqid 2771 . . 3 (join‘𝐾) = (join‘𝐾)
3 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
4 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4ispsubsp 35553 . 2 (𝐾𝐵 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
65simprbda 486 1 ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  lecple 16156  joincjn 17152  Atomscatm 35072  PSubSpcpsubsp 35304 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-psubsp 35311 This theorem is referenced by:  psubatN  35563  paddidm  35649  paddclN  35650  paddss  35653  pmodlem1  35654  pmod1i  35656  pmodl42N  35659  elpcliN  35701  pclidN  35704  pclbtwnN  35705  pclunN  35706  pclun2N  35707  pclfinN  35708  polssatN  35716  psubclsubN  35748
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