Theorem cdleme22e 40340

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.)

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)
cdleme22e.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme22e.f	⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme22e.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)))
cdleme22e.o	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme22e	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑁 ≤ (𝑂 ∨ 𝑉))

Proof of Theorem cdleme22e

Step	Hyp	Ref	Expression
1		cdleme22e.n	. . 3 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)))
2		simp1l 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐾 ∈ HL)
3	2	hllatd 39360	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐾 ∈ Lat)
4		simp21l 1291	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
5		simp22l 1293	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
6		eqid 2729	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		cdleme22.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8		cdleme22.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatjcl 39363	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	2, 4, 5, 9	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11		simp1r 1199	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑊 ∈ 𝐻)
12		simp33l 1301	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑧 ∈ 𝐴)
13		cdleme22.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
14		cdleme22.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
15		cdleme22.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
16		cdleme22e.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
17		cdleme22e.f	. . . . . . 7 ⊢ 𝐹 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
18	13, 7, 14, 8, 15, 16, 17, 6	cdleme1b 40222	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹 ∈ (Base‘𝐾))
19	2, 11, 4, 5, 12, 18	syl23anc 1379	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐹 ∈ (Base‘𝐾))
20		simp23l 1295	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑆 ∈ 𝐴)
21	6, 7, 8	hlatjcl 39363	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑆 ∨ 𝑧) ∈ (Base‘𝐾))
22	2, 20, 12, 21	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑆 ∨ 𝑧) ∈ (Base‘𝐾))
23	6, 15	lhpbase 39994	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
24	11, 23	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑊 ∈ (Base‘𝐾))
25	6, 14	latmcl 18333	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
26	3, 22, 24, 25	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑆 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
27	6, 7	latjcl 18332	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
28	3, 19, 26, 27	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
29	6, 13, 14	latmle1 18357	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊))) ≤ (𝑃 ∨ 𝑄))
30	3, 10, 28, 29	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑧) ∧ 𝑊))) ≤ (𝑃 ∨ 𝑄))
31	1, 30	eqbrtrid 5123	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑁 ≤ (𝑃 ∨ 𝑄))
32		simp1 1136	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		simp21 1207	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34		simp23r 1296	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑇 ∈ 𝐴)
35		simp31 1210	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
36		simp32l 1299	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑃 ≠ 𝑄)
37		simp32r 1300	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))
38	13, 7, 14, 8, 15, 16	cdleme22a 40336	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄))) → 𝑉 = 𝑈)
39	32, 33, 5, 34, 35, 36, 37, 38	syl133anc 1395	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑉 = 𝑈)
40	39	oveq2d 7356	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑂 ∨ 𝑉) = (𝑂 ∨ 𝑈))
41		cdleme22e.o	. . . . . 6 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
42	41	oveq1i 7350	. . . . 5 ⊢ (𝑂 ∨ 𝑈) = (((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈)
43		simp21r 1292	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ¬ 𝑃 ≤ 𝑊)
44	13, 7, 14, 8, 15, 16	cdleme0a 40207	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄)) → 𝑈 ∈ 𝐴)
45	2, 11, 4, 43, 5, 36, 44	syl222anc 1388	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑈 ∈ 𝐴)
46	6, 7, 8	hlatjcl 39363	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑇 ∨ 𝑧) ∈ (Base‘𝐾))
47	2, 34, 12, 46	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑧) ∈ (Base‘𝐾))
48	6, 14	latmcl 18333	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑇 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
49	3, 47, 24, 48	syl3anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
50	6, 7	latjcl 18332	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
51	3, 19, 49, 50	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
52	13, 7, 14, 8, 15, 16	cdlemeulpq 40216	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑈 ≤ (𝑃 ∨ 𝑄))
53	2, 11, 4, 5, 52	syl22anc 838	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑈 ≤ (𝑃 ∨ 𝑄))
54	6, 13, 7, 14, 8	atmod2i1 39857	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑈 ≤ (𝑃 ∨ 𝑄)) → (((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
55	2, 45, 10, 51, 53, 54	syl131anc 1385	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
56	42, 55	eqtr2id 2777	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑂 ∨ 𝑈))
57	40, 56	eqtr4d 2767	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑂 ∨ 𝑉) = ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
58	39	oveq2d 7356	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑉) = (𝑇 ∨ 𝑈))
59	37, 58	eqtr3d 2766	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑄) = (𝑇 ∨ 𝑈))
60	6, 7, 8	hlatjcl 39363	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
61	2, 34, 45, 60	syl3anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑈) ∈ (Base‘𝐾))
62	6, 8	atbase 39285	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (Base‘𝐾))
63	12, 62	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑧 ∈ (Base‘𝐾))
64	6, 13, 7	latlej1 18341	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑇 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑇 ∨ 𝑈) ≤ ((𝑇 ∨ 𝑈) ∨ 𝑧))
65	3, 61, 63, 64	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑈) ≤ ((𝑇 ∨ 𝑈) ∨ 𝑧))
66	7, 8	hlatj32 39368	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑇 ∨ 𝑈) ∨ 𝑧) = ((𝑇 ∨ 𝑧) ∨ 𝑈))
67	2, 34, 45, 12, 66	syl13anc 1374	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑈) ∨ 𝑧) = ((𝑇 ∨ 𝑧) ∨ 𝑈))
