Theorem linepsubclN 40414

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 39826	. . . 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)
6	2, 3, 4, 5	isline2 40237	. . . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
10	9, 3	atbase 39752	. . . . . . . . 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 39752	. . . . . . . . 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 18399	. . . . . . . 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 40409	. . . . . . 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 2832	. . . . . 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 3195	. . 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 2933  ∃wrex 3062  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  joincjn 18271  Latclat 18391  Atomscatm 39726  HLchlt 39813  Linesclines 39957  pmapcpmap 39960  PSubClcpscN 40397

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-proset 18254  df-poset 18273  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18392  df-clat 18459  df-oposet 39639  df-ol 39641  df-oml 39642  df-covers 39729  df-ats 39730  df-atl 39761  df-cvlat 39785  df-hlat 39814  df-lines 39964  df-pmap 39967  df-polarityN 40366  df-psubclN 40398

This theorem is referenced by: (None)