Metamath Proof Explorer

Theorem tnglvec

Description: Augmenting a structure with a norm conserves left vector spaces. (Contributed by Thierry Arnoux, 20-May-2023)

Ref Expression
Hypothesis tnglvec.t 
|- T = ( G toNrmGrp N )
Assertion tnglvec 
|- ( N e. V -> ( G e. LVec <-> T e. LVec ) )

Proof

Step Hyp Ref Expression
1 tnglvec.t 
 |-  T = ( G toNrmGrp N )
2 eqidd 
 |-  ( N e. V -> ( Base ` G ) = ( Base ` G ) )
3 eqid 
 |-  ( Base ` G ) = ( Base ` G )
4 1 3 tngbas 
 |-  ( N e. V -> ( Base ` G ) = ( Base ` T ) )
5 eqid 
 |-  ( +g ` G ) = ( +g ` G )
6 1 5 tngplusg 
 |-  ( N e. V -> ( +g ` G ) = ( +g ` T ) )
7 6 oveqdr 
 |-  ( ( N e. V /\ ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` T ) y ) )
8 eqidd 
 |-  ( N e. V -> ( Scalar ` G ) = ( Scalar ` G ) )
9 eqid 
 |-  ( Scalar ` G ) = ( Scalar ` G )
10 1 9 tngsca 
 |-  ( N e. V -> ( Scalar ` G ) = ( Scalar ` T ) )
11 eqid 
 |-  ( Base ` ( Scalar ` G ) ) = ( Base ` ( Scalar ` G ) )
12 eqid 
 |-  ( .s ` G ) = ( .s ` G )
13 1 12 tngvsca 
 |-  ( N e. V -> ( .s ` G ) = ( .s ` T ) )
14 13 oveqdr 
 |-  ( ( N e. V /\ ( x e. ( Base ` ( Scalar ` G ) ) /\ y e. ( Base ` G ) ) ) -> ( x ( .s ` G ) y ) = ( x ( .s ` T ) y ) )
15 2 4 7 8 10 11 14 lvecpropd 
 |-  ( N e. V -> ( G e. LVec <-> T e. LVec ) )