Metamath Proof Explorer

Theorem tngngpim

Description: The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021)

Ref Expression
Hypotheses tngngpim.t 
|- T = ( G toNrmGrp N )
tngngpim.n 
|- N = ( norm ` G )
tngngpim.x 
|- X = ( Base ` G )
tngngpim.d 
|- D = ( dist ` T )
Assertion tngngpim 
|- ( G e. NrmGrp -> D : ( X X. X ) --> RR )

Proof

Step Hyp Ref Expression
1 tngngpim.t 
 |-  T = ( G toNrmGrp N )
2 tngngpim.n 
 |-  N = ( norm ` G )
3 tngngpim.x 
 |-  X = ( Base ` G )
4 tngngpim.d 
 |-  D = ( dist ` T )
5 3 2 nmf 
 |-  ( G e. NrmGrp -> N : X --> RR )
6 2 oveq2i 
 |-  ( G toNrmGrp N ) = ( G toNrmGrp ( norm ` G ) )
7 1 6 eqtri 
 |-  T = ( G toNrmGrp ( norm ` G ) )
8 7 nrmtngnrm 
 |-  ( G e. NrmGrp -> ( T e. NrmGrp /\ ( norm ` T ) = ( norm ` G ) ) )
9 1 3 4 tngngp2 
 |-  ( N : X --> RR -> ( T e. NrmGrp <-> ( G e. Grp /\ D e. ( Met ` X ) ) ) )
10 simpr 
 |-  ( ( G e. Grp /\ D e. ( Met ` X ) ) -> D e. ( Met ` X ) )
11 9 10 syl6bi 
 |-  ( N : X --> RR -> ( T e. NrmGrp -> D e. ( Met ` X ) ) )
12 11 com12 
 |-  ( T e. NrmGrp -> ( N : X --> RR -> D e. ( Met ` X ) ) )
13 12 adantr 
 |-  ( ( T e. NrmGrp /\ ( norm ` T ) = ( norm ` G ) ) -> ( N : X --> RR -> D e. ( Met ` X ) ) )
14 8 13 syl 
 |-  ( G e. NrmGrp -> ( N : X --> RR -> D e. ( Met ` X ) ) )
15 metf 
 |-  ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
16 14 15 syl6 
 |-  ( G e. NrmGrp -> ( N : X --> RR -> D : ( X X. X ) --> RR ) )
17 5 16 mpd 
 |-  ( G e. NrmGrp -> D : ( X X. X ) --> RR )