Theorem nlmnrg

Description: The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypothesis	nlmnrg.1	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	nlmnrg	$$⊢W\in \mathrm{NrmMod}\to F\in \mathrm{NrmRing}$$

Proof

Step	Hyp	Ref	Expression
1		nlmnrg.1	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
3		eqid	$$⊢\mathrm{norm}\left(W\right)=\mathrm{norm}\left(W\right)$$
4		eqid	$$⊢{\cdot }_W={\cdot }_W$$
5		eqid	$$⊢{\mathrm{Base}}_F={\mathrm{Base}}_F$$
6		eqid	$$⊢\mathrm{norm}\left(F\right)=\mathrm{norm}\left(F\right)$$
7	2 3 4 1 5 6	isnlm	$$⊢W\in \mathrm{NrmMod}\leftrightarrow \left(\left(W\in \mathrm{NrmGrp}\wedge W\in \mathrm{LMod}\wedge F\in \mathrm{NrmRing}\right)\wedge \forall x\in {\mathrm{Base}}_F\forall y\in {\mathrm{Base}}_W\mathrm{norm}\left(W\right)\left(x{\cdot }_{W}y\right)=\mathrm{norm}\left(F\right)\left(x\right)\mathrm{norm}\left(W\right)\left(y\right)\right)$$
8	7	simplbi	$$⊢W\in \mathrm{NrmMod}\to \left(W\in \mathrm{NrmGrp}\wedge W\in \mathrm{LMod}\wedge F\in \mathrm{NrmRing}\right)$$
9	8	simp3d	$$⊢W\in \mathrm{NrmMod}\to F\in \mathrm{NrmRing}$$