Theorem dalawlem12 40345

Description: Lemma for dalaw 40349. Second part of dalawlem13 40346. (Contributed by NM, 17-Sep-2012.)

Hypotheses

Ref	Expression
dalawlem.l	⊢ ≤ = (le‘𝐾)
dalawlem.j	⊢ ∨ = (join‘𝐾)
dalawlem.m	⊢ ∧ = (meet‘𝐾)
dalawlem.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
dalawlem12	⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))))

Proof of Theorem dalawlem12

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		dalawlem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		simp11 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝐾 ∈ HL)
4	3	hllatd 39827	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝐾 ∈ Lat)
5		simp21 1208	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
6		simp22 1209	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
7		dalawlem.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
8		dalawlem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
9	1, 7, 8	hlatjcl 39830	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	3, 5, 6, 9	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11		simp31 1211	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑆 ∈ 𝐴)
12		simp32 1212	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑇 ∈ 𝐴)
13	1, 7, 8	hlatjcl 39830	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
14	3, 11, 12, 13	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
15		dalawlem.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
16	1, 15	latmcl 18400	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾))
17	4, 10, 14, 16	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾))
18	1, 8	atbase 39752	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
19	11, 18	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑆 ∈ (Base‘𝐾))
20	1, 7	latjcl 18399	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾))
21	4, 10, 19, 20	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾))
22	1, 8	atbase 39752	. . . . . . 7 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
23	12, 22	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑇 ∈ (Base‘𝐾))
24	1, 15	latmcl 18400	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∈ (Base‘𝐾))
25	4, 21, 23, 24	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∈ (Base‘𝐾))
26	1, 7	latjcl 18399	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) ∈ (Base‘𝐾))
27	4, 25, 19, 26	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) ∈ (Base‘𝐾))
28	1, 8	atbase 39752	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
29	6, 28	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑄 ∈ (Base‘𝐾))
30		simp33 1213	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑈 ∈ 𝐴)
31	1, 7, 8	hlatjcl 39830	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
32	3, 12, 30, 31	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
33	1, 15	latmcl 18400	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))
34	4, 29, 32, 33	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))
35	1, 7, 8	hlatjcl 39830	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑈 ∨ 𝑆) ∈ (Base‘𝐾))
36	3, 30, 11, 35	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑈 ∨ 𝑆) ∈ (Base‘𝐾))
37	1, 7	latjcl 18399	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾) ∧ (𝑈 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∈ (Base‘𝐾))
38	4, 34, 36, 37	syl3anc 1374	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∈ (Base‘𝐾))
39	1, 2, 7	latlej1 18408	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
40	4, 10, 19, 39	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
41	1, 7, 8	hlatjcl 39830	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑇 ∨ 𝑆) ∈ (Base‘𝐾))
42	3, 12, 11, 41	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑇 ∨ 𝑆) ∈ (Base‘𝐾))
43	1, 2, 15	latmlem1 18429	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆) → ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ 𝑆)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ (𝑇 ∨ 𝑆))))
44	4, 10, 21, 42, 43	syl13anc 1375	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆) → ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ 𝑆)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ (𝑇 ∨ 𝑆))))
45	40, 44	mpd 15	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ 𝑆)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ (𝑇 ∨ 𝑆)))
46	7, 8	hlatjcom 39831	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑆))
47	3, 11, 12, 46	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑆))
48	47	oveq2d 7377	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) = ((𝑃 ∨ 𝑄) ∧ (𝑇 ∨ 𝑆)))
49	1, 2, 7	latlej2 18409	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
50	4, 10, 19, 49	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
51	1, 2, 7, 15, 8	atmod2i2 40325	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) ∧ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) = (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ (𝑇 ∨ 𝑆)))
52	3, 12, 21, 19, 50, 51	syl131anc 1386	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) = (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ (𝑇 ∨ 𝑆)))
53	45, 48, 52	3brtr4d 5118	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆))
54		hlol 39824	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
55	3, 54	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝐾 ∈ OL)
56	1, 7, 8	hlatjcl 39830	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	3, 5, 11, 56	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
58	1, 7	latjcl 18399	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
59	4, 29, 57, 58	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
60	1, 7, 8	hlatjcl 39830	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
