Theorem 3atlem1 37497

Description: Lemma for 3at 37504. (Contributed by NM, 22-Jun-2012.)

Hypotheses

Ref	Expression
3at.l	⊢ ≤ = (le‘𝐾)
3at.j	⊢ ∨ = (join‘𝐾)
3at.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
3atlem1	⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))

Proof of Theorem 3atlem1

Step	Hyp	Ref	Expression
1		simp11 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝐾 ∈ HL)
2		simp131 1307	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑆 ∈ 𝐴)
3		simp132 1308	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑇 ∈ 𝐴)
4		simp133 1309	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑈 ∈ 𝐴)
5		3at.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
6		3at.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
7	5, 6	hlatjass 37384	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈)))
8	1, 2, 3, 4, 7	syl13anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈)))
9		simp121 1304	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ∈ 𝐴)
10		simp122 1305	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ∈ 𝐴)
11		simp123 1306	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ∈ 𝐴)
12	5, 6	hlatjass 37384	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
13	1, 9, 10, 11, 12	syl13anc 1371	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
14		simp3 1137	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
15	13, 14	eqbrtrrd 5098	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ (𝑄 ∨ 𝑅)) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
16	1	hllatd 37378	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝐾 ∈ Lat)
17		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 6	atbase 37303	. . . . . . . . . 10 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
19	9, 18	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ∈ (Base‘𝐾))
20	17, 5, 6	hlatjcl 37381	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
21	1, 10, 11, 20	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
22	17, 5, 6	hlatjcl 37381	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
23	1, 2, 3, 22	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
24	17, 6	atbase 37303	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
25	4, 24	syl 17	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑈 ∈ (Base‘𝐾))
26	17, 5	latjcl 18157	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))
27	16, 23, 25, 26	syl3anc 1370	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))
28		3at.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
29	17, 28, 5	latjle12 18168	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))) → ((𝑃 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ (𝑄 ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) ↔ (𝑃 ∨ (𝑄 ∨ 𝑅)) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
30	16, 19, 21, 27, 29	syl13anc 1371	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ (𝑄 ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) ↔ (𝑃 ∨ (𝑄 ∨ 𝑅)) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
31	15, 30	mpbird 256	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ (𝑄 ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
32	31	simpld 495	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
33	32, 8	breqtrd 5100	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)))
34	17, 5, 6	hlatjcl 37381	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
35	1, 3, 4, 34	syl3anc 1370	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
36		simp22 1206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ¬ 𝑃 ≤ (𝑇 ∨ 𝑈))
37	17, 28, 5, 6	hlexchb2 37399	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈)) → (𝑃 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)) ↔ (𝑃 ∨ (𝑇 ∨ 𝑈)) = (𝑆 ∨ (𝑇 ∨ 𝑈))))
38	1, 9, 2, 35, 36, 37	syl131anc 1382	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)) ↔ (𝑃 ∨ (𝑇 ∨ 𝑈)) = (𝑆 ∨ (𝑇 ∨ 𝑈))))
39	33, 38	mpbid 231	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ (𝑇 ∨ 𝑈)) = (𝑆 ∨ (𝑇 ∨ 𝑈)))
40	5, 6	hlatj12 37385	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ (𝑇 ∨ 𝑈)) = (𝑇 ∨ (𝑃 ∨ 𝑈)))
41	1, 9, 3, 4, 40	syl13anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ (𝑇 ∨ 𝑈)) = (𝑇 ∨ (𝑃 ∨ 𝑈)))
42	8, 39, 41	3eqtr2d 2784	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑇 ∨ (𝑃 ∨ 𝑈)))
43	5, 6	hlatj12 37385	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ (𝑃 ∨ 𝑅)))
44	1, 9, 10, 11, 43	syl13anc 1371	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ (𝑃 ∨ 𝑅)))
45	15, 44, 42	3brtr3d 5105	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ∨ (𝑃 ∨ 𝑅)) ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)))
