Theorem dalawlem12 40081

Description: Lemma for dalaw 40085. Second part of dalawlem13 40082. (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 2734	. . . 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 39563	. . . 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 39566	. . . . . 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 39566	. . . . . 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 18361	. . . . 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 39488	. . . . . . . 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 18360	. . . . . . 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 39488	. . . . . . 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 18361	. . . . . 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 18360	. . . . 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 39488	. . . . . . 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 39566	. . . . . . 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 18361	. . . . . 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 39566	. . . . . 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 18360	. . . . 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 18369	. . . . . . 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 39566	. . . . . . . 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 18390	. . . . . . 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 39567	. . . . . . 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 7372	. . . . 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 18370	. . . . . . 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 40061	. . . . . 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 5128	. . . 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 39560	. . . . . . . . . . 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 39566	. . . . . . . . . . . 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 18360	. . . . . . . . . . 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 39566	. . . . . . . . . . 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 39428	. . . . . . . . . 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 39569	. . . . . . . . . . . 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 39570	. . . . . . . . . . . 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 2770	. . . . . . . . . 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 39575	. . . . . . . . . . . 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 18389	. . . . . . . . . . . 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 7374	. . . . . . . . 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 2770	. . . . . . . 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 39574	. . . . . . . . . . . 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 40056	. . . . . . . . . . 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 39575	. . . . . . . . . . . 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 7371	. . . . . . . . . . . . 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 39567	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 5122	. . . . . . . . . . 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 18361	. . . . . . . . . . . . 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 39566	. . . . . . . . . . . . 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 18371	. . . . . . . . . . . 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 5120	. . . . . . . . 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 39574	. . . . . . . . . 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 18361	. . . . . . . . . . 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 18392	. . . . . . . . . 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 5118	. . . . . . 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 39575	. . . . . . . . . 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 40056	. . . . . . . . 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 39488	. . . . . . . . . 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 18368	. . . . . . . . 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 2771	. . . . . . 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 5122	. . . . . 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 18360	. . . . . . . 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 18374	. . . . . . 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 18404	. . . . . 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 5122	. . . 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 18367	. . 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 18385	. . . 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 18387	. . . 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 39488	. . . . . 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 18400	. . . . 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 40059	. . . 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 39568	. . . . . . 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 7372	. . . . . 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 2771	. . . . 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 7371	. . . 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 18384	. . . . . 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 39567	. . . . . . . 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 7371	. . . . . . 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 2769	. . . . . 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 7371	. . . . 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 2769	. . . 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 7374	. . 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 2771	. 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 5122	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 1541   ∈ wcel 2113   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  lecple 17182  joincjn 18232  meetcmee 18233  Latclat 18352  OLcol 39373  Atomscatm 39462  HLchlt 39549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-proset 18215  df-poset 18234  df-plt 18249  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-p0 18344  df-lat 18353  df-clat 18420  df-oposet 39375  df-ol 39377  df-oml 39378  df-covers 39465  df-ats 39466  df-atl 39497  df-cvlat 39521  df-hlat 39550  df-psubsp 39702  df-pmap 39703  df-padd 39995

This theorem is referenced by:  dalawlem13  40082