Theorem tngip

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

	Ref	Expression
Hypotheses	tngbas.t	|- T = ( G toNrmGrp N )
	tngip.2	|- ., = ( .i ` G )
Assertion	tngip	|- ( N e. V -> ., = ( .i ` T ) )

Proof

Step	Hyp	Ref	Expression
1		tngbas.t	|- T = ( G toNrmGrp N )
2		tngip.2	|- ., = ( .i ` G )
3		df-ip	|- .i = Slot 8
4		8nn	|- 8 e. NN
5		8lt9	|- 8 < 9
6	1 3 4 5	tnglem	|- ( N e. V -> ( .i ` G ) = ( .i ` T ) )
7	2 6	syl5eq	|- ( N e. V -> ., = ( .i ` T ) )