68	6, 8	atbase 39285	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
69	45, 68	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑈 ∈ (Base‘𝐾))
70	6, 7	latj32 18378	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑧 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))) → ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
71	3, 63, 69, 49, 70	syl13anc 1374	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
72	6, 7	latj32 18378	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝐹 ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) = ((𝐹 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
73	3, 19, 49, 69, 72	syl13anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) = ((𝐹 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
74	6, 7, 8	hlatjcl 39363	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑃 ∨ 𝑧) ∈ (Base‘𝐾))
75	2, 4, 12, 74	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑧) ∈ (Base‘𝐾))
76	13, 7, 8	hlatlej1 39371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑧))
77	2, 4, 12, 76	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑃 ≤ (𝑃 ∨ 𝑧))
78	6, 13, 7, 14, 8	atmod3i1 39860	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ 𝑧)) → (𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ 𝑧) ∧ (𝑃 ∨ 𝑊)))
79	2, 4, 75, 24, 77, 78	syl131anc 1385	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ 𝑧) ∧ (𝑃 ∨ 𝑊)))
80		eqid 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1.‘𝐾) = (1.‘𝐾)
81	13, 7, 80, 8, 15	lhpjat2 40017	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
82	2, 11, 33, 81	syl21anc 837	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
83	82	oveq2d 7356	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑧) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ 𝑧) ∧ (1.‘𝐾)))
84		hlol 39357	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
85	2, 84	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝐾 ∈ OL)
86	6, 14, 80	olm11 39223	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑧) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑧))
87	85, 75, 86	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑧))
88	79, 83, 87	3eqtrd 2768	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = (𝑃 ∨ 𝑧))
89	88	oveq1d 7355	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄) = ((𝑃 ∨ 𝑧) ∨ 𝑄))
90	16	oveq2i 7351	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ∨ 𝑈) = (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))
91	13, 7, 8	hlatlej2 39372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
92	2, 4, 5, 91	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑄 ≤ (𝑃 ∨ 𝑄))
93	6, 13, 7, 14, 8	atmod3i1 39860	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑄 ≤ (𝑃 ∨ 𝑄)) → (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)))
94	2, 5, 10, 24, 92, 93	syl131anc 1385	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑄 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)))
95	90, 94	eqtrid 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑄 ∨ 𝑈) = ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)))
96		simp22 1208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
97	13, 7, 80, 8, 15	lhpjat2 40017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∨ 𝑊) = (1.‘𝐾))
98	2, 11, 96, 97	syl21anc 837	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑄 ∨ 𝑊) = (1.‘𝐾))
99	98	oveq2d 7356	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)))
100	6, 14, 80	olm11 39223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑄))
101	85, 10, 100	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑄))
102	95, 99, 101	3eqtrd 2768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑄 ∨ 𝑈) = (𝑃 ∨ 𝑄))
103	102	oveq1d 7355	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
104	6, 8	atbase 39285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
105	4, 104	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑃 ∈ (Base‘𝐾))
106	6, 14	latmcl 18333	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
107	3, 75, 24, 106	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))
108	6, 8	atbase 39285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
109	5, 108	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑄 ∈ (Base‘𝐾))
110	6, 7	latj32 18378	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄) = ((𝑃 ∨ 𝑄) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
111	3, 105, 107, 109, 110	syl13anc 1374	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄) = ((𝑃 ∨ 𝑄) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
112	103, 111	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑃 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑄))
113	7, 8	hlatj32 39368	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑃 ∨ 𝑧) ∨ 𝑄))
114	2, 4, 5, 12, 113	syl13anc 1374	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑃 ∨ 𝑧) ∨ 𝑄))
115	89, 112, 114	3eqtr4rd 2775	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
116	6, 7	latj32 18378	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
117	3, 109, 69, 107, 116	syl13anc 1374	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑄 ∨ 𝑈) ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) = ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
118	115, 117	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) = ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
119	118	oveq2d 7356	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑧 ∨ 𝑈) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
120	6, 7	latjcl 18332	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾))
121	3, 10, 63, 120	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾))
122	6, 13, 7	latlej2 18342	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
123	3, 10, 63, 122	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑧 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
124	6, 13, 7, 14, 8	atmod1i1 39853	. . . . . . . . . . . . 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)))
125	2, 12, 69, 121, 123, 124	syl131anc 1385	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))) = ((𝑧 ∨ 𝑈) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)))
126	17	oveq1i 7350	. . . . . . . . . . . . 13 ⊢ (𝐹 ∨ 𝑈) = (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈)
127	6, 7, 8	hlatjcl 39363	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑧 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑧 ∨ 𝑈) ∈ (Base‘𝐾))
