Theorem shsva

Description: Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	shsva	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to C{+}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢C{+}_{ℎ}D=C{+}_{ℎ}D$$
2		rspceov	$$⊢\left(C\in A\wedge D\in B\wedge C{+}_{ℎ}D=C{+}_{ℎ}D\right)\to \exists x\in A\exists y\in BC{+}_{ℎ}D=x{+}_{ℎ}y$$
3	1 2	mp3an3	$$⊢\left(C\in A\wedge D\in B\right)\to \exists x\in A\exists y\in BC{+}_{ℎ}D=x{+}_{ℎ}y$$
4		shsel	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(C{+}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)\leftrightarrow \exists x\in A\exists y\in BC{+}_{ℎ}D=x{+}_{ℎ}y\right)$$
5	3 4	syl5ibr	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(\left(C\in A\wedge D\in B\right)\to C{+}_{ℎ}D\in \left(A{+}_{\mathscr{H}}B\right)\right)$$