Theorem tngsca

Description: The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
	tngsca.2	$$⊢F=\mathrm{Scalar}\left(G\right)$$
Assertion	tngsca	$$⊢N\in V\to F=\mathrm{Scalar}\left(T\right)$$

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
2		tngsca.2	$$⊢F=\mathrm{Scalar}\left(G\right)$$
3		df-sca	$$⊢\mathrm{Scalar}=\mathrm{Slot}5$$
4		5nn	$$⊢5\in \mathbb{N}$$
5		5lt9	$$⊢5<9$$
6	1 3 4 5	tnglem	$$⊢N\in V\to \mathrm{Scalar}\left(G\right)=\mathrm{Scalar}\left(T\right)$$
7	2 6	syl5eq	$$⊢N\in V\to F=\mathrm{Scalar}\left(T\right)$$