Theorem cphnlm

Description: A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Assertion	cphnlm	$$⊢W\in \mathrm{CPreHil}\to W\in \mathrm{NrmMod}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
2		eqid	$$⊢{\cdot }_{𝑖}\left(W\right)={\cdot }_{𝑖}\left(W\right)$$
3		eqid	$$⊢\mathrm{norm}\left(W\right)=\mathrm{norm}\left(W\right)$$
4		eqid	$$⊢\mathrm{Scalar}\left(W\right)=\mathrm{Scalar}\left(W\right)$$
5		eqid	$$⊢{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}={\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}$$
6	1 2 3 4 5	iscph	$$⊢W\in \mathrm{CPreHil}\leftrightarrow \left(\left(W\in \mathrm{PreHil}\wedge W\in \mathrm{NrmMod}\wedge \mathrm{Scalar}\left(W\right)={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}\right)\wedge \surd \left[{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}\cap \left[0,\mathrm{+\infty }\right)\right]\subseteq {\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}\wedge \mathrm{norm}\left(W\right)=\left(x\in {\mathrm{Base}}_W⟼\sqrt{x{\cdot }_{𝑖}\left(W\right)x}\right)\right)$$
7	6	simp1bi	$$⊢W\in \mathrm{CPreHil}\to \left(W\in \mathrm{PreHil}\wedge W\in \mathrm{NrmMod}\wedge \mathrm{Scalar}\left(W\right)={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}{\mathrm{Base}}_{\mathrm{Scalar}\left(W\right)}\right)$$
8	7	simp2d	$$⊢W\in \mathrm{CPreHil}\to W\in \mathrm{NrmMod}$$