61	3, 6, 12, 60	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
62	1, 15	latmassOLD 39692	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾))) → (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∧ 𝑇) = ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ ((𝑄 ∨ 𝑇) ∧ 𝑇)))
63	55, 59, 61, 23, 62	syl13anc 1375	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∧ 𝑇) = ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ ((𝑄 ∨ 𝑇) ∧ 𝑇)))
64	7, 8	hlatjass 39833	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
65	3, 5, 6, 11, 64	syl13anc 1375	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
66	7, 8	hlatj12 39834	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑆)) = (𝑄 ∨ (𝑃 ∨ 𝑆)))
67	3, 5, 6, 11, 66	syl13anc 1375	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑆)) = (𝑄 ∨ (𝑃 ∨ 𝑆)))
68	65, 67	eqtr2d 2773	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ 𝑆))
69	2, 7, 8	hlatlej2 39839	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑇 ≤ (𝑄 ∨ 𝑇))
70	3, 6, 12, 69	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑇 ≤ (𝑄 ∨ 𝑇))
71	1, 2, 15	latleeqm2 18428	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → (𝑇 ≤ (𝑄 ∨ 𝑇) ↔ ((𝑄 ∨ 𝑇) ∧ 𝑇) = 𝑇))
72	4, 23, 61, 71	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑇 ≤ (𝑄 ∨ 𝑇) ↔ ((𝑄 ∨ 𝑇) ∧ 𝑇) = 𝑇))
73	70, 72	mpbid 232	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∨ 𝑇) ∧ 𝑇) = 𝑇)
74	68, 73	oveq12d 7379	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ ((𝑄 ∨ 𝑇) ∧ 𝑇)) = (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇))
75	63, 74	eqtr2d 2773	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) = (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∧ 𝑇))
76	2, 7, 8	hlatlej1 39838	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑇))
77	3, 6, 12, 76	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑄 ≤ (𝑄 ∨ 𝑇))
78	1, 2, 7, 15, 8	atmod1i1 40320	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) ∧ 𝑄 ≤ (𝑄 ∨ 𝑇)) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))) = ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)))
79	3, 6, 57, 61, 77, 78	syl131anc 1386	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))) = ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)))
80	2, 7, 8	hlatlej2 39839	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑈 ∨ 𝑄))
81	3, 30, 6, 80	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑄 ≤ (𝑈 ∨ 𝑄))
82		simp13 1207	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈))
83		simp12 1206	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑄 = 𝑅)
84	83	oveq1d 7376	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ 𝑈) = (𝑅 ∨ 𝑈))
85	7, 8	hlatjcom 39831	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ∨ 𝑈) = (𝑈 ∨ 𝑄))
86	3, 6, 30, 85	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ 𝑈) = (𝑈 ∨ 𝑄))
87	84, 86	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑅 ∨ 𝑈) = (𝑈 ∨ 𝑄))
88	82, 87	breqtrd 5112	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄))
89	1, 15	latmcl 18400	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
90	4, 57, 61, 89	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
91	1, 7, 8	hlatjcl 39830	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑈 ∨ 𝑄) ∈ (Base‘𝐾))
92	3, 30, 6, 91	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑈 ∨ 𝑄) ∈ (Base‘𝐾))
93	1, 2, 7	latjle12 18410	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾) ∧ (𝑈 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑈 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄)) ↔ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))) ≤ (𝑈 ∨ 𝑄)))
94	4, 29, 90, 92, 93	syl13anc 1375	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ≤ (𝑈 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄)) ↔ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))) ≤ (𝑈 ∨ 𝑄)))
95	81, 88, 94	mpbi2and 713	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))) ≤ (𝑈 ∨ 𝑄))
96	79, 95	eqbrtrrd 5110	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄))
97	2, 7, 8	hlatlej1 39838	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑇 ≤ (𝑇 ∨ 𝑈))
98	3, 12, 30, 97	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑇 ≤ (𝑇 ∨ 𝑈))
99	1, 15	latmcl 18400	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
100	4, 59, 61, 99	syl3anc 1374	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
101	1, 2, 15	latmlem12 18431	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾) ∧ (𝑈 ∨ 𝑄) ∈ (Base‘𝐾)) ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))) → ((((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄) ∧ 𝑇 ≤ (𝑇 ∨ 𝑈)) → (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∧ 𝑇) ≤ ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈))))
102	4, 100, 92, 23, 32, 101	syl122anc 1382	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄) ∧ 𝑇 ≤ (𝑇 ∨ 𝑈)) → (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∧ 𝑇) ≤ ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈))))
103	96, 98, 102	mp2and 700	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑄 ∨ (𝑃 ∨ 𝑆)) ∧ (𝑄 ∨ 𝑇)) ∧ 𝑇) ≤ ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈)))
104	75, 103	eqbrtrd 5108	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ≤ ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈)))
105	2, 7, 8	hlatlej2 39839	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑈 ≤ (𝑇 ∨ 𝑈))
106	3, 12, 30, 105	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑈 ≤ (𝑇 ∨ 𝑈))
107	1, 2, 7, 15, 8	atmod1i1 40320	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) ∧ 𝑈 ≤ (𝑇 ∨ 𝑈)) → (𝑈 ∨ (𝑄 ∧ (𝑇 ∨ 𝑈))) = ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈)))
108	3, 30, 29, 32, 106, 107	syl131anc 1386	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑈 ∨ (𝑄 ∧ (𝑇 ∨ 𝑈))) = ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈)))
