Theorem tngbas

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

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

Proof

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