Definition df-cvlat 39431

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 39374	. 2 class CvLat
2		va	. . . . . . . . . . 11 setvar 𝑎
3	2	cv 1540	. . . . . . . . . 10 class 𝑎
4		vc	. . . . . . . . . . 11 setvar 𝑐
5	4	cv 1540	. . . . . . . . . 10 class 𝑐
6		vk	. . . . . . . . . . . 12 setvar 𝑘
7	6	cv 1540	. . . . . . . . . . 11 class 𝑘
8		cple 17168	. . . . . . . . . . 11 class le
9	7, 8	cfv 6481	. . . . . . . . . 10 class (le‘𝑘)
10	3, 5, 9	wbr 5089	. . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
11	10	wn 3	. . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12		vb	. . . . . . . . . . 11 setvar 𝑏
13	12	cv 1540	. . . . . . . . . 10 class 𝑏
14		cjn 18217	. . . . . . . . . . 11 class join
15	7, 14	cfv 6481	. . . . . . . . . 10 class (join‘𝑘)
16	5, 13, 15	co 7346	. . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
17	3, 16, 9	wbr 5089	. . . . . . . 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 7346	. . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
20	13, 19, 9	wbr 5089	. . . . . . 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 17120	. . . . . . 7 class Base
23	7, 22	cfv 6481	. . . . . 6 class (Base‘𝑘)
24	21, 4, 23	wral 3047	. . . . 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 39372	. . . . . 6 class Atoms
26	7, 25	cfv 6481	. . . . 5 class (Atoms‘𝑘)
27	24, 12, 26	wral 3047	. . . 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 3047	. . 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 39373	. . 3 class AtLat
30	28, 6, 29	crab 3395	. 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 1541	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  39432