Theorem latpos

Description: A lattice is a poset. (Contributed by NM, 17-Sep-2011)

	Ref	Expression
Assertion	latpos	⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )

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	islat	⊢ ( 𝐾 ∈ Lat ↔ ( 𝐾 ∈ Poset ∧ ( dom ( join ‘ 𝐾 ) = ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ∧ dom ( meet ‘ 𝐾 ) = ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) )
5	4	simplbi	⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )