Theorem tngvsca

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

	Ref	Expression
Hypotheses	tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
	tngvsca.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
Assertion	tngvsca	$$⊢N\in V\to \underset{\dot{}}{·}={\cdot }_T$$

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	$$⊢T=G\mathrm{toNrmGrp}N$$
2		tngvsca.2	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		df-vsca	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}6$$
4		6nn	$$⊢6\in \mathbb{N}$$
5		6lt9	$$⊢6<9$$
6	1 3 4 5	tnglem	$$⊢N\in V\to {\cdot }_G={\cdot }_T$$
7	2 6	syl5eq	$$⊢N\in V\to \underset{\dot{}}{·}={\cdot }_T$$