Theorem tngbas

Description: The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)

	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		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		tsetndxnbasendx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$
6		dsndxnbasendx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {\mathrm{Base}}_{\mathrm{ndx}}$$
7	6	necomi	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{dist}\left(\mathrm{ndx}\right)$$
8	1 3 5 7	tnglem	$$⊢N\in V\to {\mathrm{Base}}_G={\mathrm{Base}}_T$$
9	2 8	syl5eq	$$⊢N\in V\to B={\mathrm{Base}}_T$$