Definition df-cvlat 35097

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 35040	. 2 class CvLat
2		va	. . . . . . . . . . 11 setvar 𝑎
3	2	cv 1636	. . . . . . . . . 10 class 𝑎
4		vc	. . . . . . . . . . 11 setvar 𝑐
5	4	cv 1636	. . . . . . . . . 10 class 𝑐
6		vk	. . . . . . . . . . . 12 setvar 𝑘
7	6	cv 1636	. . . . . . . . . . 11 class 𝑘
8		cple 16156	. . . . . . . . . . 11 class le
9	7, 8	cfv 6097	. . . . . . . . . 10 class (le‘𝑘)
10	3, 5, 9	wbr 4844	. . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
11	10	wn 3	. . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12		vb	. . . . . . . . . . 11 setvar 𝑏
13	12	cv 1636	. . . . . . . . . 10 class 𝑏
14		cjn 17145	. . . . . . . . . . 11 class join
15	7, 14	cfv 6097	. . . . . . . . . 10 class (join‘𝑘)
16	5, 13, 15	co 6870	. . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
17	3, 16, 9	wbr 4844	. . . . . . . 8 wff 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)
18	11, 17	wa 384	. . . . . . 7 wff (¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏))
19	5, 3, 15	co 6870	. . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
20	13, 19, 9	wbr 4844	. . . . . . 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 16064	. . . . . . 7 class Base
23	7, 22	cfv 6097	. . . . . 6 class (Base‘𝑘)
24	21, 4, 23	wral 3096	. . . . 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 35038	. . . . . 6 class Atoms
26	7, 25	cfv 6097	. . . . 5 class (Atoms‘𝑘)
27	24, 12, 26	wral 3096	. . . 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 3096	. . 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 35039	. . 3 class AtLat
30	28, 6, 29	crab 3100	. 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 1637	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  35098