Theorem chseli

Description: Membership in subspace sum. (Contributed by NM, 19-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
	chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	chseli	$$⊢C\in \left(A{+}_{\mathscr{H}}B\right)\leftrightarrow \exists x\in A\exists y\in BC=x{+}_{ℎ}y$$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$$⊢A\in {\mathbf{C}}_{\mathscr{H}}$$
2		chjcl.2	$$⊢B\in {\mathbf{C}}_{\mathscr{H}}$$
3	1	chshii	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
4	2	chshii	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
5	3 4	shseli	$$⊢C\in \left(A{+}_{\mathscr{H}}B\right)\leftrightarrow \exists x\in A\exists y\in BC=x{+}_{ℎ}y$$