Theorem clatglbcl

Description: Any subset of the base set has a GLB in a complete lattice. (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	clatglbcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	clatglbcl.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
Assertion	clatglbcl	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		clatglbcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		clatglbcl.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
3		eqid	⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
4	1 3 2	clatlem	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ( ( lub ‘ 𝐾 ) ‘ 𝑆 ) ∈ 𝐵 ∧ ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 ) )
5	4	simprd	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝐺 ‘ 𝑆 ) ∈ 𝐵 )