Theorem spansnscl

Description: The subspace sum of a closed subspace and a one-dimensional subspace is closed. (Contributed by NM, 17-Dec-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansnscl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathscr{H}\right)\to A{+}_{\mathscr{H}}\mathrm{span}\left(\left\{B\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$

Proof

Step	Hyp	Ref	Expression
1		spansnj	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathscr{H}\right)\to A{+}_{\mathscr{H}}\mathrm{span}\left(\left\{B\right\}\right)=A{\vee }_{\mathscr{H}}\mathrm{span}\left(\left\{B\right\}\right)$$
2		spansnch	$$⊢B\in \mathscr{H}\to \mathrm{span}\left(\left\{B\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
3		chjcl	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge \mathrm{span}\left(\left\{B\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}\right)\to A{\vee }_{\mathscr{H}}\mathrm{span}\left(\left\{B\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
4	2 3	sylan2	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathscr{H}\right)\to A{\vee }_{\mathscr{H}}\mathrm{span}\left(\left\{B\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$
5	1 4	eqeltrd	$$⊢\left(A\in {\mathbf{C}}_{\mathscr{H}}\wedge B\in \mathscr{H}\right)\to A{+}_{\mathscr{H}}\mathrm{span}\left(\left\{B\right\}\right)\in {\mathbf{C}}_{\mathscr{H}}$$