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Metamath Proof Explorer

Theorem tnggrpr

Description: If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021)

Ref Expression
Hypothesis tngngp3.t 
|- T = ( G toNrmGrp N )
Assertion tnggrpr 
|- ( ( N e. V /\ T e. NrmGrp ) -> G e. Grp )

Proof

Step Hyp Ref Expression
1 tngngp3.t 
 |-  T = ( G toNrmGrp N )
2 eqid 
 |-  ( Base ` G ) = ( Base ` G )
3 1 2 tngbas 
 |-  ( N e. V -> ( Base ` G ) = ( Base ` T ) )
4 eqidd 
 |-  ( N e. V -> ( Base ` G ) = ( Base ` G ) )
5 eqid 
 |-  ( +g ` G ) = ( +g ` G )
6 1 5 tngplusg 
 |-  ( N e. V -> ( +g ` G ) = ( +g ` T ) )
7 6 eqcomd 
 |-  ( N e. V -> ( +g ` T ) = ( +g ` G ) )
8 7 oveqdr 
 |-  ( ( N e. V /\ ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) ) -> ( x ( +g ` T ) y ) = ( x ( +g ` G ) y ) )
9 3 4 8 grppropd 
 |-  ( N e. V -> ( T e. Grp <-> G e. Grp ) )
10 9 biimpd 
 |-  ( N e. V -> ( T e. Grp -> G e. Grp ) )
11 ngpgrp 
 |-  ( T e. NrmGrp -> T e. Grp )
12 10 11 impel 
 |-  ( ( N e. V /\ T e. NrmGrp ) -> G e. Grp )