Theorem shsvai

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

	Ref	Expression
Hypotheses	shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shsvai	$$⊢\left(C\in A\wedge D\in B\right)\to C{+}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shsva	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to C{+}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$
4	1 2 3	mp2an	$$⊢\left(C\in A\wedge D\in B\right)\to C{+}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)$$