Theorem cdleme15b 40269

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, (p ∨ s₁) ∧ (q ∨ s₁)=s₁. We represent s₁ with 𝐶. (Contributed by NM, 10-Oct-2012.)

Hypotheses

Ref	Expression
cdleme12.l	⊢ ≤ = (le‘𝐾)
cdleme12.j	⊢ ∨ = (join‘𝐾)
cdleme12.m	⊢ ∧ = (meet‘𝐾)
cdleme12.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme12.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme12.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme12.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme12.g	⊢ 𝐺 = ((𝑇 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑇) ∧ 𝑊)))
cdleme15.c	⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
cdleme15.x	⊢ 𝑋 = ((𝑃 ∨ 𝑇) ∧ 𝑊)

Assertion

Ref	Expression
cdleme15b	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝐶) ∧ (𝑄 ∨ 𝐶)) = 𝐶)

Proof of Theorem cdleme15b

Step	Hyp	Ref	Expression
1		cdleme15.c	. . . . . . 7 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
2	1	oveq2i 7398	. . . . . 6 ⊢ (𝑃 ∨ 𝐶) = (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))
3		simp11l 1285	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ HL)
4		simp12l 1287	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ∈ 𝐴)
5		simp21l 1291	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑆 ∈ 𝐴)
6		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		cdleme12.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
8		cdleme12.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatjcl 39360	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
10	3, 4, 5, 9	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
11		simp11r 1286	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑊 ∈ 𝐻)
12		cdleme12.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
13	6, 12	lhpbase 39992	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
14	11, 13	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑊 ∈ (Base‘𝐾))
15		cdleme12.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
16	15, 7, 8	hlatlej1 39368	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑆))
17	3, 4, 5, 16	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ≤ (𝑃 ∨ 𝑆))
18		cdleme12.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
19	6, 15, 7, 18, 8	atmod3i1 39858	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ 𝑆)) → (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)))
20	3, 4, 10, 14, 17, 19	syl131anc 1385	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)))
21	2, 20	eqtrid 2776	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∨ 𝐶) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)))
22	21	oveq1d 7402	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝐶) ∧ 𝑄) = (((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)) ∧ 𝑄))
23		hlol 39354	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
24	3, 23	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ OL)
25	3	hllatd 39357	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ Lat)
26	6, 8	atbase 39282	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
27	4, 26	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ∈ (Base‘𝐾))
28	6, 7	latjcl 18398	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑊) ∈ (Base‘𝐾))
29	25, 27, 14, 28	syl3anc 1373	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∨ 𝑊) ∈ (Base‘𝐾))
30		simp13l 1289	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑄 ∈ 𝐴)
31	6, 8	atbase 39282	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
32	30, 31	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑄 ∈ (Base‘𝐾))
33	6, 18	latmrot 39225	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ ((𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑊) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)) ∧ 𝑄) = ((𝑄 ∧ (𝑃 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑊)))
34	24, 10, 29, 32, 33	syl13anc 1374	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)) ∧ 𝑄) = ((𝑄 ∧ (𝑃 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑊)))
35		simp31 1210	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
36		simp23l 1295	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑃 ≠ 𝑄)
37	36	necomd 2980	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝑄 ≠ 𝑃)
38	15, 7, 8	hlatexch1 39389	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝑄)))
39	3, 30, 5, 4, 37, 38	syl131anc 1385	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑄 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝑄)))
40	35, 39	mtod 198	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
41		hlatl 39353	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
42	3, 41	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐾 ∈ AtLat)
43		eqid 2729	. . . . . . . . 9 ⊢ (0.‘𝐾) = (0.‘𝐾)
44	6, 15, 18, 43, 8	atnle 39310	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → (¬ 𝑄 ≤ (𝑃 ∨ 𝑆) ↔ (𝑄 ∧ (𝑃 ∨ 𝑆)) = (0.‘𝐾)))
45	42, 30, 10, 44	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (¬ 𝑄 ≤ (𝑃 ∨ 𝑆) ↔ (𝑄 ∧ (𝑃 ∨ 𝑆)) = (0.‘𝐾)))
46	40, 45	mpbid 232	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑄 ∧ (𝑃 ∨ 𝑆)) = (0.‘𝐾))
47	46	oveq1d 7402	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑄 ∧ (𝑃 ∨ 𝑆)) ∧ (𝑃 ∨ 𝑊)) = ((0.‘𝐾) ∧ (𝑃 ∨ 𝑊)))
48	6, 18, 43	olm02 39230	. . . . . 6 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑊) ∈ (Base‘𝐾)) → ((0.‘𝐾) ∧ (𝑃 ∨ 𝑊)) = (0.‘𝐾))
49	24, 29, 48	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((0.‘𝐾) ∧ (𝑃 ∨ 𝑊)) = (0.‘𝐾))
50	34, 47, 49	3eqtrrd 2769	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (0.‘𝐾) = (((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)) ∧ 𝑄))
51	22, 50	eqtr4d 2767	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝐶) ∧ 𝑄) = (0.‘𝐾))
52	51	oveq1d 7402	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (((𝑃 ∨ 𝐶) ∧ 𝑄) ∨ 𝐶) = ((0.‘𝐾) ∨ 𝐶))
53	6, 7, 18, 8, 12, 1	cdleme9b 40246	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻)) → 𝐶 ∈ (Base‘𝐾))
54	3, 4, 5, 11, 53	syl13anc 1374	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐶 ∈ (Base‘𝐾))
55	6, 7	latjcl 18398	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝐶) ∈ (Base‘𝐾))
56	25, 27, 54, 55	syl3anc 1373	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (𝑃 ∨ 𝐶) ∈ (Base‘𝐾))
57	6, 15, 7	latlej2 18408	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝐶 ≤ (𝑃 ∨ 𝐶))
58	25, 27, 54, 57	syl3anc 1373	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → 𝐶 ≤ (𝑃 ∨ 𝐶))
59	6, 15, 7, 18, 8	atmod2i2 39856	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ (𝑃 ∨ 𝐶) ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) ∧ 𝐶 ≤ (𝑃 ∨ 𝐶)) → (((𝑃 ∨ 𝐶) ∧ 𝑄) ∨ 𝐶) = ((𝑃 ∨ 𝐶) ∧ (𝑄 ∨ 𝐶)))
60	3, 30, 56, 54, 58, 59	syl131anc 1385	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → (((𝑃 ∨ 𝐶) ∧ 𝑄) ∨ 𝐶) = ((𝑃 ∨ 𝐶) ∧ (𝑄 ∨ 𝐶)))
61	6, 7, 43	olj02 39219	. . 3 ⊢ ((𝐾 ∈ OL ∧ 𝐶 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝐶) = 𝐶)
62	24, 54, 61	syl2anc 584	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((0.‘𝐾) ∨ 𝐶) = 𝐶)
63	52, 60, 62	3eqtr3d 2772	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊) ∧ (𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇)) ∧ (¬ 𝑆 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑈 ≤ (𝑆 ∨ 𝑇))) → ((𝑃 ∨ 𝐶) ∧ (𝑄 ∨ 𝐶)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  meetcmee 18273  0.cp0 18382  Latclat 18390  OLcol 39167  Atomscatm 39256  AtLatcal 39257  HLchlt 39343  LHypclh 39978

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982

This theorem is referenced by:  cdleme15c  40270