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Theorem psubclsubN 34244
 Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclsub.s 𝑆 = (PSubSp‘𝐾)
psubclsub.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclsubN ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)

Proof of Theorem psubclsubN
StepHypRef Expression
1 eqid 2610 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
2 psubclsub.c . . 3 𝐶 = (PSubCl‘𝐾)
31, 2psubcli2N 34243 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)
4 eqid 2610 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
54, 1, 2psubcliN 34242 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
65simpld 474 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7 psubclsub.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
84, 7, 1polsubN 34211 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
96, 8syldan 486 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
104, 7psubssat 34058 . . . 4 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
119, 10syldan 486 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
124, 7, 1polsubN 34211 . . 3 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
1311, 12syldan 486 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
143, 13eqeltrrd 2689 1 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ‘cfv 5804  Atomscatm 33568  HLchlt 33655  PSubSpcpsubsp 33800  ⊥𝑃cpolN 34206  PSubClcpscN 34238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-undef 7286  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-polarityN 34207  df-psubclN 34239 This theorem is referenced by:  pclfinclN  34254
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