Theorem cdleme22b 37978

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 1198	. . . . 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 1270	. . . . . 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 1271	. . . . . 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 1272	. . . . . 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 2738	. . . . . . 7 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
8	5, 6, 7	llni2 37149	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑆 ≠ 𝑇) → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾))
9	1, 2, 3, 4, 8	syl31anc 1374	. . . . 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 37151	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 ∨ 𝑇) ∈ 𝐴)
11	1, 9, 10	syl2anc 587	. . . 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 2738	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
13	12, 7	llnn0 37153	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → (𝑆 ∨ 𝑇) ≠ (0.‘𝐾))
14	1, 9, 13	syl2anc 587	. . . 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 515	. . 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 2935	. . . . 5 ⊢ ((𝑆 ∨ 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾))
17	16	anbi2i 626	. . . 4 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
18		pm4.56 988	. . . 4 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
19	17, 18	bitri 278	. . 3 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
20	15, 19	sylib 221	. 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 1283	. . . . . . 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 1202	. . . . . . . 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 37012	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑇 ≤ (𝑇 ∨ 𝑉))
25	1, 3, 22, 24	syl3anc 1372	. . . . . . 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 37001	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 6	atbase 36926	. . . . . . . . 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 36926	. . . . . . . . 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 37004	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑇 ∨ 𝑉) ∈ (Base‘𝐾))
33	1, 3, 22, 32	syl3anc 1372	. . . . . . . 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 17788	. . . . . . . 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 1373	. . . . . . 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 712	. . . . . 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 1284	. . . . . . 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 1207	. . . . . . . . 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 1208	. . . . . . . . 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 37004	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
44	1, 41, 42, 43	syl3anc 1372	. . . . . . . 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 17788	. . . . . . . 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 1373	. . . . . . 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 712	. . . . 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 37004	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
50	1, 2, 3, 49	syl3anc 1372	. . . . . . 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 17804	. . . . . . 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 1373	. . . . . 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 712	. . . 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 36999	. . . . . . . 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 771	. . . . . 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 773	. . . . . 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 36932	. . . . . 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 1374	. . . . 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 5034	. . . . . . . . 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 36824	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ≤ (0.‘𝐾) ↔ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
69	58, 50, 68	syl2anc 587	. . . . . . . 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	syl5ib 247	. . . . . . 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 873	. . . . 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 1282	. . . . 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 37163	. . . . 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 1386	. . . 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 859	. . 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 201	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 209   ∧ wa 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2934   class class class wbr 5030  ‘cfv 6339  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  meetcmee 17671  0.cp0 17763  Latclat 17771  OPcops 36809  Atomscatm 36900  HLchlt 36987  LLinesclln 37128  LHypclh 37621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135

This theorem is referenced by:  cdleme22cN  37979  cdleme27a  38004