Theorem obsrcl

Description: Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015)

	Ref	Expression
Assertion	obsrcl	$$⊢B\in \mathrm{OBasis}\left(W\right)\to W\in \mathrm{PreHil}$$

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	$$⊢\mathrm{Scalar}\left(W\right)=\mathrm{Scalar}\left(W\right)$$
4		eqid	$$⊢1_{\mathrm{Scalar}\left(W\right)}=1_{\mathrm{Scalar}\left(W\right)}$$
5		eqid	$$⊢0_{\mathrm{Scalar}\left(W\right)}=0_{\mathrm{Scalar}\left(W\right)}$$
6		eqid	$$⊢\mathrm{ocv}\left(W\right)=\mathrm{ocv}\left(W\right)$$
7		eqid	$$⊢0_W=0_W$$
8	1 2 3 4 5 6 7	isobs	$$⊢B\in \mathrm{OBasis}\left(W\right)\leftrightarrow \left(W\in \mathrm{PreHil}\wedge B\subseteq {\mathrm{Base}}_W\wedge \left(\forall x\in B\forall y\in Bx{\cdot }_{𝑖}\left(W\right)y=if\left(x=y,1_{\mathrm{Scalar}\left(W\right)},0_{\mathrm{Scalar}\left(W\right)}\right)\wedge \mathrm{ocv}\left(W\right)\left(B\right)=\left\{0_W\right\}\right)\right)$$
9	8	simp1bi	$$⊢B\in \mathrm{OBasis}\left(W\right)\to W\in \mathrm{PreHil}$$