Theorem shsvai

Description: Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shsvai	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 +ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3		shsva	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 +ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) ) )
4	1 2 3	mp2an	⊢ ( ( 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ) → ( 𝐶 +ℎ 𝐷 ) ∈ ( 𝐴 +ℋ 𝐵 ) )