Step |
Hyp |
Ref |
Expression |
1 |
|
nmpropd.1 |
|- ( ph -> ( Base ` K ) = ( Base ` L ) ) |
2 |
|
nmpropd.2 |
|- ( ph -> ( +g ` K ) = ( +g ` L ) ) |
3 |
|
nmpropd.3 |
|- ( ph -> ( dist ` K ) = ( dist ` L ) ) |
4 |
|
eqidd |
|- ( ph -> x = x ) |
5 |
|
eqidd |
|- ( ph -> ( Base ` K ) = ( Base ` K ) ) |
6 |
2
|
oveqdr |
|- ( ( ph /\ ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) |
7 |
5 1 6
|
grpidpropd |
|- ( ph -> ( 0g ` K ) = ( 0g ` L ) ) |
8 |
3 4 7
|
oveq123d |
|- ( ph -> ( x ( dist ` K ) ( 0g ` K ) ) = ( x ( dist ` L ) ( 0g ` L ) ) ) |
9 |
1 8
|
mpteq12dv |
|- ( ph -> ( x e. ( Base ` K ) |-> ( x ( dist ` K ) ( 0g ` K ) ) ) = ( x e. ( Base ` L ) |-> ( x ( dist ` L ) ( 0g ` L ) ) ) ) |
10 |
|
eqid |
|- ( norm ` K ) = ( norm ` K ) |
11 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
12 |
|
eqid |
|- ( 0g ` K ) = ( 0g ` K ) |
13 |
|
eqid |
|- ( dist ` K ) = ( dist ` K ) |
14 |
10 11 12 13
|
nmfval |
|- ( norm ` K ) = ( x e. ( Base ` K ) |-> ( x ( dist ` K ) ( 0g ` K ) ) ) |
15 |
|
eqid |
|- ( norm ` L ) = ( norm ` L ) |
16 |
|
eqid |
|- ( Base ` L ) = ( Base ` L ) |
17 |
|
eqid |
|- ( 0g ` L ) = ( 0g ` L ) |
18 |
|
eqid |
|- ( dist ` L ) = ( dist ` L ) |
19 |
15 16 17 18
|
nmfval |
|- ( norm ` L ) = ( x e. ( Base ` L ) |-> ( x ( dist ` L ) ( 0g ` L ) ) ) |
20 |
9 14 19
|
3eqtr4g |
|- ( ph -> ( norm ` K ) = ( norm ` L ) ) |