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Theorem linepsubclN 37266
 Description: A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
linepsubcl.n 𝑁 = (Lines‘𝐾)
linepsubcl.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
linepsubclN ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)

Proof of Theorem linepsubclN
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hllat 36678 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 eqid 2798 . . . . 5 (join‘𝐾) = (join‘𝐾)
3 eqid 2798 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
4 linepsubcl.n . . . . 5 𝑁 = (Lines‘𝐾)
5 eqid 2798 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
62, 3, 4, 5isline2 37089 . . . 4 (𝐾 ∈ Lat → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)))))
71, 6syl 17 . . 3 (𝐾 ∈ HL → (𝑋𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)))))
81adantr 484 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝐾 ∈ Lat)
9 eqid 2798 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 36604 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1110ad2antrl 727 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
129, 3atbase 36604 . . . . . . . . 9 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
1312ad2antll 728 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
149, 2latjcl 17656 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
158, 11, 13, 14syl3anc 1368 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
16 linepsubcl.c . . . . . . . 8 𝐶 = (PSubCl‘𝐾)
179, 5, 16pmapsubclN 37261 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
1815, 17syldan 594 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19 eleq1a 2885 . . . . . 6 (((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶 → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋𝐶))
2018, 19syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋𝐶))
2120adantld 494 . . . 4 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
2221rexlimdvva 3253 . . 3 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
237, 22sylbid 243 . 2 (𝐾 ∈ HL → (𝑋𝑁𝑋𝐶))
2423imp 410 1 ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  joincjn 17549  Latclat 17650  Atomscatm 36578  HLchlt 36665  Linesclines 36809  pmapcpmap 36812  PSubClcpscN 37249 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-riotaBAD 36268 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-undef 7925  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36491  df-ol 36493  df-oml 36494  df-covers 36581  df-ats 36582  df-atl 36613  df-cvlat 36637  df-hlat 36666  df-lines 36816  df-pmap 36819  df-polarityN 37218  df-psubclN 37250 This theorem is referenced by: (None)
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