Theorem tngplusg

Description: The group addition 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	$$⊢T=G\mathrm{toNrmGrp}N$$
	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		plusgid	$$⊢{+}_{𝑔}=\mathrm{Slot}{+}_{\mathrm{ndx}}$$
4		tsetndxnplusgndx	$$⊢\mathrm{TopSet}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$
5	4	necomi	$$⊢{+}_{\mathrm{ndx}}\ne \mathrm{TopSet}\left(\mathrm{ndx}\right)$$
6		dsndxnplusgndx	$$⊢\mathrm{dist}\left(\mathrm{ndx}\right)\ne {+}_{\mathrm{ndx}}$$
7	6	necomi	$$⊢{+}_{\mathrm{ndx}}\ne \mathrm{dist}\left(\mathrm{ndx}\right)$$
8	1 3 5 7	tnglem	$$⊢N\in V\to {+}_G={+}_T$$
9	2 8	eqtrid	$$⊢N\in V\to \underset{\dot{}}{+}={+}_T$$