Theorem tngmulr

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

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

Proof

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