Theorem shsub1i

Description: Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shsub1i	⊢ 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3	1 2	shsel1i	⊢ ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐴 +ℋ 𝐵 ) )
4	3	ssriv	⊢ 𝐴 ⊆ ( 𝐴 +ℋ 𝐵 )