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Theorem linepsubclN 36618
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 36030 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 eqid 2795 . . . . 5 (join‘𝐾) = (join‘𝐾)
3 eqid 2795 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
4 linepsubcl.n . . . . 5 𝑁 = (Lines‘𝐾)
5 eqid 2795 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
62, 3, 4, 5isline2 36441 . . . 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 481 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝐾 ∈ Lat)
9 eqid 2795 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 35956 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1110ad2antrl 724 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
129, 3atbase 35956 . . . . . . . . 9 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
1312ad2antll 725 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
149, 2latjcl 17490 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
158, 11, 13, 14syl3anc 1364 . . . . . . 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 36613 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
1815, 17syldan 591 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19 eleq1a 2878 . . . . . 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 491 . . . 4 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
2221rexlimdvva 3257 . . 3 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
237, 22sylbid 241 . 2 (𝐾 ∈ HL → (𝑋𝑁𝑋𝐶))
2423imp 407 1 ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wne 2984  wrex 3106  cfv 6225  (class class class)co 7016  Basecbs 16312  joincjn 17383  Latclat 17484  Atomscatm 35930  HLchlt 36017  Linesclines 36161  pmapcpmap 36164  PSubClcpscN 36601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-riotaBAD 35620
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-undef 7790  df-proset 17367  df-poset 17385  df-plt 17397  df-lub 17413  df-glb 17414  df-join 17415  df-meet 17416  df-p0 17478  df-p1 17479  df-lat 17485  df-clat 17547  df-oposet 35843  df-ol 35845  df-oml 35846  df-covers 35933  df-ats 35934  df-atl 35965  df-cvlat 35989  df-hlat 36018  df-lines 36168  df-pmap 36171  df-polarityN 36570  df-psubclN 36602
This theorem is referenced by: (None)
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