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Theorem psubclsubN 36956
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 2818 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
2 psubclsub.c . . 3 𝐶 = (PSubCl‘𝐾)
31, 2psubcli2N 36955 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋)
4 eqid 2818 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
54, 1, 2psubcliN 36954 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
65simpld 495 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7 psubclsub.s . . . . . 6 𝑆 = (PSubSp‘𝐾)
84, 7, 1polsubN 36923 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
96, 8syldan 591 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆)
104, 7psubssat 36770 . . . 4 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ∈ 𝑆) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
119, 10syldan 591 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
124, 7, 1polsubN 36923 . . 3 ((𝐾 ∈ HL ∧ ((⊥𝑃𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
1311, 12syldan 591 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐶) → ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) ∈ 𝑆)
143, 13eqeltrrd 2911 1 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wss 3933  cfv 6348  Atomscatm 36279  HLchlt 36366  PSubSpcpsubsp 36512  𝑃cpolN 36918  PSubClcpscN 36950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-riotaBAD 35969
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-undef 7928  df-proset 17526  df-poset 17544  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-psubsp 36519  df-pmap 36520  df-polarityN 36919  df-psubclN 36951
This theorem is referenced by:  pclfinclN  36966
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