Theorem cdleme22eALTN 37483

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115. 𝐹, 𝑁, 𝑂 represent f(z), f_z(s), f_z(t) respectively. When t ∨ v = p ∨ q, f_z(s) ≤ f_z(t) ∨ v. (Contributed by NM, 6-Dec-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme22.l	⊢ ≤ = (le‘𝐾)
cdleme22.j	⊢ ∨ = (join‘𝐾)
cdleme22.m	⊢ ∧ = (meet‘𝐾)
cdleme22.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme22.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme22eALT.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme22eALT.f	⊢ 𝐹 = ((𝑦 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ 𝑊)))
cdleme22eALT.g	⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme22eALT.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)))
cdleme22eALT.o	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme22eALTN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑁 ≤ (𝑂 ∨ 𝑉))

Proof of Theorem cdleme22eALTN

Step	Hyp	Ref	Expression
1		cdleme22eALT.n	. . 3 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)))
2		simp11 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝐾 ∈ HL)
3	2	hllatd 36502	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝐾 ∈ Lat)
4		simp21l 1286	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑃 ∈ 𝐴)
5		simp22l 1288	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑄 ∈ 𝐴)
6		eqid 2823	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		cdleme22.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8		cdleme22.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatjcl 36505	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	2, 4, 5, 9	syl3anc 1367	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11		simp12 1200	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑊 ∈ 𝐻)
12		simp3ll 1240	. . . . . . 7 ⊢ ((𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑦 ∈ 𝐴)
13	12	3ad2ant3 1131	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑦 ∈ 𝐴)
14		cdleme22.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
15		cdleme22.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
16		cdleme22.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
17		cdleme22eALT.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
18		cdleme22eALT.f	. . . . . . 7 ⊢ 𝐹 = ((𝑦 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑦) ∧ 𝑊)))
19	14, 7, 15, 8, 16, 17, 18, 6	cdleme1b 37364	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹 ∈ (Base‘𝐾))
20	2, 11, 4, 5, 13, 19	syl23anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝐹 ∈ (Base‘𝐾))
21		simp31 1205	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑆 ∈ 𝐴)
22	6, 7, 8	hlatjcl 36505	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑆 ∨ 𝑦) ∈ (Base‘𝐾))
23	2, 21, 13, 22	syl3anc 1367	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑆 ∨ 𝑦) ∈ (Base‘𝐾))
24	6, 16	lhpbase 37136	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
25	11, 24	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑊 ∈ (Base‘𝐾))
26	6, 15	latmcl 17664	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑦) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑦) ∧ 𝑊) ∈ (Base‘𝐾))
27	3, 23, 25, 26	syl3anc 1367	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑆 ∨ 𝑦) ∧ 𝑊) ∈ (Base‘𝐾))
28	6, 7	latjcl 17663	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑦) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)) ∈ (Base‘𝐾))
29	3, 20, 27, 28	syl3anc 1367	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)) ∈ (Base‘𝐾))
30	6, 14, 15	latmle1 17688	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊))) ≤ (𝑃 ∨ 𝑄))
31	3, 10, 29, 30	syl3anc 1367	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑦) ∧ 𝑊))) ≤ (𝑃 ∨ 𝑄))
32	1, 31	eqbrtrid 5103	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑁 ≤ (𝑃 ∨ 𝑄))
33		simp21 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34		simp13 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑇 ∈ 𝐴)
35		simp321 1319	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑉 ∈ 𝐴)
36		simp322 1320	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑉 ≤ 𝑊)
37	35, 36	jca 514	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
38		simp23 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑃 ≠ 𝑄)
39		simp323 1321	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))
40	14, 7, 15, 8, 16, 17	cdleme22a 37478	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝑉 = 𝑈)
41	2, 11, 33, 5, 34, 37, 38, 39, 40	syl233anc 1395	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑉 = 𝑈)
42	41	oveq2d 7174	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑂 ∨ 𝑉) = (𝑂 ∨ 𝑈))
43		cdleme22eALT.o	. . . . 5 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
44	43	oveq1i 7168	. . . 4 ⊢ (𝑂 ∨ 𝑈) = (((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈)
45		simp21r 1287	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ¬ 𝑃 ≤ 𝑊)
