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Theorem linepsubclN 39953
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 39364 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 eqid 2737 . . . . 5 (join‘𝐾) = (join‘𝐾)
3 eqid 2737 . . . . 5 (Atoms‘𝐾) = (Atoms‘𝐾)
4 linepsubcl.n . . . . 5 𝑁 = (Lines‘𝐾)
5 eqid 2737 . . . . 5 (pmap‘𝐾) = (pmap‘𝐾)
62, 3, 4, 5isline2 39776 . . . 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 2737 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 39290 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1110ad2antrl 728 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
129, 3atbase 39290 . . . . . . . . 9 (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
1312ad2antll 729 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
149, 2latjcl 18484 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
158, 11, 13, 14syl3anc 1373 . . . . . . 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 39948 . . . . . . 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 2836 . . . . . 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 490 . . . 4 ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
2221rexlimdvva 3213 . . 3 (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝𝑞𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋𝐶))
237, 22sylbid 240 . 2 (𝐾 ∈ HL → (𝑋𝑁𝑋𝐶))
2423imp 406 1 ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cfv 6561  (class class class)co 7431  Basecbs 17247  joincjn 18357  Latclat 18476  Atomscatm 39264  HLchlt 39351  Linesclines 39496  pmapcpmap 39499  PSubClcpscN 39936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-lines 39503  df-pmap 39506  df-polarityN 39905  df-psubclN 39937
This theorem is referenced by: (None)
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