Theorem tdrgunit 24466

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 24465	. 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 1570   ∈ wcel 2145  ‘cfv 6531  (class class class)co 7412   ↾_s cress 17388  mulGrpcmgp 20340  Unitcui 20565  DivRingcdr 20960  TopGrpctgp 24370  TopRingctrg 24455  TopDRingctdrg 24456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-tdrg 24460

This theorem is used by:  invrcn2  24479