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Theorem linepsubclN 34255
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 33668 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 eqid 2610 . . . . 5 (join‘𝐾) = (join‘𝐾)
3 eqid 2610 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
4 linepsubcl.n . . . . 5 𝑁 = (Lines‘𝐾)
5 eqid 2610 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
62, 3, 4, 5isline2 34078 . . . 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 480 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝐾 ∈ Lat)
9 eqid 2610 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 33594 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1110ad2antrl 760 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
129, 3atbase 33594 . . . . . . . . 9 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
1312ad2antll 761 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
149, 2latjcl 16874 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
158, 11, 13, 14syl3anc 1318 . . . . . . 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 34250 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
1815, 17syldan 486 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19 eleq1a 2683 . . . . . 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 482 . . . 4 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
2221rexlimdvva 3020 . . 3 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
237, 22sylbid 229 . 2 (𝐾 ∈ HL → (𝑋𝑁𝑋𝐶))
2423imp 444 1 ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wrex 2897  cfv 5804  (class class class)co 6549  Basecbs 15695  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655  Linesclines 33798  pmapcpmap 33801  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-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-lines 33805  df-pmap 33808  df-polarityN 34207  df-psubclN 34239
This theorem is referenced by: (None)
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