Theorem dalawlem12 40546

Description: Lemma for dalaw 40550. Second part of dalawlem13 40547. (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 2769	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		dalawlem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		simp11 1220	. . . . 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 40028	. . . 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 1223	. . . . . 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 1224	. . . . . 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 40031	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	3, 5, 6, 9	syl3anc 1396	. . . . 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 1226	. . . . . 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 1227	. . . . . 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 40031	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
14	3, 11, 12, 13	syl3anc 1396	. . . . 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 18496	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ∈ (Base‘𝐾))
17	4, 10, 14, 16	syl3anc 1396	. . . 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 39953	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
19	11, 18	syl 18	. . . . . . 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 18495	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾))
21	4, 10, 19, 20	syl3anc 1396	. . . . . 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 39953	. . . . . . 7 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
23	12, 22	syl 18	. . . . . 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 18496	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾)) → (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ∈ (Base‘𝐾))
25	4, 21, 23, 24	syl3anc 1396	. . . . 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 18495	. . . . 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 1396	. . . 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 39953	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
29	6, 28	syl 18	. . . . . 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 1228	. . . . . . 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 40031	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
32	3, 12, 30, 31	syl3anc 1396	. . . . . 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 18496	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))
34	4, 29, 32, 33	syl3anc 1396	. . . . 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 40031	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑈 ∨ 𝑆) ∈ (Base‘𝐾))
36	3, 30, 11, 35	syl3anc 1396	. . . . 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 18495	. . . . 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 1396	. . . 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 18504	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
40	4, 10, 19, 39	syl3anc 1396	. . . . . 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 40031	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑇 ∨ 𝑆) ∈ (Base‘𝐾))
42	3, 12, 11, 41	syl3anc 1396	. . . . . . 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 18525	. . . . . . 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 1397	. . . . . 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 16	. . . . 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 40032	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) = (𝑇 ∨ 𝑆))
47	3, 11, 12, 46	syl3anc 1396	. . . . . 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 7427	. . . . 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 18505	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑆))
50	4, 10, 19, 49	syl3anc 1396	. . . . . 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 40526	. . . . . 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 1408	. . . . 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 5147	. . . 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 40025	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
55	3, 54	syl 18	. . . . . . . . . 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 40031	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
57	3, 5, 11, 56	syl3anc 1396	. . . . . . . . . . 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 18495	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → (𝑄 ∨ (𝑃 ∨ 𝑆)) ∈ (Base‘𝐾))
59	4, 29, 57, 58	syl3anc 1396	. . . . . . . . . 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 40031	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
61	3, 6, 12, 60	syl3anc 1396	. . . . . . . . . 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 39893	. . . . . . . . . 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 1397	. . . . . . . . 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 40034	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
65	3, 5, 6, 11, 64	syl13anc 1397	. . . . . . . . . . 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 40035	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑆)) = (𝑄 ∨ (𝑃 ∨ 𝑆)))
67	3, 5, 6, 11, 66	syl13anc 1397	. . . . . . . . . . 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 2805	. . . . . . . . . 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 40040	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑇 ≤ (𝑄 ∨ 𝑇))
70	3, 6, 12, 69	syl3anc 1396	. . . . . . . . . . 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 18524	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → (𝑇 ≤ (𝑄 ∨ 𝑇) ↔ ((𝑄 ∨ 𝑇) ∧ 𝑇) = 𝑇))
72	4, 23, 61, 71	syl3anc 1396	. . . . . . . . . . 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 235	. . . . . . . . . 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 7429	. . . . . . . . 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 2805	. . . . . . . 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 40039	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑇))
77	3, 6, 12, 76	syl3anc 1396	. . . . . . . . . . 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 40521	. . . . . . . . . . 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 1408	. . . . . . . . . 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 40040	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑈 ∨ 𝑄))
81	3, 30, 6, 80	syl3anc 1396	. . . . . . . . . . 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 1222	. . . . . . . . . . . 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 1221	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . 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 40032	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ∨ 𝑈) = (𝑈 ∨ 𝑄))
86	3, 6, 30, 85	syl3anc 1396	. . . . . . . . . . . . 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 2806	. . . . . . . . . . . 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 5141	. . . . . . . . . . 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 18496	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ (Base‘𝐾))
90	4, 57, 61, 89	syl3anc 1396	. . . . . . . . . . . 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 40031	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑈 ∨ 𝑄) ∈ (Base‘𝐾))
92	3, 30, 6, 91	syl3anc 1396	. . . . . . . . . . . 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 18506	. . . . . . . . . . . 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 1397	. . . . . . . . . . 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 724	. . . . . . . . . 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 5139	. . . . . . . . 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 40039	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑇 ≤ (𝑇 ∨ 𝑈))
98	3, 12, 30, 97	syl3anc 1396	. . . . . . . . 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 18496	. . . . . . . . . . 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 1396	. . . . . . . . . 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 18527	. . . . . . . . . 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 1404	. . . . . . . . 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 711	. . . . . . . 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 5137	. . . . . . 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 40040	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑈 ≤ (𝑇 ∨ 𝑈))
106	3, 12, 30, 105	syl3anc 1396	. . . . . . . . 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 40521	. . . . . . . . 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 1408	. . . . . . . 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 39953	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
110	30, 109	syl 18	. . . . . . . . 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 18503	. . . . . . . . 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 1396	. . . . . . . 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 2806	. . . . . . 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 5141	. . . . . 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 18495	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∧ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑄 ∧ (𝑇 ∨ 𝑈)) ∨ 𝑈) ∈ (Base‘𝐾))
116	4, 34, 110, 115	syl3anc 1396	. . . . . . 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 18509	. . . . . . 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 1397	. . . . . 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 16	. . . . 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 18539	. . . . . 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 1397	. . . . 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 5141	. . . 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 18502	. . 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 18520	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄))
125	4, 10, 14, 124	syl3anc 1396	. . 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 18522	. . . 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 1397	. . 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 724	. 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 39953	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
130	5, 129	syl 18	. . . . 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 18535	. . . . 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 1397	. . . 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 40524	. . . 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 1417	. . 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 40033	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
136	3, 6, 135	syl2anc 595	. . . . . 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 7427	. . . . . 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 2806	. . . . 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 7426	. . . 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 18519	. . . . . 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 1396	. . . . 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 40032	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
143	3, 5, 6, 142	syl3anc 1396	. . . . . . 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 7426	. . . . . . 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 2804	. . . . . 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 7426	. . . . 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 2804	. . . 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 7429	. . 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 2806	. 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 5141	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 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  joincjn 18367  meetcmee 18368  Latclat 18487  OLcol 39838  Atomscatm 39927  HLchlt 40014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39840  df-ol 39842  df-oml 39843  df-covers 39930  df-ats 39931  df-atl 39962  df-cvlat 39986  df-hlat 40015  df-psubsp 40167  df-pmap 40168  df-padd 40460

This theorem is referenced by:  dalawlem13  40547