Theorem linepsubclN 39908

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 39319	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2		eqid 2740	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2740	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		linepsubcl.n	. . . . 5 ⊢ 𝑁 = (Lines‘𝐾)
5		eqid 2740	. . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
6	2, 3, 4, 5	isline2 39731	. . . 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 2740	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
10	9, 3	atbase 39245	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
11	10	ad2antrl 727	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
12	9, 3	atbase 39245	. . . . . . . . 9 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
13	12	ad2antll 728	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
14	9, 2	latjcl 18509	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
15	8, 11, 13, 14	syl3anc 1371	. . . . . . 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 39903	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
18	15, 17	syldan 590	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19		eleq1a 2839	. . . . . 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 3219	. . 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 2108   ≠ wne 2946  ∃wrex 3076  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  joincjn 18381  Latclat 18501  Atomscatm 39219  HLchlt 39306  Linesclines 39451  pmapcpmap 39454  PSubClcpscN 39891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-p1 18496  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-lines 39458  df-pmap 39461  df-polarityN 39860  df-psubclN 39892

This theorem is referenced by: (None)