Metamath Proof Explorer


Theorem tngsca

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

Ref Expression
Hypotheses tngbas.t
|- T = ( G toNrmGrp N )
tngsca.2
|- F = ( Scalar ` G )
Assertion tngsca
|- ( N e. V -> F = ( Scalar ` T ) )

Proof

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