46	14, 7, 15, 8, 16, 17	cdleme0a 37349	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
47	2, 11, 4, 45, 5, 38, 46	syl222anc 1382	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑈 ∈ 𝐴)
48		simp3rl 1242	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑧 ∈ 𝐴)
49	48	3ad2ant3 1131	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑧 ∈ 𝐴)
50		cdleme22eALT.g	. . . . . . . 8 ⊢ 𝐺 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
51	14, 7, 15, 8, 16, 17, 50, 6	cdleme1b 37364	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐺 ∈ (Base‘𝐾))
52	2, 11, 4, 5, 49, 51	syl23anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝐺 ∈ (Base‘𝐾))
53	6, 7, 8	hlatjcl 36505	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑇 ∨ 𝑧) ∈ (Base‘𝐾))
54	2, 34, 49, 53	syl3anc 1367	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑧) ∈ (Base‘𝐾))
55	6, 15	latmcl 17664	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑇 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
56	3, 54, 25, 55	syl3anc 1367	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
57	6, 7	latjcl 17663	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐺 ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
58	3, 52, 56, 57	syl3anc 1367	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
59	14, 7, 15, 8, 16, 17	cdlemeulpq 37358	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑈 ≤ (𝑃 ∨ 𝑄))
60	2, 11, 4, 5, 59	syl22anc 836	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑈 ≤ (𝑃 ∨ 𝑄))
61	6, 14, 7, 15, 8	atmod2i1 36999	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑈 ≤ (𝑃 ∨ 𝑄)) → (((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
62	2, 47, 10, 58, 60, 61	syl131anc 1379	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (((𝑃 ∨ 𝑄) ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
63	44, 62	syl5req 2871	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑂 ∨ 𝑈))
64	41	oveq2d 7174	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑉) = (𝑇 ∨ 𝑈))
65	39, 64	eqtr3d 2860	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑄) = (𝑇 ∨ 𝑈))
66	6, 7, 8	hlatjcl 36505	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
67	2, 34, 47, 66	syl3anc 1367	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
68	6, 8	atbase 36427	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (Base‘𝐾))
69	49, 68	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑧 ∈ (Base‘𝐾))
70	6, 14, 7	latlej1 17672	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑇 ∨ 𝑈) ≤ ((𝑇 ∨ 𝑈) ∨ 𝑧))
71	3, 67, 69, 70	syl3anc 1367	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑈) ≤ ((𝑇 ∨ 𝑈) ∨ 𝑧))
72	7, 8	hlatj32 36510	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑇 ∨ 𝑈) ∨ 𝑧) = ((𝑇 ∨ 𝑧) ∨ 𝑈))
73	2, 34, 47, 49, 72	syl13anc 1368	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑈) ∨ 𝑧) = ((𝑇 ∨ 𝑧) ∨ 𝑈))
74	6, 8	atbase 36427	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
75	47, 74	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑈 ∈ (Base‘𝐾))
76	6, 7	latj32 17709	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))) → ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
77	3, 69, 75, 56, 76	syl13anc 1368	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
78	6, 7	latj32 17709	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) = ((𝐺 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
79	3, 52, 56, 75, 78	syl13anc 1368	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) = ((𝐺 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
80	6, 7, 8	hlatjcl 36505	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑃 ∨ 𝑧) ∈ (Base‘𝐾))
81	2, 4, 49, 80	syl3anc 1367	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑧) ∈ (Base‘𝐾))
82	14, 7, 8	hlatlej1 36513	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑧))
83	2, 4, 49, 82	syl3anc 1367	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑃 ≤ (𝑃 ∨ 𝑧))
84	6, 14, 7, 15, 8	atmod3i1 37002	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ 𝑧)) → (𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ 𝑧) ∧ (𝑃 ∨ 𝑊)))
85	2, 4, 81, 25, 83, 84	syl131anc 1379	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ 𝑧) ∧ (𝑃 ∨ 𝑊)))
86		eqid 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1.‘𝐾) = (1.‘𝐾)
87	14, 7, 86, 8, 16	lhpjat2 37159	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
88	2, 11, 33, 87	syl21anc 835	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
89	88	oveq2d 7174	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑧) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ 𝑧) ∧ (1.‘𝐾)))
90		hlol 36499	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
91	2, 90	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝐾 ∈ OL)
92	6, 15, 86	olm11 36365	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑧) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑧))
93	91, 81, 92	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑧))
94	85, 89, 93	3eqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = (𝑃 ∨ 𝑧))
95	94	oveq1d 7173	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄) = ((𝑃 ∨ 𝑧) ∨ 𝑄))
96	17	oveq2i 7169	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ∨ 𝑈) = (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))
97	14, 7, 8	hlatlej2 36514	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
98	2, 4, 5, 97	syl3anc 1367	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑄 ≤ (𝑃 ∨ 𝑄))
99	6, 14, 7, 15, 8	atmod3i1 37002	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑄 ≤ (𝑃 ∨ 𝑄)) → (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)))
100	2, 5, 10, 25, 98, 99	syl131anc 1379	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)))
101	96, 100	syl5eq 2870	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑄 ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)))
102		simp22 1203	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
103	14, 7, 86, 8, 16	lhpjat2 37159	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∨ 𝑊) = (1.‘𝐾))
104	2, 11, 102, 103	syl21anc 835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑄 ∨ 𝑊) = (1.‘𝐾))
105	104	oveq2d 7174	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)))
106	6, 15, 86	olm11 36365	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑄))
107	91, 10, 106	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑄))
108	101, 105, 107	3eqtrd 2862	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑄 ∨ 𝑈) = (𝑃 ∨ 𝑄))
109	108	oveq1d 7173	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
110	6, 8	atbase 36427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
111	4, 110	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑃 ∈ (Base‘𝐾))
112	6, 15	latmcl 17664	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
113	3, 81, 25, 112	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
114	6, 8	atbase 36427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
115	5, 114	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑄 ∈ (Base‘𝐾))
116	6, 7	latj32 17709	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄) = ((𝑃 ∨ 𝑄) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
117	3, 111, 113, 115, 116	syl13anc 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄) = ((𝑃 ∨ 𝑄) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
118	109, 117	eqtr4d 2861	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄))
119	7, 8	hlatj32 36510	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑃 ∨ 𝑧) ∨ 𝑄))
120	2, 4, 5, 49, 119	syl13anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑃 ∨ 𝑧) ∨ 𝑄))
121	95, 118, 120	3eqtr4rd 2869	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
122	6, 7	latj32 17709	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
123	3, 115, 75, 113, 122	syl13anc 1368	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
124	121, 123	eqtrd 2858	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
125	124	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑧 ∨ 𝑈) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
126	6, 7	latjcl 17663	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾))
127	3, 10, 69, 126	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾))
128	6, 14, 7	latlej2 17673	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
129	3, 10, 69, 128	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑧 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
130	6, 14, 7, 15, 8	atmod1i1 36995	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑧 ∈ 𝐴 ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾)) ∧ 𝑧 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧)) → (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))) = ((𝑧 ∨ 𝑈) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)))
131	2, 49, 75, 127, 129, 130	syl131anc 1379	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))) = ((𝑧 ∨ 𝑈) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)))
132	50	oveq1i 7168	. . . . . . . . . . . . 13 ⊢ (𝐺 ∨ 𝑈) = (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈)
133	6, 7, 8	hlatjcl 36505	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑧 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑧 ∨ 𝑈) ∈ (Base‘𝐾))
134	2, 49, 47, 133	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑧 ∨ 𝑈) ∈ (Base‘𝐾))
135	6, 7	latjcl 17663	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
136	3, 115, 113, 135	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
137	14, 7, 8	hlatlej2 36514	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑧 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑈 ≤ (𝑧 ∨ 𝑈))
138	2, 49, 47, 137	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑈 ≤ (𝑧 ∨ 𝑈))
139	6, 14, 7, 15, 8	atmod2i1 36999	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ (𝑧 ∨ 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑈 ≤ (𝑧 ∨ 𝑈)) → (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