109	1, 8	atbase 39752	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
110	30, 109	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑈 ∈ (Base‘𝐾))
111	1, 7	latjcom 18407	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾)) → (𝑈 ∨ (𝑄 ∧ (𝑇 ∨ 𝑈))) = ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈))
112	4, 110, 34, 111	syl3anc 1374	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑈 ∨ (𝑄 ∧ (𝑇 ∨ 𝑈))) = ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈))
113	108, 112	eqtr3d 2774	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑈 ∨ 𝑄) ∧ (𝑇 ∨ 𝑈)) = ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈))
114	104, 113	breqtrd 5112	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈))
115	1, 7	latjcl 18399	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∈ (Base‘𝐾))
116	4, 34, 110, 115	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∈ (Base‘𝐾))
117	1, 2, 7	latjlej1 18413	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∈ (Base‘𝐾) ∧ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) ≤ (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∨ 𝑆)))
118	4, 25, 116, 19, 117	syl13anc 1375	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) ≤ (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∨ 𝑆)))
119	114, 118	mpd 15	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) ≤ (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∨ 𝑆))
120	1, 7	latjass 18443	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∨ 𝑆) = ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)))
121	4, 34, 110, 19, 120	syl13anc 1375	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∨ 𝑆) = ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)))
122	119, 121	breqtrd 5112	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∨ 𝑆) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)))
123	1, 2, 4, 17, 27, 38, 53, 122	lattrd 18406	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)))
124	1, 2, 15	latmle1 18424	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄))
125	4, 10, 14, 124	syl3anc 1374	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄))
126	1, 2, 15	latlem12 18426	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾) ∧ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑄))))
127	4, 17, 38, 10, 126	syl13anc 1375	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑄))))
128	123, 125, 127	mpbi2and 713	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑄)))
129	1, 8	atbase 39752	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
130	5, 129	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾))
131	1, 2, 7, 15	latmlej12 18439	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → (𝑄 ∧ (𝑇 ∨ 𝑈)) ≤ (𝑃 ∨ 𝑄))
132	4, 29, 32, 130, 131	syl13anc 1375	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∧ (𝑇 ∨ 𝑈)) ≤ (𝑃 ∨ 𝑄))
133	1, 2, 7, 15, 8	llnmod1i2 40323	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) ∧ (𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑈 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄))) = (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑄)))
134	3, 34, 10, 30, 11, 132, 133	syl321anc 1395	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑈 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄))) = (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑄)))
135	7, 8	hlatjidm 39832	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
136	3, 6, 135	syl2anc 585	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ 𝑄) = 𝑄)
137	83	oveq2d 7377	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ 𝑄) = (𝑄 ∨ 𝑅))
138	136, 137	eqtr3d 2774	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → 𝑄 = (𝑄 ∨ 𝑅))
139	138	oveq1d 7376	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∧ (𝑇 ∨ 𝑈)) = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)))
140	1, 15	latmcom 18423	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑈 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝑈 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ (𝑈 ∨ 𝑆)))
141	4, 36, 10, 140	syl3anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑈 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ (𝑈 ∨ 𝑆)))
142	7, 8	hlatjcom 39831	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
143	3, 5, 6, 142	syl3anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
144	83	oveq1d 7376	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑄 ∨ 𝑃) = (𝑅 ∨ 𝑃))
145	143, 144	eqtrd 2772	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑃))
146	145	oveq1d 7376	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑈 ∨ 𝑆)) = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))
147	141, 146	eqtrd 2772	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑈 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄)) = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))
148	139, 147	oveq12d 7379	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑈 ∨ 𝑆) ∧ (𝑃 ∨ 𝑄))) = (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))))
149	134, 148	eqtr3d 2774	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑄)) = (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))))
150	128, 149	breqtrd 5112	1 ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  lecple 17221  joincjn 18271  meetcmee 18272  Latclat 18391  OLcol 39637  Atomscatm 39726  HLchlt 39813

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-mpo 7366  df-1st 7936  df-2nd 7937  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-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-psubsp 39966  df-pmap 39967  df-padd 40259

This theorem is referenced by:  dalawlem13  40346