Theorem 3dim3 40090

Description: Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)

Hypotheses

Ref	Expression
3dim0.j	⊢ ∨ = (join‘𝐾)
3dim0.l	⊢ ≤ = (le‘𝐾)
3dim0.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
3dim3	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ≤ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠

Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem 3dim3

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3dim0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
2		3dim0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		3dim0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 2, 3	3dim2 40089	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)))
5	4	3adant3r1 1196	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)))
6		simpl2l 1240	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → 𝑣 ∈ 𝐴)
7		simp3l 1215	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ¬ 𝑣 ≤ (𝑄 ∨ 𝑅))
8		simp1l 1211	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝐾 ∈ HL)
9		simp1r2 1284	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑄 ∈ 𝐴)
10	1, 3	hlatjidm 39990	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
11	8, 9, 10	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → (𝑄 ∨ 𝑄) = 𝑄)
12	11	oveq1d 7411	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ((𝑄 ∨ 𝑄) ∨ 𝑅) = (𝑄 ∨ 𝑅))
13	12	breq2d 5112	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → (𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ (𝑄 ∨ 𝑅)))
14	7, 13	mtbird 327	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅))
15		oveq1 7403	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	oveq1d 7411	. . . . . . . . . 10 ⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅))
17	16	breq2d 5112	. . . . . . . . 9 ⊢ (𝑃 = 𝑄 → (𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
18	17	notbid 320	. . . . . . . 8 ⊢ (𝑃 = 𝑄 → (¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
19	18	biimparc 483	. . . . . . 7 ⊢ ((¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅) ∧ 𝑃 = 𝑄) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
20	14, 19	sylan 589	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
21		breq1 5103	. . . . . . . 8 ⊢ (𝑠 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
22	21	notbid 320	. . . . . . 7 ⊢ (𝑠 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
23	22	rspcev 3581	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
24	6, 20, 23	syl2anc 593	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 = 𝑄) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
25		simp2l 1213	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑣 ∈ 𝐴)
26	25	ad2antrr 736	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → 𝑣 ∈ 𝐴)
27	7	ad2antrr 736	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ¬ 𝑣 ≤ (𝑄 ∨ 𝑅))
28	1, 3	hlatjass 39991	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
29	28	3ad2ant1 1146	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
30	29	ad2antrr 736	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
31	8	hllatd 39985	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝐾 ∈ Lat)
32		simp1r1 1283	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑃 ∈ 𝐴)
33		eqid 2762	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
34	33, 3	atbase 39910	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
35	32, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑃 ∈ (Base‘𝐾))
36		simp1r3 1285	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑅 ∈ 𝐴)
37	33, 1, 3	hlatjcl 39988	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
38	8, 9, 36, 37	syl3anc 1390	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
39	31, 35, 38	3jca 1141	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → (𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)))
40	39	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)))
41	33, 2, 1	latleeqj1 18483	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ 𝑅)))
42	40, 41	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ 𝑅)))
43	42	biimpa 480	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ 𝑅))
44	30, 43	eqtrd 2797	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑄 ∨ 𝑅))
45	44	breq2d 5112	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → (𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ (𝑄 ∨ 𝑅)))
46	27, 45	mtbird 327	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
47	26, 46, 23	syl2anc 593	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
48		simpl2r 1241	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → 𝑤 ∈ 𝐴)
49	48	ad2antrr 736	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → 𝑤 ∈ 𝐴)
50	8, 32, 9	3jca 1141	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
51	50	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
52	36, 25	jca 519	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → (𝑅 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴))
53	52	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → (𝑅 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴))
54		simpl3r 1243	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))
55	54	ad2antrr 736	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))
56		simplr 778	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
57		simpr 488	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))
58	1, 2, 3	3dimlem3a 40081	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
59	51, 53, 55, 56, 57, 58	syl113anc 1401	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
60		breq1 5103	. . . . . . . . . 10 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
61	60	notbid 320	. . . . . . . . 9 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
62	61	rspcev 3581	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
63	49, 59, 62	syl2anc 593	. . . . . . 7 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
64		simpl2l 1240	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → 𝑣 ∈ 𝐴)
65	64	ad2antrr 736	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → 𝑣 ∈ 𝐴)
66	50	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
67	52	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → (𝑅 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴))
68		simpl3l 1242	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑣 ≤ (𝑄 ∨ 𝑅))
69	68	ad2antrr 736	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑣 ≤ (𝑄 ∨ 𝑅))
70		simplr 778	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
71		simpr 488	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))
72	1, 2, 3	3dimlem4a 40084	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
73	66, 67, 69, 70, 71, 72	syl113anc 1401	. . . . . . . 8 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
74	65, 73, 23	syl2anc 593	. . . . . . 7 ⊢ ((((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
75	63, 74	pm2.61dan 822	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
76	47, 75	pm2.61dan 822	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) ∧ 𝑃 ≠ 𝑄) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
77	24, 76	pm2.61dane 3044	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
78	77	3exp 1132	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → ((¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))))
79	78	rexlimdvv 3218	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
80	5, 79	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  lecple 17293  joincjn 18343  Latclat 18463  Atomscatm 39884  HLchlt 39971

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 39797  df-ol 39799  df-oml 39800  df-covers 39887  df-ats 39888  df-atl 39919  df-cvlat 39943  df-hlat 39972

This theorem is referenced by:  lvolex3N  40159  dalem18  40302  dvh4dimat  42059