Step |
Hyp |
Ref |
Expression |
1 |
|
tngnrg.t |
|- T = ( R toNrmGrp F ) |
2 |
|
tngnrg.a |
|- A = ( AbsVal ` R ) |
3 |
2
|
abvrcl |
|- ( F e. A -> R e. Ring ) |
4 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
5 |
3 4
|
syl |
|- ( F e. A -> R e. Grp ) |
6 |
|
eqid |
|- ( -g ` R ) = ( -g ` R ) |
7 |
1 6
|
tngds |
|- ( F e. A -> ( F o. ( -g ` R ) ) = ( dist ` T ) ) |
8 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
9 |
8 2 6
|
abvmet |
|- ( F e. A -> ( F o. ( -g ` R ) ) e. ( Met ` ( Base ` R ) ) ) |
10 |
7 9
|
eqeltrrd |
|- ( F e. A -> ( dist ` T ) e. ( Met ` ( Base ` R ) ) ) |
11 |
2 8
|
abvf |
|- ( F e. A -> F : ( Base ` R ) --> RR ) |
12 |
|
eqid |
|- ( dist ` T ) = ( dist ` T ) |
13 |
1 8 12
|
tngngp2 |
|- ( F : ( Base ` R ) --> RR -> ( T e. NrmGrp <-> ( R e. Grp /\ ( dist ` T ) e. ( Met ` ( Base ` R ) ) ) ) ) |
14 |
11 13
|
syl |
|- ( F e. A -> ( T e. NrmGrp <-> ( R e. Grp /\ ( dist ` T ) e. ( Met ` ( Base ` R ) ) ) ) ) |
15 |
5 10 14
|
mpbir2and |
|- ( F e. A -> T e. NrmGrp ) |
16 |
|
reex |
|- RR e. _V |
17 |
1 8 16
|
tngnm |
|- ( ( R e. Grp /\ F : ( Base ` R ) --> RR ) -> F = ( norm ` T ) ) |
18 |
5 11 17
|
syl2anc |
|- ( F e. A -> F = ( norm ` T ) ) |
19 |
|
eqidd |
|- ( F e. A -> ( Base ` R ) = ( Base ` R ) ) |
20 |
1 8
|
tngbas |
|- ( F e. A -> ( Base ` R ) = ( Base ` T ) ) |
21 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
22 |
1 21
|
tngplusg |
|- ( F e. A -> ( +g ` R ) = ( +g ` T ) ) |
23 |
22
|
oveqdr |
|- ( ( F e. A /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` T ) y ) ) |
24 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
25 |
1 24
|
tngmulr |
|- ( F e. A -> ( .r ` R ) = ( .r ` T ) ) |
26 |
25
|
oveqdr |
|- ( ( F e. A /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` T ) y ) ) |
27 |
19 20 23 26
|
abvpropd |
|- ( F e. A -> ( AbsVal ` R ) = ( AbsVal ` T ) ) |
28 |
2 27
|
syl5eq |
|- ( F e. A -> A = ( AbsVal ` T ) ) |
29 |
18 28
|
eleq12d |
|- ( F e. A -> ( F e. A <-> ( norm ` T ) e. ( AbsVal ` T ) ) ) |
30 |
29
|
ibi |
|- ( F e. A -> ( norm ` T ) e. ( AbsVal ` T ) ) |
31 |
|
eqid |
|- ( norm ` T ) = ( norm ` T ) |
32 |
|
eqid |
|- ( AbsVal ` T ) = ( AbsVal ` T ) |
33 |
31 32
|
isnrg |
|- ( T e. NrmRing <-> ( T e. NrmGrp /\ ( norm ` T ) e. ( AbsVal ` T ) ) ) |
34 |
15 30 33
|
sylanbrc |
|- ( F e. A -> T e. NrmRing ) |