Step |
Hyp |
Ref |
Expression |
1 |
|
opprbas.1 |
⊢ 𝑂 = ( oppr ‘ 𝑅 ) |
2 |
|
opprneg.2 |
⊢ 𝑁 = ( invg ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
6 |
3 4 5 2
|
grpinvfval |
⊢ 𝑁 = ( 𝑥 ∈ ( Base ‘ 𝑅 ) ↦ ( ℩ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝑅 ) ) ) |
7 |
1 3
|
opprbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 ) |
8 |
1 4
|
oppradd |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑂 ) |
9 |
1 5
|
oppr0 |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑂 ) |
10 |
|
eqid |
⊢ ( invg ‘ 𝑂 ) = ( invg ‘ 𝑂 ) |
11 |
7 8 9 10
|
grpinvfval |
⊢ ( invg ‘ 𝑂 ) = ( 𝑥 ∈ ( Base ‘ 𝑅 ) ↦ ( ℩ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝑦 ( +g ‘ 𝑅 ) 𝑥 ) = ( 0g ‘ 𝑅 ) ) ) |
12 |
6 11
|
eqtr4i |
⊢ 𝑁 = ( invg ‘ 𝑂 ) |