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 toNrmGrp N )
	tngbas.2	|- B = ( Base ` G )
Assertion	tngbas	|- ( N e. V -> B = ( Base ` T ) )

Proof

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