Theorem cdleme22b 40965

Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 5th line on p. 115. Show that t ∨ v =/= p ∨ q and s ≤ p ∨ q implies ¬ t ≤ p ∨ q. (Contributed by NM, 2-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)

Assertion

Ref	Expression
cdleme22b	⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))

Proof of Theorem cdleme22b

Step	Hyp	Ref	Expression
1		simp1l 1211	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ HL)
2		simp1r1 1283	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ∈ 𝐴)
3		simp1r2 1284	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑇 ∈ 𝐴)
4		simp1r3 1285	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ≠ 𝑇)
5		cdleme22.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
6		cdleme22.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
7		eqid 2762	. . . . . . 7 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
8	5, 6, 7	llni2 40136	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑆 ≠ 𝑇) → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾))
9	1, 2, 3, 4, 8	syl31anc 1392	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾))
10	6, 7	llnneat 40138	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 ∨ 𝑇) ∈ 𝐴)
11	1, 9, 10	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ¬ (𝑆 ∨ 𝑇) ∈ 𝐴)
12		eqid 2762	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
13	12, 7	llnn0 40140	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → (𝑆 ∨ 𝑇) ≠ (0.‘𝐾))
14	1, 9, 13	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) ≠ (0.‘𝐾))
15	11, 14	jca 519	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)))
16		df-ne 2958	. . . . 5 ⊢ ((𝑆 ∨ 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾))
17	16	anbi2i 632	. . . 4 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
18		pm4.56 1002	. . . 4 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
19	17, 18	bitri 277	. . 3 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
20	15, 19	sylib 220	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
21		simp3r2 1296	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ≤ (𝑇 ∨ 𝑉))
22		simp3l 1215	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑉 ∈ 𝐴)
23		cdleme22.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
24	23, 5, 6	hlatlej1 39999	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑇 ≤ (𝑇 ∨ 𝑉))
25	1, 3, 22, 24	syl3anc 1390	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑇 ≤ (𝑇 ∨ 𝑉))
26	1	hllatd 39988	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ Lat)
27		eqid 2762	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 6	atbase 39913	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
29	2, 28	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ∈ (Base‘𝐾))
30	27, 6	atbase 39913	. . . . . . . . 9 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
31	3, 30	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑇 ∈ (Base‘𝐾))
32	27, 5, 6	hlatjcl 39991	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑇 ∨ 𝑉) ∈ (Base‘𝐾))
33	1, 3, 22, 32	syl3anc 1390	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑇 ∨ 𝑉) ∈ (Base‘𝐾))
34	27, 23, 5	latjle12 18482	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑉) ∈ (Base‘𝐾))) → ((𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑇 ≤ (𝑇 ∨ 𝑉)) ↔ (𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉)))
35	26, 29, 31, 33, 34	syl13anc 1391	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ((𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑇 ≤ (𝑇 ∨ 𝑉)) ↔ (𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉)))
36	21, 25, 35	mpbi2and 722	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉))
37	36	adantr 484	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉))
38		simp3r3 1297	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑆 ≤ (𝑃 ∨ 𝑄))
39	38	adantr 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ≤ (𝑃 ∨ 𝑄))
40		simpr 488	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≤ (𝑃 ∨ 𝑄))
41		simp21 1220	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑃 ∈ 𝐴)
42		simp22 1221	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝑄 ∈ 𝐴)
43	27, 5, 6	hlatjcl 39991	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
44	1, 41, 42, 43	syl3anc 1390	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
45	27, 23, 5	latjle12 18482	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄)))
46	26, 29, 31, 44, 45	syl13anc 1391	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ((𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄)))
47	46	adantr 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → ((𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄)))
48	39, 40, 47	mpbi2and 722	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄))
49	27, 5, 6	hlatjcl 39991	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
50	1, 2, 3, 49	syl3anc 1390	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
51		cdleme22.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
52	27, 23, 51	latlem12 18498	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑇 ∨ 𝑉) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉) ∧ (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))))
53	26, 50, 33, 44, 52	syl13anc 1391	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (((𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉) ∧ (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))))
54	53	adantr 484	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (((𝑆 ∨ 𝑇) ≤ (𝑇 ∨ 𝑉) ∧ (𝑆 ∨ 𝑇) ≤ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))))
55	37, 48, 54	mpbi2and 722	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))
56	55	ex 416	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑇 ≤ (𝑃 ∨ 𝑄) → (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))))
57		hlop 39986	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
58	1, 57	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ OP)
59	58	adantr 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → 𝐾 ∈ OP)
60	50	adantr 484	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
61		simprl 780	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
62		simprr 782	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))
63	27, 23, 12, 6	leat3 39919	. . . . . 6 ⊢ (((𝐾 ∈ OP ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
64	59, 60, 61, 62, 63	syl31anc 1392	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
65	64	exp32 424	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 → ((𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))))
66		breq2 5104	. . . . . . . . 9 ⊢ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) → ((𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ (0.‘𝐾)))
67	66	biimpa 480	. . . . . . . 8 ⊢ ((((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))) → (𝑆 ∨ 𝑇) ≤ (0.‘𝐾))
68	27, 23, 12	ople0 39811	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ≤ (0.‘𝐾) ↔ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
69	58, 50, 68	syl2anc 593	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ((𝑆 ∨ 𝑇) ≤ (0.‘𝐾) ↔ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
70	67, 69	imbitrid 246	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ((((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))) → (𝑆 ∨ 𝑇) = (0.‘𝐾)))
71	70	imp 410	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → (𝑆 ∨ 𝑇) = (0.‘𝐾))
72	71	olcd 885	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) ∧ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)))) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
73	72	exp32 424	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) → ((𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))))
74		simp3r1 1295	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄))
75	5, 51, 12, 6	2atmat0 40150	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄))) → (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∨ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾)))
76	1, 3, 22, 41, 42, 74, 75	syl33anc 1404	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∨ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾)))
77	65, 73, 76	mpjaod 871	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ((𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾))))
78	56, 77	syld 47	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → (𝑇 ≤ (𝑃 ∨ 𝑄) → ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾))))
79	20, 78	mtod 200	1 ⊢ (((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ≠ 𝑇)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑉 ∈ 𝐴 ∧ ((𝑇 ∨ 𝑉) ≠ (𝑃 ∨ 𝑄) ∧ 𝑆 ≤ (𝑇 ∨ 𝑉) ∧ 𝑆 ≤ (𝑃 ∨ 𝑄)))) → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  lecple 17293  joincjn 18343  meetcmee 18344  0.cp0 18453  Latclat 18463  OPcops 39796  Atomscatm 39887  HLchlt 39974  LLinesclln 40115  LHypclh 40608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-proset 18326  df-poset 18345  df-plt 18360  df-lub 18376  df-glb 18377  df-join 18378  df-meet 18379  df-p0 18455  df-p1 18456  df-lat 18464  df-clat 18531  df-oposet 39800  df-ol 39802  df-oml 39803  df-covers 39890  df-ats 39891  df-atl 39922  df-cvlat 39946  df-hlat 39975  df-llines 40122

This theorem is referenced by:  cdleme22cN  40966  cdleme27a  40991