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| Description: The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| istrg.1 | ⊢ 𝑀 = (mulGrp‘𝑅) | 
| istdrg.1 | ⊢ 𝑈 = (Unit‘𝑅) | 
| Ref | Expression | 
|---|---|
| tdrgunit | ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | istrg.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
| 2 | istdrg.1 | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
| 3 | 1, 2 | istdrg 24174 | . 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp)) | 
| 4 | 3 | simp3bi 1148 | 1 ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 mulGrpcmgp 20137 Unitcui 20355 DivRingcdr 20729 TopGrpctgp 24079 TopRingctrg 24164 TopDRingctdrg 24165 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-tdrg 24169 | 
| This theorem is referenced by: invrcn2 24188 | 
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