Metamath Proof Explorer


Theorem dlatl

Description: A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015)

Ref Expression
Assertion dlatl ( 𝐾 ∈ DLat → 𝐾 ∈ Lat )

Proof

Step Hyp Ref Expression
1 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
2 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
3 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
4 1 2 3 isdlat ( 𝐾 ∈ DLat ↔ ( 𝐾 ∈ Lat ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) ( 𝑦 ( join ‘ 𝐾 ) 𝑧 ) ) = ( ( 𝑥 ( meet ‘ 𝐾 ) 𝑦 ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑧 ) ) ) )
5 4 simplbi ( 𝐾 ∈ DLat → 𝐾 ∈ Lat )