Definition df-cvlat 39033

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 38976	. 2 class CvLat
2		va	. . . . . . . . . . 11 setvar 𝑎
3	2	cv 1533	. . . . . . . . . 10 class 𝑎
4		vc	. . . . . . . . . . 11 setvar 𝑐
5	4	cv 1533	. . . . . . . . . 10 class 𝑐
6		vk	. . . . . . . . . . . 12 setvar 𝑘
7	6	cv 1533	. . . . . . . . . . 11 class 𝑘
8		cple 17268	. . . . . . . . . . 11 class le
9	7, 8	cfv 6546	. . . . . . . . . 10 class (le‘𝑘)
10	3, 5, 9	wbr 5145	. . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
11	10	wn 3	. . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12		vb	. . . . . . . . . . 11 setvar 𝑏
13	12	cv 1533	. . . . . . . . . 10 class 𝑏
14		cjn 18331	. . . . . . . . . . 11 class join
15	7, 14	cfv 6546	. . . . . . . . . 10 class (join‘𝑘)
16	5, 13, 15	co 7416	. . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
17	3, 16, 9	wbr 5145	. . . . . . . 8 wff 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)
18	11, 17	wa 394	. . . . . . 7 wff (¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏))
19	5, 3, 15	co 7416	. . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
20	13, 19, 9	wbr 5145	. . . . . . 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 17208	. . . . . . 7 class Base
23	7, 22	cfv 6546	. . . . . 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 38974	. . . . . 6 class Atoms
26	7, 25	cfv 6546	. . . . 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 38975	. . 3 class AtLat
30	28, 6, 29	crab 3419	. 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 1534	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  39034