Theorem linepsubclN 40397

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 39809	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2		eqid 2736	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
3		eqid 2736	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		linepsubcl.n	. . . . 5 ⊢ 𝑁 = (Lines‘𝐾)
5		eqid 2736	. . . . 5 ⊢ (pmap‘𝐾) = (pmap‘𝐾)
6	2, 3, 4, 5	isline2 40220	. . . 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 2736	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
10	9, 3	atbase 39735	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
11	10	ad2antrl 729	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑝 ∈ (Base‘𝐾))
12	9, 3	atbase 39735	. . . . . . . . 9 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
13	12	ad2antll 730	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → 𝑞 ∈ (Base‘𝐾))
14	9, 2	latjcl 18405	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
15	8, 11, 13, 14	syl3anc 1374	. . . . . . 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 40392	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
18	15, 17	syldan 592	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((pmap‘𝐾)‘(𝑝(join‘𝐾)𝑞)) ∈ 𝐶)
19		eleq1a 2831	. . . . . 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 3194	. . 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 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  joincjn 18277  Latclat 18397  Atomscatm 39709  HLchlt 39796  Linesclines 39940  pmapcpmap 39943  PSubClcpscN 40380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-p1 18390  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-lines 39947  df-pmap 39950  df-polarityN 40349  df-psubclN 40381

This theorem is referenced by: (None)