Metamath Proof Explorer


Theorem tngplusg

Description: The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by AV, 31-Oct-2024)

Ref Expression
Hypotheses tngbas.t
|- T = ( G toNrmGrp N )
tngplusg.2
|- .+ = ( +g ` G )
Assertion tngplusg
|- ( N e. V -> .+ = ( +g ` T ) )

Proof

Step Hyp Ref Expression
1 tngbas.t
 |-  T = ( G toNrmGrp N )
2 tngplusg.2
 |-  .+ = ( +g ` G )
3 plusgid
 |-  +g = Slot ( +g ` ndx )
4 tsetndxnplusgndx
 |-  ( TopSet ` ndx ) =/= ( +g ` ndx )
5 4 necomi
 |-  ( +g ` ndx ) =/= ( TopSet ` ndx )
6 dsndxnplusgndx
 |-  ( dist ` ndx ) =/= ( +g ` ndx )
7 6 necomi
 |-  ( +g ` ndx ) =/= ( dist ` ndx )
8 1 3 5 7 tnglem
 |-  ( N e. V -> ( +g ` G ) = ( +g ` T ) )
9 2 8 eqtrid
 |-  ( N e. V -> .+ = ( +g ` T ) )