Theorem 3dim3 39451

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 39450	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)))
5	4	3adant3r1 1183	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑣 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣)))
6		simpl2l 1227	. . . . . 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 1202	. . . . . . . 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 1198	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝐾 ∈ HL)
9		simp1r2 1271	. . . . . . . . . . 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 39350	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
11	8, 9, 10	syl2anc 584	. . . . . . . . . 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 7368	. . . . . . . . 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 5107	. . . . . . . 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 325	. . . . . . 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 7360	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	oveq1d 7368	. . . . . . . . . 10 ⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅))
17	16	breq2d 5107	. . . . . . . . 9 ⊢ (𝑃 = 𝑄 → (𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
18	17	notbid 318	. . . . . . . 8 ⊢ (𝑃 = 𝑄 → (¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅)))
19	18	biimparc 479	. . . . . . 7 ⊢ ((¬ 𝑣 ≤ ((𝑄 ∨ 𝑄) ∨ 𝑅) ∧ 𝑃 = 𝑄) → ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
20	14, 19	sylan 580	. . . . . 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 5098	. . . . . . . 8 ⊢ (𝑠 = 𝑣 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
22	21	notbid 318	. . . . . . 7 ⊢ (𝑠 = 𝑣 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
23	22	rspcev 3579	. . . . . 6 ⊢ ((𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
24	6, 20, 23	syl2anc 584	. . . . 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 1200	. . . . . . . 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 726	. . . . . . 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 726	. . . . . . . 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 39351	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
29	28	3ad2ant1 1133	. . . . . . . . . . 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 726	. . . . . . . . . 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 39345	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝐾 ∈ Lat)
32		simp1r1 1270	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (¬ 𝑣 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑤 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑣))) → 𝑃 ∈ 𝐴)
33		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
34	33, 3	atbase 39270	. . . . . . . . . . . . . . 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 1272	. . . . . . . . . . . . . . 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 39348	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
38	8, 9, 36, 37	syl3anc 1373	. . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 18375	. . . . . . . . . . . 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 476	. . . . . . . . . 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 2764	. . . . . . . . 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 5107	. . . . . . . 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 325	. . . . . . 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 584	. . . . . 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 1228	. . . . . . . . 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 726	. . . . . . . 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 1128	. . . . . . . . . 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 730	. . . . . . . . 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 511	. . . . . . . . . 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 730	. . . . . . . . 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 1230	. . . . . . . . . 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 726	. . . . . . . . 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 768	. . . . . . . . 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 484	. . . . . . . . 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 39442	. . . . . . . . 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 1384	. . . . . . . 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 5098	. . . . . . . . . 10 ⊢ (𝑠 = 𝑤 → (𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
61	60	notbid 318	. . . . . . . . 9 ⊢ (𝑠 = 𝑤 → (¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
62	61	rspcev 3579	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝐴 ∧ ¬ 𝑤 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
63	49, 59, 62	syl2anc 584	. . . . . . 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 1227	. . . . . . . . 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 726	. . . . . . . 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 730	. . . . . . . . 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 730	. . . . . . . . 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 1229	. . . . . . . . . 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 726	. . . . . . . . 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 768	. . . . . . . . 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 484	. . . . . . . . 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 39445	. . . . . . . . 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 1384	. . . . . . . 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 584	. . . . . . 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 812	. . . . . 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 812	. . . . 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 3012	. . . 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 1119	. . 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 3185	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  lecple 17186  joincjn 18235  Latclat 18355  Atomscatm 39244  HLchlt 39331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-proset 18218  df-poset 18237  df-plt 18252  df-lub 18268  df-glb 18269  df-join 18270  df-meet 18271  df-p0 18347  df-p1 18348  df-lat 18356  df-clat 18423  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332

This theorem is referenced by:  lvolex3N  39520  dalem18  39663  dvh4dimat  41420