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Definition df-cvlat 34089
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 34032 . 2 class CvLat
2 va . . . . . . . . . . 11 setvar 𝑎
32cv 1479 . . . . . . . . . 10 class 𝑎
4 vc . . . . . . . . . . 11 setvar 𝑐
54cv 1479 . . . . . . . . . 10 class 𝑐
6 vk . . . . . . . . . . . 12 setvar 𝑘
76cv 1479 . . . . . . . . . . 11 class 𝑘
8 cple 15869 . . . . . . . . . . 11 class le
97, 8cfv 5847 . . . . . . . . . 10 class (le‘𝑘)
103, 5, 9wbr 4613 . . . . . . . . 9 wff 𝑎(le‘𝑘)𝑐
1110wn 3 . . . . . . . 8 wff ¬ 𝑎(le‘𝑘)𝑐
12 vb . . . . . . . . . . 11 setvar 𝑏
1312cv 1479 . . . . . . . . . 10 class 𝑏
14 cjn 16865 . . . . . . . . . . 11 class join
157, 14cfv 5847 . . . . . . . . . 10 class (join‘𝑘)
165, 13, 15co 6604 . . . . . . . . 9 class (𝑐(join‘𝑘)𝑏)
173, 16, 9wbr 4613 . . . . . . . 8 wff 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)
1811, 17wa 384 . . . . . . 7 wff 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏))
195, 3, 15co 6604 . . . . . . . 8 class (𝑐(join‘𝑘)𝑎)
2013, 19, 9wbr 4613 . . . . . . 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 15781 . . . . . . 7 class Base
237, 22cfv 5847 . . . . . 6 class (Base‘𝑘)
2421, 4, 23wral 2907 . . . . 5 wff 𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))
25 catm 34030 . . . . . 6 class Atoms
267, 25cfv 5847 . . . . 5 class (Atoms‘𝑘)
2724, 12, 26wral 2907 . . . 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 2907 . . 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 34031 . . 3 class AtLat
3028, 6, 29crab 2911 . 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 1480 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  34090
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