Step |
Hyp |
Ref |
Expression |
1 |
|
resvbas.1 |
⊢ 𝐻 = ( 𝐺 ↾v 𝐴 ) |
2 |
|
resv0g.2 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
eqidd |
⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
5 |
1 4
|
resvbas |
⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
7 |
1 6
|
resvplusg |
⊢ ( 𝐴 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) ) |
8 |
7
|
oveqdr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐻 ) 𝑦 ) ) |
9 |
3 5 8
|
grpidpropd |
⊢ ( 𝐴 ∈ 𝑉 → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
10 |
2 9
|
syl5eq |
⊢ ( 𝐴 ∈ 𝑉 → 0 = ( 0g ‘ 𝐻 ) ) |