Metamath Proof Explorer

Theorem 0nsg

Description: The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015)

Ref Expression
Hypothesis 0nsg.z 
|- .0. = ( 0g ` G )
Assertion 0nsg 
|- ( G e. Grp -> { .0. } e. ( NrmSGrp ` G ) )

Proof

Step Hyp Ref Expression
1 0nsg.z 
 |-  .0. = ( 0g ` G )
2 1 0subg 
 |-  ( G e. Grp -> { .0. } e. ( SubGrp ` G ) )
3 elsni 
 |-  ( y e. { .0. } -> y = .0. )
4 3 ad2antll 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> y = .0. )
5 4 oveq2d 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` G ) .0. ) )
6 eqid 
 |-  ( Base ` G ) = ( Base ` G )
7 eqid 
 |-  ( +g ` G ) = ( +g ` G )
8 6 7 1 grprid 
 |-  ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( +g ` G ) .0. ) = x )
9 8 adantrr 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) .0. ) = x )
10 5 9 eqtrd 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( +g ` G ) y ) = x )
11 10 oveq1d 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = ( x ( -g ` G ) x ) )
12 eqid 
 |-  ( -g ` G ) = ( -g ` G )
13 6 1 12 grpsubid 
 |-  ( ( G e. Grp /\ x e. ( Base ` G ) ) -> ( x ( -g ` G ) x ) = .0. )
14 13 adantrr 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( x ( -g ` G ) x ) = .0. )
15 11 14 eqtrd 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. )
16 ovex 
 |-  ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. _V
17 16 elsn 
 |-  ( ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } <-> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) = .0. )
18 15 17 sylibr 
 |-  ( ( G e. Grp /\ ( x e. ( Base ` G ) /\ y e. { .0. } ) ) -> ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } )
19 18 ralrimivva 
 |-  ( G e. Grp -> A. x e. ( Base ` G ) A. y e. { .0. } ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } )
20 6 7 12 isnsg3 
 |-  ( { .0. } e. ( NrmSGrp ` G ) <-> ( { .0. } e. ( SubGrp ` G ) /\ A. x e. ( Base ` G ) A. y e. { .0. } ( ( x ( +g ` G ) y ) ( -g ` G ) x ) e. { .0. } ) )
21 2 19 20 sylanbrc 
 |-  ( G e. Grp -> { .0. } e. ( NrmSGrp ` G ) )