Theorem clatlubcl

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

	Ref	Expression
Hypotheses	clatlubcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	clatlubcl.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
Assertion	clatlubcl	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		clatlubcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		clatlubcl.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
3		eqid	⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 )
4	1 2 3	clatlem	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 ∧ ( ( glb ‘ 𝐾 ) ‘ 𝑆 ) ∈ 𝐵 ) )
5	4	simpld	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 )