Theorem dalawlem12 39876

Description: Lemma for dalaw 39880. Second part of dalawlem13 39877. (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 2729	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		dalawlem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		simp11 1204	. . . . 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 39357	. . . 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 1207	. . . . . 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 1208	. . . . . 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 39360	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	3, 5, 6, 9	syl3anc 1373	. . . . 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 1210	. . . . . 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 1211	. . . . . 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 39360	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
14	3, 11, 12, 13	syl3anc 1373	. . . . 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 18399	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾))
17	4, 10, 14, 16	syl3anc 1373	. . . 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 39282	. . . . . . . 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 18398	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾))
21	4, 10, 19, 20	syl3anc 1373	. . . . . 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 39282	. . . . . . 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 18399	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∈ (Base‘𝐾))
25	4, 21, 23, 24	syl3anc 1373	. . . . 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 18398	. . . . 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 1373	. . . 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 39282	. . . . . . 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 1212	. . . . . . 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 39360	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
32	3, 12, 30, 31	syl3anc 1373	. . . . . 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 18399	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))
34	4, 29, 32, 33	syl3anc 1373	. . . . 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 39360	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑈 ∨ 𝑆) ∈ (Base‘𝐾))
36	3, 30, 11, 35	syl3anc 1373	. . . . 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 18398	. . . . 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 1373	. . . 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 18407	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
40	4, 10, 19, 39	syl3anc 1373	. . . . . 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 39360	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑇 ∨ 𝑆) ∈ (Base‘𝐾))
42	3, 12, 11, 41	syl3anc 1373	. . . . . . 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 18428	. . . . . . 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 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) → ((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 39361	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑆))
47	3, 11, 12, 46	syl3anc 1373	. . . . . 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 7403	. . . . 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 18408	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
50	4, 10, 19, 49	syl3anc 1373	. . . . . 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 39856	. . . . . 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 1385	. . . . 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 5139	. . . 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 39354	. . . . . . . . . . 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 39360	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	3, 5, 11, 56	syl3anc 1373	. . . . . . . . . . 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 18398	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
59	4, 29, 57, 58	syl3anc 1373	. . . . . . . . . 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 39360	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
61	3, 6, 12, 60	syl3anc 1373	. . . . . . . . . 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 39222	. . . . . . . . . 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 1374	. . . . . . . . 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 39363	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
65	3, 5, 6, 11, 64	syl13anc 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 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
66	7, 8	hlatj12 39364	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑆)) = (𝑄 ∨ (𝑃 ∨ 𝑆)))
67	3, 5, 6, 11, 66	syl13anc 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 ∨ (𝑄 ∨ 𝑆)) = (𝑄 ∨ (𝑃 ∨ 𝑆)))
68	65, 67	eqtr2d 2765	. . . . . . . . . 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 39369	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑇 ≤ (𝑄 ∨ 𝑇))
70	3, 6, 12, 69	syl3anc 1373	. . . . . . . . . . 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 18427	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → (𝑇 ≤ (𝑄 ∨ 𝑇) ↔ ((𝑄 ∨ 𝑇) ∧ 𝑇) = 𝑇))
72	4, 23, 61, 71	syl3anc 1373	. . . . . . . . . . 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 7405	. . . . . . . . 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 2765	. . . . . . . 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 39368	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑇))
77	3, 6, 12, 76	syl3anc 1373	. . . . . . . . . . 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 39851	. . . . . . . . . . 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 1385	. . . . . . . . . 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 39369	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑈 ∨ 𝑄))
81	3, 30, 6, 80	syl3anc 1373	. . . . . . . . . . 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 1206	. . . . . . . . . . . 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 1205	. . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . 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 39361	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ∨ 𝑈) = (𝑈 ∨ 𝑄))
86	3, 6, 30, 85	syl3anc 1373	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 5133	. . . . . . . . . . 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 18399	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
90	4, 57, 61, 89	syl3anc 1373	. . . . . . . . . . . 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 39360	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑈 ∨ 𝑄) ∈ (Base‘𝐾))
92	3, 30, 6, 91	syl3anc 1373	. . . . . . . . . . . 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 18409	. . . . . . . . . . . 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 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) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑄)) ↔ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))) ≤ (𝑈 ∨ 𝑄)))
95	81, 88, 94	mpbi2and 712	. . . . . . . . . 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 5131	. . . . . . . . 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 39368	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑇 ≤ (𝑇 ∨ 𝑈))
98	3, 12, 30, 97	syl3anc 1373	. . . . . . . . 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 18399	. . . . . . . . . . 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 1373	. . . . . . . . . 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 18430	. . . . . . . . . 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 1381	. . . . . . . . 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 699	. . . . . . . 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 5129	. . . . . . 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 39369	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑈 ≤ (𝑇 ∨ 𝑈))
106	3, 12, 30, 105	syl3anc 1373	. . . . . . . . 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 39851	. . . . . . . . 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 1385	. . . . . . . 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 39282	. . . . . . . . . 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 18406	. . . . . . . . 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 1373	. . . . . . . 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 2766	. . . . . . 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 5133	. . . . . 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 18398	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∈ (Base‘𝐾))
116	4, 34, 110, 115	syl3anc 1373	. . . . . . 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 18412	. . . . . . 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 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) ∧ 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 18442	. . . . . 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 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)) ∨ 𝑈) ∨ 𝑆) = ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ (𝑈 ∨ 𝑆)))
122	119, 121	breqtrd 5133	. . . 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 18405	. . 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 18423	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄))
125	4, 10, 14, 124	syl3anc 1373	. . 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 18425	. . . 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 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)) ≤ ((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 712	. 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 39282	. . . . . 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 18438	. . . . 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 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)) ≤ (𝑃 ∨ 𝑄))
133	1, 2, 7, 15, 8	llnmod1i2 39854	. . . 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 1394	. . 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 39362	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
136	3, 6, 135	syl2anc 584	. . . . . 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 7403	. . . . . 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 2766	. . . . 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 7402	. . . 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 18422	. . . . . 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 1373	. . . . 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 39361	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
143	3, 5, 6, 142	syl3anc 1373	. . . . . . 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 7402	. . . . . . 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 2764	. . . . . 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 7402	. . . . 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 2764	. . . 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 7405	. . 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 2766	. 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 5133	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 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  Latclat 18390  OLcol 39167  Atomscatm 39256  HLchlt 39343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-psubsp 39497  df-pmap 39498  df-padd 39790

This theorem is referenced by:  dalawlem13  39877