Step |
Hyp |
Ref |
Expression |
1 |
|
0nsg.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
1
|
0subg |
⊢ ( 𝐺 ∈ Grp → { 0 } ∈ ( SubGrp ‘ 𝐺 ) ) |
3 |
|
elsni |
⊢ ( 𝑦 ∈ { 0 } → 𝑦 = 0 ) |
4 |
3
|
ad2antll |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → 𝑦 = 0 ) |
5 |
4
|
oveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 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 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 0 ) = 𝑥 ) |
9 |
8
|
adantrr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 0 ) = 𝑥 ) |
10 |
5 9
|
eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = 𝑥 ) |
11 |
10
|
oveq1d |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) = ( 𝑥 ( -g ‘ 𝐺 ) 𝑥 ) ) |
12 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
13 |
6 1 12
|
grpsubid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( -g ‘ 𝐺 ) 𝑥 ) = 0 ) |
14 |
13
|
adantrr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → ( 𝑥 ( -g ‘ 𝐺 ) 𝑥 ) = 0 ) |
15 |
11 14
|
eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) = 0 ) |
16 |
|
ovex |
⊢ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ V |
17 |
16
|
elsn |
⊢ ( ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ { 0 } ↔ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) = 0 ) |
18 |
15 17
|
sylibr |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ { 0 } ) ) → ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ { 0 } ) |
19 |
18
|
ralrimivva |
⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ { 0 } ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ { 0 } ) |
20 |
6 7 12
|
isnsg3 |
⊢ ( { 0 } ∈ ( NrmSGrp ‘ 𝐺 ) ↔ ( { 0 } ∈ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) ∀ 𝑦 ∈ { 0 } ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ( -g ‘ 𝐺 ) 𝑥 ) ∈ { 0 } ) ) |
21 |
2 19 20
|
sylanbrc |
⊢ ( 𝐺 ∈ Grp → { 0 } ∈ ( NrmSGrp ‘ 𝐺 ) ) |