Theorem hhssbnOLD

Description: Obsolete version of cssbn : Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	hhssbnOLD.1	$$⊢W=⟨⟨{{+}_{ℎ}\upharpoonright }_{\left(H\times H\right)},{{\cdot }_{ℎ}\upharpoonright }_{\left(ℂ\times H\right)}⟩,{{norm}_{ℎ}\upharpoonright }_H⟩$$
	hhssbnOLD.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	hhssbnOLD	$$⊢W\in \mathrm{CBan}$$

Proof

Step	Hyp	Ref	Expression
1		hhssbnOLD.1	$$⊢W=⟨⟨{{+}_{ℎ}\upharpoonright }_{\left(H\times H\right)},{{\cdot }_{ℎ}\upharpoonright }_{\left(ℂ\times H\right)}⟩,{{norm}_{ℎ}\upharpoonright }_H⟩$$
2		hhssbnOLD.2	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
3	2	chshii	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}$$
4	1 3	hhssnv	$$⊢W\in \mathrm{NrmCVec}$$
5		eqid	$$⊢\mathrm{IndMet}\left(W\right)=\mathrm{IndMet}\left(W\right)$$
6	1 5 2	hhsscms	$$⊢\mathrm{IndMet}\left(W\right)\in \mathrm{CMet}\left(H\right)$$
7	1 3	hhssba	$$⊢H=\mathrm{BaseSet}\left(W\right)$$
8	7 5	iscbn	$$⊢W\in \mathrm{CBan}\leftrightarrow \left(W\in \mathrm{NrmCVec}\wedge \mathrm{IndMet}\left(W\right)\in \mathrm{CMet}\left(H\right)\right)$$
9	4 6 8	mpbir2an	$$⊢W\in \mathrm{CBan}$$