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Theorem tdrgunit 22191
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 → (𝑀s 𝑈) ∈ TopGrp)

Proof of Theorem tdrgunit
StepHypRef Expression
1 istrg.1 . . 3 𝑀 = (mulGrp‘𝑅)
2 istdrg.1 . . 3 𝑈 = (Unit‘𝑅)
31, 2istdrg 22190 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp))
43simp3bi 1141 1 (𝑅 ∈ TopDRing → (𝑀s 𝑈) ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6032  (class class class)co 6794  s cress 16066  mulGrpcmgp 18698  Unitcui 18848  DivRingcdr 18958  TopGrpctgp 22096  TopRingctrg 22180  TopDRingctdrg 22181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5995  df-fv 6040  df-ov 6797  df-tdrg 22185
This theorem is referenced by:  invrcn2  22204
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