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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)𝑐𝑎(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
StepHypRef Expression
1 clc 38976 . 2 class CvLat
2 va . . . . . . . . . . 11 setvar 𝑎
32cv 1533 . . . . . . . . . 10 class 𝑎
4 vc . . . . . . . . . . 11 setvar 𝑐
54cv 1533 . . . . . . . . . 10 class 𝑐
6 vk . . . . . . . . . . . 12 setvar 𝑘
76cv 1533 . . . . . . . . . . 11 class 𝑘
8 cple 17268 . . . . . . . . . . 11 class le
97, 8cfv 6546 . . . . . . . . . 10 class (le‘𝑘)
103, 5, 9wbr 5145 . . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
1110wn 3 . . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12 vb . . . . . . . . . . 11 setvar 𝑏
1312cv 1533 . . . . . . . . . 10 class 𝑏
14 cjn 18331 . . . . . . . . . . 11 class join
157, 14cfv 6546 . . . . . . . . . 10 class (join‘𝑘)
165, 13, 15co 7416 . . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
173, 16, 9wbr 5145 . . . . . . . 8 wff 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)
1811, 17wa 394 . . . . . . 7 wff 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏))
195, 3, 15co 7416 . . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
2013, 19, 9wbr 5145 . . . . . . 7 wff 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎)
2118, 20wi 4 . . . . . 6 wff ((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
22 cbs 17208 . . . . . . 7 class Base
237, 22cfv 6546 . . . . . 6 class (Base‘𝑘)
2421, 4, 23wral 3051 . . . . 5 wff 𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
25 catm 38974 . . . . . 6 class Atoms
267, 25cfv 6546 . . . . 5 class (Atoms‘𝑘)
2724, 12, 26wral 3051 . . . 4 wff 𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
2827, 2, 26wral 3051 . . 3 wff 𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
29 cal 38975 . . 3 class AtLat
3028, 6, 29crab 3419 . 2 class {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}
311, 30wceq 1534 1 wff CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))}
Colors of variables: wff setvar class
This definition is referenced by:  iscvlat  39034
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