46	17, 6	atbase 37303	. . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
47	10, 46	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ∈ (Base‘𝐾))
48	17, 5, 6	hlatjcl 37381	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
49	1, 9, 11, 48	syl3anc 1370	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
50	17, 6	atbase 37303	. . . . . . . . 9 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
51	3, 50	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑇 ∈ (Base‘𝐾))
52	17, 5, 6	hlatjcl 37381	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
53	1, 9, 4, 52	syl3anc 1370	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
54	17, 5	latjcl 18157	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑇 ∨ (𝑃 ∨ 𝑈)) ∈ (Base‘𝐾))
55	16, 51, 53, 54	syl3anc 1370	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑇 ∨ (𝑃 ∨ 𝑈)) ∈ (Base‘𝐾))
56	17, 28, 5	latjle12 18168	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ (𝑇 ∨ (𝑃 ∨ 𝑈)) ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)) ∧ (𝑃 ∨ 𝑅) ≤ (𝑇 ∨ (𝑃 ∨ 𝑈))) ↔ (𝑄 ∨ (𝑃 ∨ 𝑅)) ≤ (𝑇 ∨ (𝑃 ∨ 𝑈))))
57	16, 47, 49, 55, 56	syl13anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑄 ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)) ∧ (𝑃 ∨ 𝑅) ≤ (𝑇 ∨ (𝑃 ∨ 𝑈))) ↔ (𝑄 ∨ (𝑃 ∨ 𝑅)) ≤ (𝑇 ∨ (𝑃 ∨ 𝑈))))
58	45, 57	mpbird 256	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)) ∧ (𝑃 ∨ 𝑅) ≤ (𝑇 ∨ (𝑃 ∨ 𝑈))))
59	58	simpld 495	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)))
60		simp23 1207	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑈))
61	17, 28, 5, 6	hlexchb2 37399	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) → (𝑄 ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)) ↔ (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑇 ∨ (𝑃 ∨ 𝑈))))
62	1, 10, 3, 53, 60, 61	syl131anc 1382	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ≤ (𝑇 ∨ (𝑃 ∨ 𝑈)) ↔ (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑇 ∨ (𝑃 ∨ 𝑈))))
63	59, 62	mpbid 231	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑇 ∨ (𝑃 ∨ 𝑈)))
64	17, 5	latj13 18204	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑈 ∨ (𝑃 ∨ 𝑄)))
65	16, 47, 19, 25, 64	syl13anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑈 ∨ (𝑃 ∨ 𝑄)))
66	42, 63, 65	3eqtr2d 2784	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑈 ∨ (𝑃 ∨ 𝑄)))
67	17, 5, 6	hlatjcl 37381	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
68	1, 9, 10, 67	syl3anc 1370	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
69	17, 6	atbase 37303	. . . . . . . 8 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
70	11, 69	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ∈ (Base‘𝐾))
71	17, 28, 5	latjle12 18168	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ 𝑅 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
72	16, 68, 70, 27, 71	syl13anc 1371	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (((𝑃 ∨ 𝑄) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ 𝑅 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
73	14, 72	mpbird 256	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ 𝑅 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
74	73	simprd 496	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
75	74, 66	breqtrd 5100	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ≤ (𝑈 ∨ (𝑃 ∨ 𝑄)))
76		simp21 1205	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
77	17, 28, 5, 6	hlexchb2 37399	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ≤ (𝑈 ∨ (𝑃 ∨ 𝑄)) ↔ (𝑅 ∨ (𝑃 ∨ 𝑄)) = (𝑈 ∨ (𝑃 ∨ 𝑄))))
78	1, 11, 4, 68, 76, 77	syl131anc 1382	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑅 ≤ (𝑈 ∨ (𝑃 ∨ 𝑄)) ↔ (𝑅 ∨ (𝑃 ∨ 𝑄)) = (𝑈 ∨ (𝑃 ∨ 𝑄))))
79	75, 78	mpbid 231	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑅 ∨ (𝑃 ∨ 𝑄)) = (𝑈 ∨ (𝑃 ∨ 𝑄)))
80	17, 5	latjcom 18165	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑅 ∨ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
81	16, 70, 68, 80	syl3anc 1370	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑅 ∨ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
82	66, 79, 81	3eqtr2rd 2785	1 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑃 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  Latclat 18149  Atomscatm 37277  HLchlt 37364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-proset 18013  df-poset 18031  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-lat 18150  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365

This theorem is referenced by:  3atlem3  37499