Theorem shub1

Description: Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shub1	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shsub1	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 ) )
2		shslej	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 +ℋ 𝐵 ) ⊆ ( 𝐴 ∨ℋ 𝐵 ) )
3	1 2	sstrd	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) )