140	2, 47, 134, 136, 138, 139	syl131anc 1379	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
141	132, 140	syl5eq 2870	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝐺 ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
142	125, 131, 141	3eqtr4rd 2869	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝐺 ∨ 𝑈) = (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))))
143	6, 14, 7	latlej1 17672	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
144	3, 10, 69, 143	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
145	6, 14, 3, 75, 10, 127, 60, 144	lattrd 17670	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑈 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
146	6, 14, 15	latleeqm1 17691	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾)) → (𝑈 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧) ↔ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = 𝑈))
147	3, 75, 127, 146	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑈 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧) ↔ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = 𝑈))
148	145, 147	mpbid 234	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = 𝑈)
149	148	oveq2d 7174	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))) = (𝑧 ∨ 𝑈))
150	142, 149	eqtrd 2858	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝐺 ∨ 𝑈) = (𝑧 ∨ 𝑈))
151	150	oveq1d 7173	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝐺 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
152	79, 151	eqtrd 2858	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
153	14, 7, 8	hlatlej2 36514	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (𝑇 ∨ 𝑧))
154	2, 34, 49, 153	syl3anc 1367	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑧 ≤ (𝑇 ∨ 𝑧))
155	6, 14, 7, 15, 8	atmod3i1 37002	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑧 ∈ 𝐴 ∧ (𝑇 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑧 ≤ (𝑇 ∨ 𝑧)) → (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (𝑧 ∨ 𝑊)))
156	2, 49, 54, 25, 154, 155	syl131anc 1379	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (𝑧 ∨ 𝑊)))
157		simp33r 1297	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))
158	14, 7, 86, 8, 16	lhpjat2 37159	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)) → (𝑧 ∨ 𝑊) = (1.‘𝐾))
159	2, 11, 157, 158	syl21anc 835	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑧 ∨ 𝑊) = (1.‘𝐾))
160	159	oveq2d 7174	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑧) ∧ (𝑧 ∨ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)))
161	6, 15, 86	olm11 36365	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ (𝑇 ∨ 𝑧) ∈ (Base‘𝐾)) → ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑇 ∨ 𝑧))
162	91, 54, 161	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑇 ∨ 𝑧))
163	156, 160, 162	3eqtrrd 2863	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑧) = (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
164	163	oveq1d 7173	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑧) ∨ 𝑈) = ((𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
165	77, 152, 164	3eqtr4rd 2869	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑧) ∨ 𝑈) = ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
166	73, 165	eqtrd 2858	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑇 ∨ 𝑈) ∨ 𝑧) = ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
167	71, 166	breqtrd 5094	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑇 ∨ 𝑈) ≤ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
168	65, 167	eqbrtrd 5090	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
169	6, 7	latjcl 17663	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ∈ (Base‘𝐾))
170	3, 58, 75, 169	syl3anc 1367	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ∈ (Base‘𝐾))
171	6, 14, 15	latleeqm1 17691	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑃 ∨ 𝑄)))
172	3, 10, 170, 171	syl3anc 1367	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑃 ∨ 𝑄)))
173	168, 172	mpbid 234	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → ((𝑃 ∨ 𝑄) ∧ ((𝐺 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑃 ∨ 𝑄))
174	42, 63, 173	3eqtr2rd 2865	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → (𝑃 ∨ 𝑄) = (𝑂 ∨ 𝑉))
175	32, 174	breqtrd 5094	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄) ∧ (𝑆 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)))) → 𝑁 ≤ (𝑂 ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  joincjn 17556  meetcmee 17557  1.cp1 17650  Latclat 17657  OLcol 36312  Atomscatm 36401  HLchlt 36488  LHypclh 37122

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-p1 17652  df-lat 17658  df-clat 17720  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489  df-psubsp 36641  df-pmap 36642  df-padd 36934  df-lhyp 37126

This theorem is referenced by:  cdleme26eALTN  37499