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 toNrmGrp N )
	tngplusg.2	|- .+ = ( +g ` G )
Assertion	tngplusg	|- ( N e. V -> .+ = ( +g ` T ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	|- T = ( G toNrmGrp N )
2		tngplusg.2	|- .+ = ( +g ` G )
3		df-plusg	|- +g = Slot 2
4		2nn	|- 2 e. NN
5		2lt9	|- 2 < 9
6	1 3 4 5	tnglem	|- ( N e. V -> ( +g ` G ) = ( +g ` T ) )
7	2 6	syl5eq	|- ( N e. V -> .+ = ( +g ` T ) )