Theorem linepsubN 40215

Description: A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
linepsub.n	⊢ 𝑁 = (Lines‘𝐾)
linepsub.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
linepsubN	⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝑆)

Proof of Theorem linepsubN

Dummy variables 𝑎 𝑏 𝑐 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4021	. . . . . . . 8 ⊢ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)
2		sseq1 3948	. . . . . . . 8 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑋 ⊆ (Atoms‘𝐾) ↔ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)))
3	1, 2	mpbiri 258	. . . . . . 7 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → 𝑋 ⊆ (Atoms‘𝐾))
4	3	a1i 11	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → 𝑋 ⊆ (Atoms‘𝐾)))
5		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2737	. . . . . . . . . 10 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7	5, 6	atbase 39752	. . . . . . . . 9 ⊢ (𝑎 ∈ (Atoms‘𝐾) → 𝑎 ∈ (Base‘𝐾))
8	5, 6	atbase 39752	. . . . . . . . 9 ⊢ (𝑏 ∈ (Atoms‘𝐾) → 𝑏 ∈ (Base‘𝐾))
9	7, 8	anim12i 614	. . . . . . . 8 ⊢ ((𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾)) → (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)))
10		eqid 2737	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
11	5, 10	latjcl 18399	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
12	11	3expb 1121	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
13	9, 12	sylan2 594	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
14		eleq2 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑝 ∈ 𝑋 ↔ 𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}))
15		breq1 5089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
16	15	elrab 3635	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
17	5, 6	atbase 39752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
18	17	anim1i 616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
19	16, 18	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
20	14, 19	biimtrdi 253	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑝 ∈ 𝑋 → (𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
21		eleq2 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑞 ∈ 𝑋 ↔ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}))
22		breq1 5089	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
23	22	elrab 3635	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
24	5, 6	atbase 39752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
25	24	anim1i 616	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
26	23, 25	sylbi 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
27	21, 26	biimtrdi 253	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑞 ∈ 𝑋 → (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
28	20, 27	anim12d 610	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
29		an4 657	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑝 ∈ (Base‘𝐾) ∧ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)) ∧ (𝑞 ∈ (Base‘𝐾) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))) ↔ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
30	28, 29	imbitrdi 251	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ((𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
31	30	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
32	31	anim2i 618	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))) → ((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
33	32	anassrs 467	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
34	5, 6	atbase 39752	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
35		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (le‘𝐾) = (le‘𝐾)
36	5, 35, 10	latjle12 18410	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾) ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
37	36	biimpd 229	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾) ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
38	37	3exp2 1356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Lat → (𝑝 ∈ (Base‘𝐾) → (𝑞 ∈ (Base‘𝐾) → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
39	38	impd 410	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
40	39	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Lat → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → ((𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)))))
41	40	imp43 427	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏))
42	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏))
43	5, 10	latjcl 18399	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
44	43	3expib 1123	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)))
45	5, 35	lattr 18404	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ (Base‘𝐾) ∧ (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
46	45	3exp2 1356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Lat → (𝑟 ∈ (Base‘𝐾) → ((𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
47	46	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Lat → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾) → (𝑟 ∈ (Base‘𝐾) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
48	44, 47	syl5d 73	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Lat → ((𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾) → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑟 ∈ (Base‘𝐾) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))))
49	48	imp41 425	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾))) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
50	49	adantlrr 722	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) ∧ 𝑟 ∈ (Base‘𝐾)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)(𝑎(join‘𝐾)𝑏)) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
51	42, 50	mpan2d 695	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) ∧ (𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏) ∧ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
52	33, 34, 51	syl2an 597	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
53		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Atoms‘𝐾))
54	52, 53	jctild 525	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
55		eleq2 2826	. . . . . . . . . . . . 13 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑟 ∈ 𝑋 ↔ 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}))
56		breq1 5089	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
57	56	elrab 3635	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
58	55, 57	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑟 ∈ 𝑋 ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
59	58	ad3antlr 732	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟 ∈ 𝑋 ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏))))
60	54, 59	sylibrd 259	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))
61	60	ralrimiva 3130	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))
62	61	ralrimivva 3181	. . . . . . . 8 ⊢ (((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) → ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))
63	62	ex 412	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾)) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋)))
64	13, 63	syldan 592	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋)))
65	4, 64	jcad 512	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} → (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))))
66	65	adantld 490	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → ((𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))))
67	66	rexlimdvva 3195	. . 3 ⊢ (𝐾 ∈ Lat → (∃𝑎 ∈ (Atoms‘𝐾)∃𝑏 ∈ (Atoms‘𝐾)(𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)}) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))))
68		linepsub.n	. . . 4 ⊢ 𝑁 = (Lines‘𝐾)
69	35, 10, 6, 68	isline 40202	. . 3 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑁 ↔ ∃𝑎 ∈ (Atoms‘𝐾)∃𝑏 ∈ (Atoms‘𝐾)(𝑎 ≠ 𝑏 ∧ 𝑋 = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)})))
70		linepsub.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
71	35, 10, 6, 70	ispsubsp 40208	. . 3 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ 𝑋))))
72	67, 69, 71	3imtr4d 294	. 2 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝑆))
73	72	imp 406	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ⊆ wss 3890   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221  joincjn 18271  Latclat 18391  Atomscatm 39726  Linesclines 39957  PSubSpcpsubsp 39959

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-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-poset 18273  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-lat 18392  df-ats 39730  df-lines 39964  df-psubsp 39966

This theorem is referenced by: (None)