Theorem shub1i

Description: Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3	1 2	shsub1i	⊢ 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 )
4	1 2	shsleji	⊢ ( 𝐴 +ℋ 𝐵 ) ⊆ ( 𝐴 ∨ℋ 𝐵 )
5	3 4	sstri	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )