Theorem linepsubN 40132

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 4034	. . . . . . . 8 ⊢ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)
2		sseq1 3961	. . . . . . . 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 39669	. . . . . . . . 9 ⊢ (𝑎 ∈ (Atoms‘𝐾) → 𝑎 ∈ (Base‘𝐾))
8	5, 6	atbase 39669	. . . . . . . . 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 18374	. . . . . . . . 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 5103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
16	15	elrab 3648	. . . . . . . . . . . . . . . . . . . 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 39669	. . . . . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
23	22	elrab 3648	. . . . . . . . . . . . . . . . . . . 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 39669	. . . . . . . . . . . . . . . . . . . . 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 39669	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
35		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (le‘𝐾) = (le‘𝐾)
36	5, 35, 10	latjle12 18385	. . . . . . . . . . . . . . . . . . . 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 18374	. . . . . . . . . . . . . . . . . 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 18379	. . . . . . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
57	56	elrab 3648	. . . . . . . . . . . . 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 40119	. . 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 40125	. . 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 3401   ⊆ wss 3903   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  Latclat 18366  Atomscatm 39643  Linesclines 39874  PSubSpcpsubsp 39876

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-poset 18248  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-lat 18367  df-ats 39647  df-lines 39881  df-psubsp 39883

This theorem is referenced by: (None)