Theorem linepsubN 37766

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 4013	. . . . . . . 8 ⊢ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)
2		sseq1 3946	. . . . . . . 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 257	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2738	. . . . . . . . . 10 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7	5, 6	atbase 37303	. . . . . . . . 9 ⊢ (𝑎 ∈ (Atoms‘𝐾) → 𝑎 ∈ (Base‘𝐾))
8	5, 6	atbase 37303	. . . . . . . . 9 ⊢ (𝑏 ∈ (Atoms‘𝐾) → 𝑏 ∈ (Base‘𝐾))
9	7, 8	anim12i 613	. . . . . . . 8 ⊢ ((𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾)) → (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)))
10		eqid 2738	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
11	5, 10	latjcl 18157	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
12	11	3expb 1119	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
13	9, 12	sylan2 593	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
14		eleq2 2827	. . . . . . . . . . . . . . . . . . 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 5077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
16	15	elrab 3624	. . . . . . . . . . . . . . . . . . . 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 37303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
18	17	anim1i 615	. . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . . . . . . 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 5077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
23	22	elrab 3624	. . . . . . . . . . . . . . . . . . . 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 37303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
25	24	anim1i 615	. . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . . 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 609	. . . . . . . . . . . . . . . . 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 653	. . . . . . . . . . . . . . . . 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	syl6ib 250	. . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 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 617	. . . . . . . . . . . . . 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 468	. . . . . . . . . . . . 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 37303	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
35		eqid 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (le‘𝐾) = (le‘𝐾)
36	5, 35, 10	latjle12 18168	. . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . 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 1353	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . 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 428	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 18157	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
44	43	3expib 1121	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)))
45	5, 35	lattr 18162	. . . . . . . . . . . . . . . . . . 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 1353	. . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . 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 718	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 485	. . . . . . . . . . . 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 526	. . . . . . . . . . 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 2827	. . . . . . . . . . . . 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 5077	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
57	56	elrab 3624	. . . . . . . . . . . . 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 728	. . . . . . . . . . 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 258	. . . . . . . . . 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 3103	. . . . . . . . 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 3123	. . . . . . . 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 413	. . . . . . 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 591	. . . . . 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 513	. . . . 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 491	. . . 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 3223	. . 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 37753	. . 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 37759	. . 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 407	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  Latclat 18149  Atomscatm 37277  Linesclines 37508  PSubSpcpsubsp 37510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-poset 18031  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-lat 18150  df-ats 37281  df-lines 37515  df-psubsp 37517

This theorem is referenced by: (None)