Theorem 3atlem2 39976

Description: Lemma for 3at 39982. (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
3atlem2	⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑈))

Proof of Theorem 3atlem2

Step	Hyp	Ref	Expression
1		simp3 1144	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
2		simp11 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝐾 ∈ HL)
3	2	hllatd 39856	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝐾 ∈ Lat)
4		simp121 1312	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ∈ 𝐴)
5		simp122 1313	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ∈ 𝐴)
6		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		3at.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
8		3at.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatjcl 39859	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	2, 4, 5, 9	syl3anc 1379	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11		simp123 1314	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ∈ 𝐴)
12	6, 8	atbase 39781	. . . . . . . 8 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
13	11, 12	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ∈ (Base‘𝐾))
14		simp131 1315	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑆 ∈ 𝐴)
15		simp132 1316	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑇 ∈ 𝐴)
16	6, 7, 8	hlatjcl 39859	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
17	2, 14, 15, 16	syl3anc 1379	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
18		simp133 1317	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑈 ∈ 𝐴)
19	6, 8	atbase 39781	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
20	18, 19	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑈 ∈ (Base‘𝐾))
21	6, 7	latjcl 18396	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))
22	3, 17, 20, 21	syl3anc 1379	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))
23		3at.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
24	6, 23, 7	latjle12 18407	. . . . . . 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)))
25	3, 10, 13, 22, 24	syl13anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 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)))
26	1, 25	mpbird 258	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∧ 𝑅 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)))
27	26	simprd 496	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈))
28	7, 8	hlatjass 39862	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈)))
29	2, 14, 15, 18, 28	syl13anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑇 ∨ 𝑈)))
30		simp22r 1300	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ≤ (𝑇 ∨ 𝑈))
31		simp22l 1299	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑃 ≠ 𝑈)
32	23, 7, 8	hlatexchb2 39886	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ 𝑃 ≠ 𝑈) → (𝑃 ≤ (𝑇 ∨ 𝑈) ↔ (𝑃 ∨ 𝑈) = (𝑇 ∨ 𝑈)))
33	2, 4, 15, 18, 31, 32	syl131anc 1391	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ≤ (𝑇 ∨ 𝑈) ↔ (𝑃 ∨ 𝑈) = (𝑇 ∨ 𝑈)))
34	30, 33	mpbid 233	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑈) = (𝑇 ∨ 𝑈))
35	34	oveq2d 7372	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑆 ∨ (𝑃 ∨ 𝑈)) = (𝑆 ∨ (𝑇 ∨ 𝑈)))
36	29, 35	eqtr4d 2777	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = (𝑆 ∨ (𝑃 ∨ 𝑈)))
37	7, 8	hlatjass 39862	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑈) = (𝑃 ∨ (𝑄 ∨ 𝑈)))
38	2, 4, 5, 18, 37	syl13anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑈) = (𝑃 ∨ (𝑄 ∨ 𝑈)))
39	7, 8	hlatj12 39863	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (𝑃 ∨ (𝑄 ∨ 𝑈)) = (𝑄 ∨ (𝑃 ∨ 𝑈)))
40	2, 4, 5, 18, 39	syl13anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ (𝑄 ∨ 𝑈)) = (𝑄 ∨ (𝑃 ∨ 𝑈)))
41	7, 8	hlatj32 39864	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑅) ∨ 𝑄))
42	2, 4, 5, 11, 41	syl13anc 1380	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑅) ∨ 𝑄))
43	1, 42, 29	3brtr3d 5103	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑅) ∨ 𝑄) ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)))
44	6, 7, 8	hlatjcl 39859	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
45	2, 4, 11, 44	syl3anc 1379	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
46	6, 8	atbase 39781	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
47	5, 46	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ∈ (Base‘𝐾))
48	6, 8	atbase 39781	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
49	14, 48	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑆 ∈ (Base‘𝐾))
50	6, 7, 8	hlatjcl 39859	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
51	2, 15, 18, 50	syl3anc 1379	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
52	6, 7	latjcl 18396	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑆 ∨ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))
53	3, 49, 51, 52	syl3anc 1379	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑆 ∨ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))
54	6, 23, 7	latjle12 18407	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑆 ∨ (𝑇 ∨ 𝑈)) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑅) ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)) ∧ 𝑄 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))) ↔ ((𝑃 ∨ 𝑅) ∨ 𝑄) ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))))
55	3, 45, 47, 53, 54	syl13anc 1380	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (((𝑃 ∨ 𝑅) ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)) ∧ 𝑄 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))) ↔ ((𝑃 ∨ 𝑅) ∨ 𝑄) ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))))
56	43, 55	mpbird 258	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑅) ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)) ∧ 𝑄 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈))))
57	56	simprd 496	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ≤ (𝑆 ∨ (𝑇 ∨ 𝑈)))
58	57, 35	breqtrrd 5100	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑄 ≤ (𝑆 ∨ (𝑃 ∨ 𝑈)))
59	6, 7, 8	hlatjcl 39859	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
60	2, 4, 18, 59	syl3anc 1379	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
61		simp23 1215	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑈))
62	6, 23, 7, 8	hlexchb2 39877	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) → (𝑄 ≤ (𝑆 ∨ (𝑃 ∨ 𝑈)) ↔ (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑆 ∨ (𝑃 ∨ 𝑈))))
63	2, 5, 14, 60, 61, 62	syl131anc 1391	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ≤ (𝑆 ∨ (𝑃 ∨ 𝑈)) ↔ (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑆 ∨ (𝑃 ∨ 𝑈))))
64	58, 63	mpbid 233	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑄 ∨ (𝑃 ∨ 𝑈)) = (𝑆 ∨ (𝑃 ∨ 𝑈)))
65	38, 40, 64	3eqtrd 2778	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑈) = (𝑆 ∨ (𝑃 ∨ 𝑈)))
66	36, 65	eqtr4d 2777	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∨ 𝑈))
67	27, 66	breqtrd 5098	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → 𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑈))
68		simp21 1213	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
69	6, 23, 7, 8	hlexchb1 39876	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑄) ∨ 𝑈)))
70	2, 11, 18, 10, 68, 69	syl131anc 1391	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → (𝑅 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑄) ∨ 𝑈)))
71	67, 70	mpbid 233	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 𝑃 ≤ (𝑇 ∨ 𝑈)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑈)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑃 ∨ 𝑄) ∨ 𝑈))
72	71, 66	eqtr4d 2777	1 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑃 ≠ 𝑈 ∧ 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  joincjn 18268  Latclat 18388  Atomscatm 39755  HLchlt 39842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18389  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843

This theorem is referenced by:  3atlem3  39977