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 )