cdleme20j - Mathbox for Norm Megill

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Theorem cdleme20j 41112

Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. 𝐷, 𝐹, 𝑌, 𝐺 represent s₂, f(s), t₂, f(t). We show s₂ ≠ t₂. (Contributed by NM, 18-Nov-2012.)

**Hypotheses**
Ref	Expression
cdleme19.l	⊢ ≤ = (le‘𝐾)
cdleme19.j	⊢ ∨ = (join‘𝐾)
cdleme19.m	⊢ ∧ = (meet‘𝐾)
cdleme19.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme19.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme19.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme19.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme19.g	⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
cdleme19.d	⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
cdleme19.y	⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊)
cdleme20.v	⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊)

**Assertion**
Ref	Expression
cdleme20j	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 ≠ 𝑌)

**Proof of Theorem cdleme20j**
Step	Hyp	Ref	Expression
1		simp33 1230	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))
2		simp11l 1303	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ HL)
3		simp22l 1311	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ∈ 𝐴)
4		simp21l 1309	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑅 ∈ 𝐴)
5		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		cdleme19.j	. . . . . . . . . . . 12 ⊢ ∨ = (join‘𝐾)
7		cdleme19.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (Atoms‘𝐾)
8	5, 6, 7	hlatjcl 40161	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
9	2, 4, 3, 8	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
10		simp11r 1304	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑊 ∈ 𝐻)
11		cdleme19.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (LHyp‘𝐾)
12	5, 11	lhpbase 40792	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
13	10, 12	syl 18	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑊 ∈ (Base‘𝐾))
14		cdleme19.l	. . . . . . . . . . . 12 ⊢ ≤ = (le‘𝐾)
15	14, 6, 7	hlatlej2 40170	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑆 ≤ (𝑅 ∨ 𝑆))
16	2, 4, 3, 15	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ≤ (𝑅 ∨ 𝑆))
17		cdleme19.m	. . . . . . . . . . 11 ⊢ ∧ = (meet‘𝐾)
18	5, 14, 6, 17, 7	atmod2i1 40655	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑆 ≤ (𝑅 ∨ 𝑆)) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) = ((𝑅 ∨ 𝑆) ∧ (𝑊 ∨ 𝑆)))
19	2, 3, 9, 13, 16, 18	syl131anc 1410	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) = ((𝑅 ∨ 𝑆) ∧ (𝑊 ∨ 𝑆)))
20		simp22 1226	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
21		eqid 2763	. . . . . . . . . . . 12 ⊢ (1.‘𝐾) = (1.‘𝐾)
22	14, 6, 21, 7, 11	lhpjat1 40814	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) → (𝑊 ∨ 𝑆) = (1.‘𝐾))
23	2, 10, 20, 22	syl21anc 850	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑊 ∨ 𝑆) = (1.‘𝐾))
24	23	oveq2d 7426	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑅 ∨ 𝑆) ∧ (𝑊 ∨ 𝑆)) = ((𝑅 ∨ 𝑆) ∧ (1.‘𝐾)))
25		hlol 40155	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
26	2, 25	syl 18	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ OL)
27	5, 17, 21	olm11 40021	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑆) ∧ (1.‘𝐾)) = (𝑅 ∨ 𝑆))
28	26, 9, 27	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑅 ∨ 𝑆) ∧ (1.‘𝐾)) = (𝑅 ∨ 𝑆))
29	19, 24, 28	3eqtrd 2802	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) = (𝑅 ∨ 𝑆))
30	29	adantr 485	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) = (𝑅 ∨ 𝑆))
31		cdleme19.d	. . . . . . . . . 10 ⊢ 𝐷 = ((𝑅 ∨ 𝑆) ∧ 𝑊)
32		simp1 1154	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
33		simp23 1227	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊))
34		simp21 1225	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
35		simp31 1228	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇))
36		simp321 1342	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
37		simp322 1343	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
38	36, 37	jca 520	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)))
