Theorem lubub

Description: The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011)

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

Proof

Step	Hyp	Ref	Expression
1		lublem.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lublem.l	⊢ ≤ = ( le ‘ 𝐾 )
3		lublem.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
4	1 2 3	lublem	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )
5	4	simpld	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ) → ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) )
6		breq1	⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ↔ 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) ) )
7	6	rspccva	⊢ ( ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) )
8	5 7	stoic3	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) )