Theorem tdrgunit 24157

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 24156	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
4	3	simp3bi 1153	1 ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  (class class class)co 7363   ↾_s cress 17198  mulGrpcmgp 20119  Unitcui 20333  DivRingcdr 20708  TopGrpctgp 24061  TopRingctrg 24146  TopDRingctdrg 24147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-tdrg 24151

This theorem is referenced by:  invrcn2  24170