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Theorem tdrgunit 24084

Description: The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

**Hypotheses**
Ref	Expression
istrg.1	⊢ 𝑀 = (mulGrp‘𝑅)
istdrg.1	⊢ 𝑈 = (Unit‘𝑅)

**Assertion**
Ref	Expression
tdrgunit	⊢ (𝑅 ∈ TopDRing → (𝑀 ↾_s 𝑈) ∈ TopGrp)

**Proof of Theorem tdrgunit**
Step	Hyp	Ref	Expression
1		istrg.1	. . 3 ⊢ 𝑀 = (mulGrp‘𝑅)
2		istdrg.1	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	istdrg 24083	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾_s 𝑈) ∈ TopGrp))
4	3	simp3bi 1145	1 ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾_s 𝑈) ∈ TopGrp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 ↾_s cress 17209 mulGrpcmgp 20074 Unitcui 20294 DivRingcdr 20624 TopGrpctgp 23988 TopRingctrg 24073 TopDRingctdrg 24074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-tdrg 24078

This theorem is referenced by: invrcn2 24097

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