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Metamath Proof Explorer
Description: The unit group of a topological division ring is a topological group.
(Contributed by Mario Carneiro, 5-Oct-2015)
|
|
Ref |
Expression |
|
Hypotheses |
istrg.1 |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
|
|
istdrg.1 |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
|
Assertion |
tdrgunit |
⊢ ( 𝑅 ∈ TopDRing → ( 𝑀 ↾s 𝑈 ) ∈ TopGrp ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
istrg.1 |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
| 2 |
|
istdrg.1 |
⊢ 𝑈 = ( Unit ‘ 𝑅 ) |
| 3 |
1 2
|
istdrg |
⊢ ( 𝑅 ∈ TopDRing ↔ ( 𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ( 𝑀 ↾s 𝑈 ) ∈ TopGrp ) ) |
| 4 |
3
|
simp3bi |
⊢ ( 𝑅 ∈ TopDRing → ( 𝑀 ↾s 𝑈 ) ∈ TopGrp ) |