Theorem 3dim3 40243

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 40242	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)))
5	4	3adant3r1 1201	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)))
6		simpl2l 1245	. . . . . 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 1220	. . . . . . . 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 1216	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝐾 ∈ HL)
9		simp1r2 1289	. . . . . . . . . . 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 40143	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
11	8, 9, 10	syl2anc 595	. . . . . . . . . 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 7425	. . . . . . . . 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 5121	. . . . . . . 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 328	. . . . . . 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 7417	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅))
17	16	breq2d 5121	. . . . . . . . 9 ⊢ (𝑃 = 𝑄 → (𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
18	17	notbid 321	. . . . . . . 8 ⊢ (𝑃 = 𝑄 → (¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
19	18	biimparc 484	. . . . . . 7 ⊢ ((¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅) ∧ 𝑃 = 𝑄) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
20	14, 19	sylan 591	. . . . . 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 5112	. . . . . . . 8 ⊢ (𝑠 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
22	21	notbid 321	. . . . . . 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 595	. . . . 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 1218	. . . . . . . 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 738	. . . . . . 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 738	. . . . . . . 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 40144	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
29	28	3ad2ant1 1151	. . . . . . . . . . 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 738	. . . . . . . . . 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 40138	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝐾 ∈ Lat)
32		simp1r1 1288	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑃 ∈ 𝐴)
33		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
34	33, 3	atbase 40063	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
35	32, 34	syl 18	. . . . . . . . . . . . . 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 1290	. . . . . . . . . . . . . . 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 40141	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
38	8, 9, 36, 37	syl3anc 1398	. . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 18502	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ (𝑃 ∨ (𝑄 ∨ 𝑅)) = (𝑄 ∨ 𝑅)))
42	40, 41	syl 18	. . . . . . . . . . 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 481	. . . . . . . . . 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 2798	. . . . . . . . 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 5121	. . . . . . . 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 328	. . . . . . 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 595	. . . . . 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 1246	. . . . . . . . 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 738	. . . . . . . 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 1146	. . . . . . . . . 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 742	. . . . . . . . 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 520	. . . . . . . . . 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 742	. . . . . . . . 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 1248	. . . . . . . . . 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 738	. . . . . . . . 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 780	. . . . . . . . 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 489	. . . . . . . . 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 40234	. . . . . . . . 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 1409	. . . . . . . 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 5112	. . . . . . . . . 10 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
61	60	notbid 321	. . . . . . . . 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 595	. . . . . . 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 1245	. . . . . . . . 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 738	. . . . . . . 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 742	. . . . . . . . 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 742	. . . . . . . . 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 1247	. . . . . . . . . 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 738	. . . . . . . . 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 780	. . . . . . . . 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 489	. . . . . . . . 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 40237	. . . . . . . . 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 1409	. . . . . . . 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 595	. . . . . . 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 824	. . . . . 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 824	. . . . 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 3045	. . . 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 1137	. . 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 3221	. 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 16	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312  joincjn 18362  Latclat 18482  Atomscatm 40037  HLchlt 40124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-p1 18475  df-lat 18483  df-clat 18550  df-oposet 39950  df-ol 39952  df-oml 39953  df-covers 40040  df-ats 40041  df-atl 40072  df-cvlat 40096  df-hlat 40125

This theorem is referenced by:  lvolex3N  40312  dalem18  40455  dvh4dimat  42212