Theorem linepsubclN 39934

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 ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐶)

Proof of Theorem linepsubclN

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hllat 39345	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2		eqid 2735	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2735	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		linepsubcl.n	. . . . 5 ⊢ 𝑁 = (Lines‘𝐾)
5		eqid 2735	. . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
6	2, 3, 4, 5	isline2 39757	. . . 4 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)))))
7	1, 6	syl 17	. . 3 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)))))
8	1	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝐾 ∈ Lat)
9		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
10	9, 3	atbase 39271	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
11	10	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
12	9, 3	atbase 39271	. . . . . . . . 9 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
13	12	ad2antll 729	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
14	9, 2	latjcl 18497	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
15	8, 11, 13, 14	syl3anc 1370	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
16		linepsubcl.c	. . . . . . . 8 ⊢ 𝐶 = (PSubCl‘𝐾)
17	9, 5, 16	pmapsubclN 39929	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
18	15, 17	syldan 591	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19		eleq1a 2834	. . . . . 6 ⊢ (((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶 → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋 ∈ 𝐶))
20	18, 19	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → (𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) → 𝑋 ∈ 𝐶))
21	20	adantld 490	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋 ∈ 𝐶))
22	21	rexlimdvva 3211	. . 3 ⊢ (𝐾 ∈ HL → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑋 = ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞))) → 𝑋 ∈ 𝐶))
23	7, 22	sylbid 240	. 2 ⊢ (𝐾 ∈ HL → (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐶))
24	23	imp 406	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  joincjn 18369  Latclat 18489  Atomscatm 39245  HLchlt 39332  Linesclines 39477  pmapcpmap 39480  PSubClcpscN 39917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-lines 39484  df-pmap 39487  df-polarityN 39886  df-psubclN 39918

This theorem is referenced by: (None)