Definition df-cvlat 39286

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 39229	. 2 class CvLat
2		va	. . . . . . . . . . 11 setvar 𝑎
3	2	cv 1539	. . . . . . . . . 10 class 𝑎
4		vc	. . . . . . . . . . 11 setvar 𝑐
5	4	cv 1539	. . . . . . . . . 10 class 𝑐
6		vk	. . . . . . . . . . . 12 setvar 𝑘
7	6	cv 1539	. . . . . . . . . . 11 class 𝑘
8		cple 17276	. . . . . . . . . . 11 class le
9	7, 8	cfv 6530	. . . . . . . . . 10 class (le‘𝑘)
10	3, 5, 9	wbr 5119	. . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
11	10	wn 3	. . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12		vb	. . . . . . . . . . 11 setvar 𝑏
13	12	cv 1539	. . . . . . . . . 10 class 𝑏
14		cjn 18321	. . . . . . . . . . 11 class join
15	7, 14	cfv 6530	. . . . . . . . . 10 class (join‘𝑘)
16	5, 13, 15	co 7403	. . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
17	3, 16, 9	wbr 5119	. . . . . . . 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 7403	. . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
20	13, 19, 9	wbr 5119	. . . . . . 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 17226	. . . . . . 7 class Base
23	7, 22	cfv 6530	. . . . . 6 class (Base‘𝑘)
24	21, 4, 23	wral 3051	. . . . 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 39227	. . . . . 6 class Atoms
26	7, 25	cfv 6530	. . . . 5 class (Atoms‘𝑘)
27	24, 12, 26	wral 3051	. . . 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 3051	. . 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 39228	. . 3 class AtLat
30	28, 6, 29	crab 3415	. 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 1540	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  39287