Theorem cdleme22f 41153

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), f_t(s) respectively. If s ≤ t ∨ v, then f_t(s) ≤ f(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)
cdleme22f.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme22f.f	⊢ 𝐹 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
cdleme22f.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme22f	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑁 ≤ (𝐹 ∨ 𝑉))

Proof of Theorem cdleme22f

Step	Hyp	Ref	Expression
1		cdleme22f.n	. 2 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)))
2		simp11l 1303	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝐾 ∈ HL)
3	2	hllatd 40171	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝐾 ∈ Lat)
4		simp12l 1305	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑃 ∈ 𝐴)
5		simp13l 1307	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑄 ∈ 𝐴)
6		eqid 2765	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		cdleme22.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8		cdleme22.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatjcl 40174	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
10	2, 4, 5, 9	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11		simp11r 1304	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑊 ∈ 𝐻)
12		simp22 1226	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑇 ∈ 𝐴)
13		cdleme22.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
14		cdleme22.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
15		cdleme22.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
16		cdleme22f.u	. . . . . . 7 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
17		cdleme22f.f	. . . . . . 7 ⊢ 𝐹 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
18	13, 7, 14, 8, 15, 16, 17, 6	cdleme1b 41033	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐹 ∈ (Base‘𝐾))
19	2, 11, 4, 5, 12, 18	syl23anc 1404	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝐹 ∈ (Base‘𝐾))
20		simp21l 1309	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑆 ∈ 𝐴)
21	6, 7, 8	hlatjcl 40174	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
22	2, 20, 12, 21	syl3anc 1398	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
23	6, 15	lhpbase 40805	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
24	11, 23	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑊 ∈ (Base‘𝐾))
25	6, 14	latmcl 18514	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ∈ (Base‘𝐾))
26	3, 22, 24, 25	syl3anc 1398	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → ((𝑆 ∨ 𝑇) ∧ 𝑊) ∈ (Base‘𝐾))
27	6, 7	latjcl 18513	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑇) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)) ∈ (Base‘𝐾))
28	3, 19, 26, 27	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)) ∈ (Base‘𝐾))
29	6, 13, 14	latmle2 18539	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊))) ≤ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)))
30	3, 10, 28, 29	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊))) ≤ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)))
31		simp21 1225	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊))
32		simp3l 1220	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑆 ≠ 𝑇)
33		simp23l 1313	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑉 ∈ 𝐴)
34		simp23r 1314	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑉 ≤ 𝑊)
35		simp3r 1221	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑆 ≤ (𝑇 ∨ 𝑉))
36	7, 8	hlatjcom 40175	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑇 ∨ 𝑉) = (𝑉 ∨ 𝑇))
37	2, 12, 33, 36	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝑇 ∨ 𝑉) = (𝑉 ∨ 𝑇))
38	35, 37	breqtrd 5139	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑆 ≤ (𝑉 ∨ 𝑇))
39		hlcvl 40166	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
40	2, 39	syl 18	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝐾 ∈ CvLat)
41	13, 7, 8	cvlatexch2 40144	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑆 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑆 ≠ 𝑇) → (𝑆 ≤ (𝑉 ∨ 𝑇) → 𝑉 ≤ (𝑆 ∨ 𝑇)))
42	40, 20, 33, 12, 32, 41	syl131anc 1410	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝑆 ≤ (𝑉 ∨ 𝑇) → 𝑉 ≤ (𝑆 ∨ 𝑇)))
43	38, 42	mpd 16	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑉 ≤ (𝑆 ∨ 𝑇))
44		eqid 2765	. . . . . 6 ⊢ ((𝑆 ∨ 𝑇) ∧ 𝑊) = ((𝑆 ∨ 𝑇) ∧ 𝑊)
45	13, 7, 14, 8, 15, 44	cdleme22aa 41146	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ∧ 𝑉 ≤ (𝑆 ∨ 𝑇))) → 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊))
46	2, 11, 31, 12, 32, 33, 34, 43, 45	syl233anc 1426	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑉 = ((𝑆 ∨ 𝑇) ∧ 𝑊))
47	46	oveq2d 7432	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → (𝐹 ∨ 𝑉) = (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊)))
48	30, 47	breqtrrd 5141	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑆 ∨ 𝑇) ∧ 𝑊))) ≤ (𝐹 ∨ 𝑉))
49	1, 48	eqbrtrid 5148	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ 𝑇 ∈ 𝐴 ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ∧ (𝑆 ≠ 𝑇 ∧ 𝑆 ≤ (𝑇 ∨ 𝑉))) → 𝑁 ≤ (𝐹 ∨ 𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  lecple 17335  joincjn 18385  meetcmee 18386  Latclat 18505  Atomscatm 40070  CvLatclc 40072  HLchlt 40157  LHypclh 40791

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18368  df-poset 18387  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-p1 18498  df-lat 18506  df-clat 18573  df-oposet 39983  df-ol 39985  df-oml 39986  df-covers 40073  df-ats 40074  df-atl 40105  df-cvlat 40129  df-hlat 40158  df-lhyp 40795

This theorem is used by:  cdleme22f2  41154  cdleme26fALTN  41169  cdleme26f  41170