Theorem phlsrng

Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Hypothesis	phlsrng.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	phlsrng	$$⊢W\in \mathrm{PreHil}\to F\in \mathrm{*-Ring}$$

Proof

Step	Hyp	Ref	Expression
1		phlsrng.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
3		eqid	$$⊢{\cdot }_{𝑖}\left(W\right)={\cdot }_{𝑖}\left(W\right)$$
4		eqid	$$⊢0_W=0_W$$
5		eqid	$$⊢{\mathrm{*}}_F={\mathrm{*}}_F$$
6		eqid	$$⊢0_F=0_F$$
7	2 1 3 4 5 6	isphl	$$⊢W\in \mathrm{PreHil}\leftrightarrow \left(W\in \mathrm{LVec}\wedge F\in \mathrm{*-Ring}\wedge \forall x\in {\mathrm{Base}}_W\left(\left(y\in {\mathrm{Base}}_W⟼y{\cdot }_{𝑖}\left(W\right)x\right)\in \left(W\mathrm{LMHom}\mathrm{ringLMod}\left(F\right)\right)\wedge \left(x{\cdot }_{𝑖}\left(W\right)x=0_F\to x=0_W\right)\wedge \forall y\in {\mathrm{Base}}_W{\left(x{\cdot }_{𝑖}\left(W\right)y\right)}^{{\mathrm{*}}_F}=y{\cdot }_{𝑖}\left(W\right)x\right)\right)$$
8	7	simp2bi	$$⊢W\in \mathrm{PreHil}\to F\in \mathrm{*-Ring}$$