Theorem tngplusg

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

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

Proof

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