Theorem 3dim3 39463

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 39462	. . 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 39362	. . . . . . . . . . 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 7402	. . . . . . . . 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 5119	. . . . . . . 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 7394	. . . . . . . . . . 11 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	oveq1d 7402	. . . . . . . . . 10 ⊢ (𝑃 = 𝑄 → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑄) ∨ 𝑅))
17	16	breq2d 5119	. . . . . . . . 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 5110	. . . . . . . 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 3588	. . . . . 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 39363	. . . . . . . . . . . 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 39357	. . . . . . . . . . . . . 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 39282	. . . . . . . . . . . . . . 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 39360	. . . . . . . . . . . . . . 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 18410	. . . . . . . . . . . 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 5119	. . . . . . . 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 39454	. . . . . . . . 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 5110	. . . . . . . . . 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 3588	. . . . . . . 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 39457	. . . . . . . . 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 3193	. 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 5107  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  lecple 17227  joincjn 18272  Latclat 18390  Atomscatm 39256  HLchlt 39343

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344

This theorem is referenced by:  lvolex3N  39532  dalem18  39675  dvh4dimat  41432