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Metamath Proof Explorer
Description: Addition operation of an opposite ring. (Contributed by Mario
Carneiro, 1-Dec-2014) (Proof shortened by AV, 6-Nov-2024)
|
|
Ref |
Expression |
|
Hypotheses |
opprbas.1 |
⊢ 𝑂 = ( oppr ‘ 𝑅 ) |
|
|
oppradd.2 |
⊢ + = ( +g ‘ 𝑅 ) |
|
Assertion |
oppradd |
⊢ + = ( +g ‘ 𝑂 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
opprbas.1 |
⊢ 𝑂 = ( oppr ‘ 𝑅 ) |
| 2 |
|
oppradd.2 |
⊢ + = ( +g ‘ 𝑅 ) |
| 3 |
|
plusgid |
⊢ +g = Slot ( +g ‘ ndx ) |
| 4 |
|
plusgndxnmulrndx |
⊢ ( +g ‘ ndx ) ≠ ( .r ‘ ndx ) |
| 5 |
1 3 4
|
opprlem |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑂 ) |
| 6 |
2 5
|
eqtri |
⊢ + = ( +g ‘ 𝑂 ) |