Theorem cdleme22b 40446

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 2731	. . . . . . 7 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
8	5, 6, 7	llni2 39617	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ 𝑆 ≠ 𝑇) → (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾))
9	1, 2, 3, 4, 8	syl31anc 1375	. . . . 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 39619	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → ¬ (𝑆 ∨ 𝑇) ∈ 𝐴)
11	1, 9, 10	syl2anc 584	. . . 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 2731	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
13	12, 7	llnn0 39621	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∨ 𝑇) ∈ (LLines‘𝐾)) → (𝑆 ∨ 𝑇) ≠ (0.‘𝐾))
14	1, 9, 13	syl2anc 584	. . . 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 511	. . 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 2929	. . . . 5 ⊢ ((𝑆 ∨ 𝑇) ≠ (0.‘𝐾) ↔ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾))
17	16	anbi2i 623	. . . 4 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)) ↔ (¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
18		pm4.56 990	. . . 4 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ ¬ (𝑆 ∨ 𝑇) = (0.‘𝐾)) ↔ ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
19	17, 18	bitri 275	. . 3 ⊢ ((¬ (𝑆 ∨ 𝑇) ∈ 𝐴 ∧ (𝑆 ∨ 𝑇) ≠ (0.‘𝐾)) ↔ ¬ ((𝑆 ∨ 𝑇) ∈ 𝐴 ∨ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
20	15, 19	sylib 218	. 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 39480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑇 ≤ (𝑇 ∨ 𝑉))
25	1, 3, 22, 24	syl3anc 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)))) → 𝑇 ≤ (𝑇 ∨ 𝑉))
26	1	hllatd 39469	. . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 6	atbase 39394	. . . . . . . . 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 39394	. . . . . . . . 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 39472	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑇 ∨ 𝑉) ∈ (Base‘𝐾))
33	1, 3, 22, 32	syl3anc 1373	. . . . . . . 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 18362	. . . . . . . 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 1374	. . . . . . 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 480	. . . . 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 480	. . . . . 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 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)) → 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 39472	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
44	1, 41, 42, 43	syl3anc 1373	. . . . . . . 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 18362	. . . . . . . 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 1374	. . . . . . 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 480	. . . . . 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 39472	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
50	1, 2, 3, 49	syl3anc 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) ∈ (Base‘𝐾))
51		cdleme22.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
52	27, 23, 51	latlem12 18378	. . . . . . 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 1374	. . . . . 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 480	. . . . 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 412	. . 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 39467	. . . . . . . 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 480	. . . . . 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 480	. . . . . 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 770	. . . . . 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 772	. . . . . 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 39400	. . . . . 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 1375	. . . . 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 420	. . . 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 5097	. . . . . . . . 9 ⊢ (((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) → ((𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) ↔ (𝑆 ∨ 𝑇) ≤ (0.‘𝐾)))
67	66	biimpa 476	. . . . . . . 8 ⊢ ((((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄)) = (0.‘𝐾) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑇 ∨ 𝑉) ∧ (𝑃 ∨ 𝑄))) → (𝑆 ∨ 𝑇) ≤ (0.‘𝐾))
68	27, 23, 12	ople0 39292	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) → ((𝑆 ∨ 𝑇) ≤ (0.‘𝐾) ↔ (𝑆 ∨ 𝑇) = (0.‘𝐾)))
69	58, 50, 68	syl2anc 584	. . . . . . . 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 244	. . . . . . 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 406	. . . . . 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 874	. . . . 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 420	. . . 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 39631	. . . . 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 1387	. . . 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 860	. . 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 198	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352  Basecbs 17126  lecple 17174  joincjn 18223  meetcmee 18224  0.cp0 18333  Latclat 18343  OPcops 39277  Atomscatm 39368  HLchlt 39455  LLinesclln 39596  LHypclh 40089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-proset 18206  df-poset 18225  df-plt 18240  df-lub 18256  df-glb 18257  df-join 18258  df-meet 18259  df-p0 18335  df-p1 18336  df-lat 18344  df-clat 18411  df-oposet 39281  df-ol 39283  df-oml 39284  df-covers 39371  df-ats 39372  df-atl 39403  df-cvlat 39427  df-hlat 39456  df-llines 39603

This theorem is referenced by:  cdleme22cN  40447  cdleme27a  40472