128	2, 12, 45, 127	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∨ 𝑈) ∈ (Base‘𝐾))
129	6, 7	latjcl 18332	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑧) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
130	3, 109, 107, 129	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾))
131	13, 7, 8	hlatlej2 39372	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑧 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → 𝑈 ≤ (𝑧 ∨ 𝑈))
132	2, 12, 45, 131	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑈 ≤ (𝑧 ∨ 𝑈))
133	6, 13, 7, 14, 8	atmod2i1 39857	. . . . . . . . . . . . . 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)))
134	2, 45, 128, 130, 132, 133	syl131anc 1385	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊))) ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
135	126, 134	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝐹 ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∧ ((𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)))
136	119, 125, 135	3eqtr4rd 2775	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝐹 ∨ 𝑈) = (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))))
137	6, 13, 7	latlej1 18341	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
138	3, 10, 63, 137	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
139	6, 13, 3, 69, 10, 121, 53, 138	lattrd 18339	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑈 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧))
140	6, 13, 14	latleeqm1 18360	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑈 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧) ∈ (Base‘𝐾)) → (𝑈 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧) ↔ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = 𝑈))
141	3, 69, 121, 140	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑈 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑧) ↔ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = 𝑈))
142	139, 141	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧)) = 𝑈)
143	142	oveq2d 7356	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∨ (𝑈 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑧))) = (𝑧 ∨ 𝑈))
144	136, 143	eqtrd 2764	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝐹 ∨ 𝑈) = (𝑧 ∨ 𝑈))
145	144	oveq1d 7355	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝐹 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
146	73, 145	eqtrd 2764	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) = ((𝑧 ∨ 𝑈) ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
147	13, 7, 8	hlatlej2 39372	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤ (𝑇 ∨ 𝑧))
148	2, 34, 12, 147	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑧 ≤ (𝑇 ∨ 𝑧))
149	6, 13, 7, 14, 8	atmod3i1 39860	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑧 ∈ 𝐴 ∧ (𝑇 ∨ 𝑧) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑧 ≤ (𝑇 ∨ 𝑧)) → (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (𝑧 ∨ 𝑊)))
150	2, 12, 47, 24, 148, 149	syl131anc 1385	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (𝑧 ∨ 𝑊)))
151		simp33 1212	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))
152	13, 7, 80, 8, 15	lhpjat2 40017	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊)) → (𝑧 ∨ 𝑊) = (1.‘𝐾))
153	2, 11, 151, 152	syl21anc 837	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∨ 𝑊) = (1.‘𝐾))
154	153	oveq2d 7356	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑧) ∧ (𝑧 ∨ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)))
155	150, 154	eqtrd 2764	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) = ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)))
156	6, 14, 80	olm11 39223	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ (𝑇 ∨ 𝑧) ∈ (Base‘𝐾)) → ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑇 ∨ 𝑧))
157	85, 47, 156	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑧) ∧ (1.‘𝐾)) = (𝑇 ∨ 𝑧))
158	155, 157	eqtr2d 2765	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑧) = (𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)))
159	158	oveq1d 7355	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑧) ∨ 𝑈) = ((𝑧 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
160	71, 146, 159	3eqtr4rd 2775	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑧) ∨ 𝑈) = ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
161	67, 160	eqtrd 2764	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑇 ∨ 𝑈) ∨ 𝑧) = ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
162	65, 161	breqtrd 5114	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑇 ∨ 𝑈) ≤ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
163	59, 162	eqbrtrd 5110	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑄) ≤ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈))
164	6, 7	latjcl 18332	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ∈ (Base‘𝐾))
165	3, 51, 69, 164	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ∈ (Base‘𝐾))
166	6, 13, 14	latleeqm1 18360	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ≤ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑃 ∨ 𝑄)))
167	3, 10, 165, 166	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ≤ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈) ↔ ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑃 ∨ 𝑄)))
168	163, 167	mpbid 232	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ ((𝐹 ∨ ((𝑇 ∨ 𝑧) ∧ 𝑊)) ∨ 𝑈)) = (𝑃 ∨ 𝑄))
169	57, 168	eqtr2d 2765	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → (𝑃 ∨ 𝑄) = (𝑂 ∨ 𝑉))
170	31, 169	breqtrd 5114	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ (𝑇 ∨ 𝑉) = (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊))) → 𝑁 ≤ (𝑂 ∨ 𝑉))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5088  ‘cfv 6476  (class class class)co 7340  Basecbs 17107  lecple 17155  joincjn 18204  meetcmee 18205  1.cp1 18315  Latclat 18324  OLcol 39170  Atomscatm 39259  HLchlt 39346  LHypclh 39980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-iin 4941  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-proset 18187  df-poset 18206  df-plt 18221  df-lub 18237  df-glb 18238  df-join 18239  df-meet 18240  df-p0 18316  df-p1 18317  df-lat 18325  df-clat 18392  df-oposet 39172  df-ol 39174  df-oml 39175  df-covers 39262  df-ats 39263  df-atl 39294  df-cvlat 39318  df-hlat 39347  df-psubsp 39499  df-pmap 39500  df-padd 39792  df-lhyp 39984

This theorem is referenced by:  cdleme26e  40355