Theorem shub2

Description: A subspace is a subset of its Hilbert lattice join with another. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shub2	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐵 ∨ℋ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		shub1	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) )
2		shjcom	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )
3	1 2	sseqtrd	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 𝐴 ⊆ ( 𝐵 ∨ℋ 𝐴 ) )