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Definition df-lat 16812
Description: Define the class of all lattices. A lattice is a poset in which the join and meet of any two elements always exists. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Assertion
Ref Expression
df-lat Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}

Detailed syntax breakdown of Definition df-lat
StepHypRef Expression
1 clat 16811 . 2 class Lat
2 vp . . . . . . . 8 setvar 𝑝
32cv 1473 . . . . . . 7 class 𝑝
4 cjn 16710 . . . . . . 7 class join
53, 4cfv 5787 . . . . . 6 class (join‘𝑝)
65cdm 5025 . . . . 5 class dom (join‘𝑝)
7 cbs 15638 . . . . . . 7 class Base
83, 7cfv 5787 . . . . . 6 class (Base‘𝑝)
98, 8cxp 5023 . . . . 5 class ((Base‘𝑝) × (Base‘𝑝))
106, 9wceq 1474 . . . 4 wff dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝))
11 cmee 16711 . . . . . . 7 class meet
123, 11cfv 5787 . . . . . 6 class (meet‘𝑝)
1312cdm 5025 . . . . 5 class dom (meet‘𝑝)
1413, 9wceq 1474 . . . 4 wff dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝))
1510, 14wa 382 . . 3 wff (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))
16 cpo 16706 . . 3 class Poset
1715, 2, 16crab 2896 . 2 class {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
181, 17wceq 1474 1 wff Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
Colors of variables: wff setvar class
This definition is referenced by:  islat  16813
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