Theorem latmcl

Description: Closure of meet operation in a lattice. ( incom analog.) (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	latmcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latmcl.m	⊢ ∧ = ( meet ‘ 𝐾 )
Assertion	latmcl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		latmcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latmcl.m	⊢ ∧ = ( meet ‘ 𝐾 )
3		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
4	1 3 2	latlem	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( join ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 ) )
5	4	simprd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )