Theorem tdrgunit 24492

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 24491	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
4	3	simp3bi 1165	1 ⊢ (𝑅 ∈ TopDRing → (𝑀 ↾s 𝑈) ∈ TopGrp)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6539  (class class class)co 7421   ↾_s cress 17414  mulGrpcmgp 20366  Unitcui 20591  DivRingcdr 20986  TopGrpctgp 24396  TopRingctrg 24481  TopDRingctdrg 24482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-ov 7424  df-tdrg 24486

This theorem is used by:  invrcn2  24505