Theorem chlub

Description: Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chlub	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		chsh	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		chsh	⊢ ( 𝐵 ∈ Cℋ → 𝐵 ∈ Sℋ )
3		shlub	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐶 ) )
4	2 3	syl3an2	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐶 ) )
5	1 4	syl3an1	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) ↔ ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐶 ) )