Theorem lublem

Description: Lemma for the least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011)

	Ref	Expression
Hypotheses	lublem.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	lublem.l	⊢ ≤ = ( le ‘ 𝐾 )
	lublem.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
Assertion	lublem	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lublem.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lublem.l	⊢ ≤ = ( le ‘ 𝐾 )
3		lublem.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
4		simpl	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → 𝐾 ∈ CLat )
5	1 3	clatlubcl2	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ∈ dom 𝑈 )
6	1 2 3 4 5	lubprop	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )