Theorem linepsubN 36330

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 3947	. . . . . . . 8 ⊢ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏)} ⊆ (Atoms‘𝐾)
2		sseq1 3883	. . . . . . . 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 250	. . . . . . 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 2779	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2779	. . . . . . . . . 10 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
7	5, 6	atbase 35867	. . . . . . . . 9 ⊢ (𝑎 ∈ (Atoms‘𝐾) → 𝑎 ∈ (Base‘𝐾))
8	5, 6	atbase 35867	. . . . . . . . 9 ⊢ (𝑏 ∈ (Atoms‘𝐾) → 𝑏 ∈ (Base‘𝐾))
9	7, 8	anim12i 603	. . . . . . . 8 ⊢ ((𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾)) → (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)))
10		eqid 2779	. . . . . . . . . 10 ⊢ (join‘𝐾) = (join‘𝐾)
11	5, 10	latjcl 17519	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾)) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
12	11	3expb 1100	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Base‘𝐾) ∧ 𝑏 ∈ (Base‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
13	9, 12	sylan2 583	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑎 ∈ (Atoms‘𝐾) ∧ 𝑏 ∈ (Atoms‘𝐾))) → (𝑎(join‘𝐾)𝑏) ∈ (Base‘𝐾))
14		eleq2 2855	. . . . . . . . . . . . . . . . . . 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 4932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑝(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
16	15	elrab 3596	. . . . . . . . . . . . . . . . . . . 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 35867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
18	17	anim1i 605	. . . . . . . . . . . . . . . . . . . 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 209	. . . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . . 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 2855	. . . . . . . . . . . . . . . . . . 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 4932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑞(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
23	22	elrab 3596	. . . . . . . . . . . . . . . . . . . 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 35867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ (Base‘𝐾))
25	24	anim1i 605	. . . . . . . . . . . . . . . . . . . 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 209	. . . . . . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . . 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 398	. . . . . . . . . . . . . . 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 607	. . . . . . . . . . . . . 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 460	. . . . . . . . . . . . 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 35867	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ (Base‘𝐾))
35		eqid 2779	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (le‘𝐾) = (le‘𝐾)
36	5, 35, 10	latjle12 17530	. . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . 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 1334	. . . . . . . . . . . . . . . . . 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 402	. . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 17519	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾))
44	43	3expib 1102	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ (Base‘𝐾) ∧ 𝑞 ∈ (Base‘𝐾)) → (𝑝(join‘𝐾)𝑞) ∈ (Base‘𝐾)))
45	5, 35	lattr 17524	. . . . . . . . . . . . . . . . . . 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 1334	. . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . 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 708	. . . . . . . . . . . . . 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 681	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 477	. . . . . . . . . . . 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 518	. . . . . . . . . . 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 2855	. . . . . . . . . . . . 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 4932	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)(𝑎(join‘𝐾)𝑏) ↔ 𝑟(le‘𝐾)(𝑎(join‘𝐾)𝑏)))
57	56	elrab 3596	. . . . . . . . . . . . 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	syl6bb 279	. . . . . . . . . . . 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 718	. . . . . . . . . . 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 251	. . . . . . . . . 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 3133	. . . . . . . . 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 3142	. . . . . . . 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 405	. . . . . . 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 582	. . . . . 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 505	. . . . 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 483	. . . 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 3240	. . 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 36317	. . 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 36323	. . 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 286	. 2 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑁 → 𝑋 ∈ 𝑆))
73	72	imp 398	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   ≠ wne 2968  ∀wral 3089  ∃wrex 3090  {crab 3093   ⊆ wss 3830   class class class wbr 4929  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  Latclat 17513  Atomscatm 35841  Linesclines 36072  PSubSpcpsubsp 36074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-poset 17414  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-lat 17514  df-ats 35845  df-lines 36079  df-psubsp 36081

This theorem is referenced by: (None)