Theorem shseli

Description: Membership in subspace sum. (Contributed by NM, 4-May-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shscl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shscl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shseli	$$⊢C\in \left(A{+}_{\mathscr{H}}B\right)\leftrightarrow \exists x\in A\exists y\in BC=x{+}_{ℎ}y$$

Proof

Step	Hyp	Ref	Expression
1		shscl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shscl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shsel	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to \left(C\in \left(A{+}_{\mathscr{H}}B\right)\leftrightarrow \exists x\in A\exists y\in BC=x{+}_{ℎ}y\right)$$
4	1 2 3	mp2an	$$⊢C\in \left(A{+}_{\mathscr{H}}B\right)\leftrightarrow \exists x\in A\exists y\in BC=x{+}_{ℎ}y$$