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Metamath Proof Explorer
Description: Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
ringgcl.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
|
|
ringgcl.2 |
⊢ 𝑋 = ran 𝐺 |
|
Assertion |
rngogcl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ringgcl.1 |
⊢ 𝐺 = ( 1st ‘ 𝑅 ) |
| 2 |
|
ringgcl.2 |
⊢ 𝑋 = ran 𝐺 |
| 3 |
1
|
rngogrpo |
⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp ) |
| 4 |
2
|
grpocl |
⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 ) |
| 5 |
3 4
|
syl3an1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 ) |