39		simp323 1344	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
40		cdleme19.u	. . . . . . . . . . . . . . . . 17 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
41		cdleme19.f	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
42		cdleme19.g	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
43		cdleme19.y	. . . . . . . . . . . . . . . . 17 ⊢ 𝑌 = ((𝑅 ∨ 𝑇) ∧ 𝑊)
44		cdleme20.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊)
45	14, 6, 17, 7, 11, 40, 41, 42, 31, 43, 44	cdleme20d 41106	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) = 𝑉)
46	32, 20, 33, 34, 35, 38, 39, 45	syl133anc 1420	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) = 𝑉)
47	2	hllatd 40158	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ Lat)
48		simp12l 1305	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ∈ 𝐴)
49		simp13l 1307	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑄 ∈ 𝐴)
50	14, 6, 17, 7, 11, 40, 41, 5	cdleme1b 41020	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐹 ∈ (Base‘𝐾))
51	2, 10, 48, 49, 3, 50	syl23anc 1404	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐹 ∈ (Base‘𝐾))
52		simp23l 1313	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑇 ∈ 𝐴)
53	14, 6, 17, 7, 11, 40, 42, 5	cdleme1b 41020	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐺 ∈ (Base‘𝐾))
54	2, 10, 48, 49, 52, 53	syl23anc 1404	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐺 ∈ (Base‘𝐾))
55	5, 6	latjcl 18490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾)) → (𝐹 ∨ 𝐺) ∈ (Base‘𝐾))
56	47, 51, 54, 55	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝐹 ∨ 𝐺) ∈ (Base‘𝐾))
57		simp22r 1312	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑆 ≤ 𝑊)
58	14, 6, 17, 7, 11, 31	cdlemeda 41092	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐷 ∈ 𝐴)
59	2, 10, 3, 57, 4, 39, 36, 58	syl223anc 1423	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 ∈ 𝐴)
60		simp23r 1314	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑇 ≤ 𝑊)
61	14, 6, 17, 7, 11, 43	cdlemeda 41092	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))) → 𝑌 ∈ 𝐴)
62	2, 10, 52, 60, 4, 39, 37, 61	syl223anc 1423	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑌 ∈ 𝐴)
63	5, 6, 7	hlatjcl 40161	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐷 ∨ 𝑌) ∈ (Base‘𝐾))
64	2, 59, 62, 63	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝐷 ∨ 𝑌) ∈ (Base‘𝐾))
65	5, 14, 17	latmle2 18516	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ (𝐹 ∨ 𝐺) ∈ (Base‘𝐾) ∧ (𝐷 ∨ 𝑌) ∈ (Base‘𝐾)) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) ≤ (𝐷 ∨ 𝑌))
66	47, 56, 64, 65	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝐹 ∨ 𝐺) ∧ (𝐷 ∨ 𝑌)) ≤ (𝐷 ∨ 𝑌))
67	46, 66	eqbrtrrd 5135	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑉 ≤ (𝐷 ∨ 𝑌))
68	67	adantr 485	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝑉 ≤ (𝐷 ∨ 𝑌))
69	6, 7	hlatjidm 40163	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴) → (𝐷 ∨ 𝐷) = 𝐷)
70	2, 59, 69	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝐷 ∨ 𝐷) = 𝐷)
71		oveq2 7418	. . . . . . . . . . . . . 14 ⊢ (𝐷 = 𝑌 → (𝐷 ∨ 𝐷) = (𝐷 ∨ 𝑌))
72	70, 71	sylan9req 2819	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝐷 = (𝐷 ∨ 𝑌))
73	68, 72	breqtrrd 5139	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝑉 ≤ 𝐷)
74		hlatl 40154	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
75	2, 74	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ AtLat)
76		simp31r 1316	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ≠ 𝑇)
77	14, 6, 17, 7, 11, 44	lhpat2 40839	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) → 𝑉 ∈ 𝐴)
78	2, 10, 3, 57, 52, 76, 77	syl222anc 1413	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑉 ∈ 𝐴)
79	14, 7	atcmp 40105	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝑉 ≤ 𝐷 ↔ 𝑉 = 𝐷))
80	75, 78, 59, 79	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑉 ≤ 𝐷 ↔ 𝑉 = 𝐷))
81	80	adantr 485	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → (𝑉 ≤ 𝐷 ↔ 𝑉 = 𝐷))
