Definition df-cvlat 39278

Description: Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)

Assertion

Ref	Expression
df-cvlat	⊢ CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}

Distinct variable group:   𝑘,𝑐,𝑎,𝑏

Detailed syntax breakdown of Definition df-cvlat

Step	Hyp	Ref	Expression
1		clc 39221	. 2 class CvLat
2		va	. . . . . . . . . . 11 setvar 𝑎
3	2	cv 1536	. . . . . . . . . 10 class 𝑎
4		vc	. . . . . . . . . . 11 setvar 𝑐
5	4	cv 1536	. . . . . . . . . 10 class 𝑐
6		vk	. . . . . . . . . . . 12 setvar 𝑘
7	6	cv 1536	. . . . . . . . . . 11 class 𝑘
8		cple 17318	. . . . . . . . . . 11 class le
9	7, 8	cfv 6573	. . . . . . . . . 10 class (le‘𝑘)
10	3, 5, 9	wbr 5166	. . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
11	10	wn 3	. . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12		vb	. . . . . . . . . . 11 setvar 𝑏
13	12	cv 1536	. . . . . . . . . 10 class 𝑏
14		cjn 18381	. . . . . . . . . . 11 class join
15	7, 14	cfv 6573	. . . . . . . . . 10 class (join‘𝑘)
16	5, 13, 15	co 7448	. . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
17	3, 16, 9	wbr 5166	. . . . . . . 8 wff 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)
18	11, 17	wa 395	. . . . . . 7 wff (¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏))
19	5, 3, 15	co 7448	. . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
20	13, 19, 9	wbr 5166	. . . . . . 7 wff 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎)
21	18, 20	wi 4	. . . . . 6 wff ((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
22		cbs 17258	. . . . . . 7 class Base
23	7, 22	cfv 6573	. . . . . 6 class (Base‘𝑘)
24	21, 4, 23	wral 3067	. . . . 5 wff ∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
25		catm 39219	. . . . . 6 class Atoms
26	7, 25	cfv 6573	. . . . 5 class (Atoms‘𝑘)
27	24, 12, 26	wral 3067	. . . 4 wff ∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
28	27, 2, 26	wral 3067	. . 3 wff ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
29		cal 39220	. . 3 class AtLat
30	28, 6, 29	crab 3443	. 2 class {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}
31	1, 30	wceq 1537	1 wff CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}

This definition is referenced by:  iscvlat  39279