Step |
Hyp |
Ref |
Expression |
1 |
|
trivnsgd.1 |
|- B = ( Base ` G ) |
2 |
|
trivnsgd.2 |
|- .0. = ( 0g ` G ) |
3 |
|
trivnsgd.3 |
|- ( ph -> G e. Grp ) |
4 |
|
trivnsgd.4 |
|- ( ph -> B = { .0. } ) |
5 |
|
nsgsubg |
|- ( x e. ( NrmSGrp ` G ) -> x e. ( SubGrp ` G ) ) |
6 |
5
|
a1i |
|- ( ph -> ( x e. ( NrmSGrp ` G ) -> x e. ( SubGrp ` G ) ) ) |
7 |
6
|
ssrdv |
|- ( ph -> ( NrmSGrp ` G ) C_ ( SubGrp ` G ) ) |
8 |
1 2 3 4
|
trivsubgsnd |
|- ( ph -> ( SubGrp ` G ) = { B } ) |
9 |
7 8
|
sseqtrd |
|- ( ph -> ( NrmSGrp ` G ) C_ { B } ) |
10 |
1
|
nsgid |
|- ( G e. Grp -> B e. ( NrmSGrp ` G ) ) |
11 |
3 10
|
syl |
|- ( ph -> B e. ( NrmSGrp ` G ) ) |
12 |
11
|
snssd |
|- ( ph -> { B } C_ ( NrmSGrp ` G ) ) |
13 |
9 12
|
eqssd |
|- ( ph -> ( NrmSGrp ` G ) = { B } ) |