Theorem 3dim3 39470

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 39469	. . 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 39369	. . . . . . . . . . 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 7405	. . . . . . . . 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 5122	. . . . . . . 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 7397	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	oveq1d 7405	. . . . . . . . . 10 ⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅))
17	16	breq2d 5122	. . . . . . . . 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 5113	. . . . . . . 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 3591	. . . . . 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 39370	. . . . . . . . . . . 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 39364	. . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
34	33, 3	atbase 39289	. . . . . . . . . . . . . . 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 39367	. . . . . . . . . . . . . . 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 18417	. . . . . . . . . . . 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 2765	. . . . . . . . 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 5122	. . . . . . . 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 39461	. . . . . . . . 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 5113	. . . . . . . . . 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 3591	. . . . . . . 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 39464	. . . . . . . . 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 3013	. . . 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 3194	. 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 2926  ∃wrex 3054   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  Latclat 18397  Atomscatm 39263  HLchlt 39350

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351

This theorem is referenced by:  lvolex3N  39539  dalem18  39682  dvh4dimat  41439