This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: A ring's addition operation is a group operation. (Contributed by Steve
Rodriguez, 9-Sep-2007) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypothesis |
ringgrp.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
|
Assertion |
rngogrpo |
⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ringgrp.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
| 2 |
1
|
rngoablo |
⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ AbelOp ) |
| 3 |
|
ablogrpo |
⊢ ( 𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp ) |
| 4 |
2 3
|
syl |
⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp ) |