82	73, 81	mpbid 235	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝑉 = 𝐷)
83	82, 44	eqtr3di 2813	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝐷 = ((𝑆 ∨ 𝑇) ∧ 𝑊))
84	31, 83	eqtr3id 2812	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → ((𝑅 ∨ 𝑆) ∧ 𝑊) = ((𝑆 ∨ 𝑇) ∧ 𝑊))
85	5, 6, 7	hlatjcl 40161	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
86	2, 3, 52, 85	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
87	5, 14, 17	latmle1 18515	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇))
88	47, 86, 13, 87	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇))
89	88	adantr 485	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇))
90	84, 89	eqbrtrd 5133	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇))
91	14, 6, 7	hlatlej1 40169	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑆 ≤ (𝑆 ∨ 𝑇))
92	2, 3, 52, 91	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ≤ (𝑆 ∨ 𝑇))
93	92	adantr 485	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝑆 ≤ (𝑆 ∨ 𝑇))
94	5, 17	latmcl 18491	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾))
95	47, 9, 13, 94	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾))
96	5, 7	atbase 40083	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
97	3, 96	syl 18	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ∈ (Base‘𝐾))
98	5, 14, 6	latjle12 18501	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (((𝑅 ∨ 𝑆) ∧ 𝑊) ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))) → ((((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)) ↔ (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) ≤ (𝑆 ∨ 𝑇)))
99	47, 95, 97, 86, 98	syl13anc 1399	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)) ↔ (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) ≤ (𝑆 ∨ 𝑇)))
100	99	adantr 485	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → ((((𝑅 ∨ 𝑆) ∧ 𝑊) ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)) ↔ (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) ≤ (𝑆 ∨ 𝑇)))
101	90, 93, 100	mpbi2and 724	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → (((𝑅 ∨ 𝑆) ∧ 𝑊) ∨ 𝑆) ≤ (𝑆 ∨ 𝑇))
102	30, 101	eqbrtrrd 5135	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → (𝑅 ∨ 𝑆) ≤ (𝑆 ∨ 𝑇))
103	5, 7	atbase 40083	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
104	4, 103	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝑅 ∈ (Base‘𝐾))
105	5, 14, 6	latjle12 18501	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))) → ((𝑅 ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)) ↔ (𝑅 ∨ 𝑆) ≤ (𝑆 ∨ 𝑇)))
106	47, 104, 97, 86, 105	syl13anc 1399	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → ((𝑅 ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)) ↔ (𝑅 ∨ 𝑆) ≤ (𝑆 ∨ 𝑇)))
107	106	adantr 485	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → ((𝑅 ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)) ↔ (𝑅 ∨ 𝑆) ≤ (𝑆 ∨ 𝑇)))
108	102, 107	mpbird 260	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → (𝑅 ≤ (𝑆 ∨ 𝑇) ∧ 𝑆 ≤ (𝑆 ∨ 𝑇)))
109	108	simpld 499	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) ∧ 𝐷 = 𝑌) → 𝑅 ≤ (𝑆 ∨ 𝑇))
110	109	ex 417	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (𝐷 = 𝑌 → 𝑅 ≤ (𝑆 ∨ 𝑇)))
111	110	necon3bd 2972	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → (¬ 𝑅 ≤ (𝑆 ∨ 𝑇) → 𝐷 ≠ 𝑌))
112	1, 111	mpd 16	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ ¬ 𝑅 ≤ (𝑆 ∨ 𝑇))) → 𝐷 ≠ 𝑌)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 lecple 17312 joincjn 18362 meetcmee 18363 1.cp1 18473 Latclat 18482 OLcol 39968 Atomscatm 40057 AtLatcal 40058 HLchlt 40144 LHypclh 40778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-proset 18345 df-poset 18364 df-plt 18379 df-lub 18395 df-glb 18396 df-join 18397 df-meet 18398 df-p0 18474 df-p1 18475 df-lat 18483 df-clat 18550 df-oposet 39970 df-ol 39972 df-oml 39973 df-covers 40060 df-ats 40061 df-atl 40092 df-cvlat 40116 df-hlat 40145 df-llines 40292 df-lplanes 40293 df-lvols 40294 df-lines 40295 df-psubsp 40297 df-pmap 40298 df-padd 40590 df-lhyp 40782

This theorem is referenced by: cdleme20l2 41115

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