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Theorem tdrgunit 21780
 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 21779 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp))
43simp3bi 1071 1 (𝑅 ∈ TopDRing → (𝑀s 𝑈) ∈ TopGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549   ↾s cress 15696  mulGrpcmgp 18312  Unitcui 18462  DivRingcdr 18570  TopGrpctgp 21685  TopRingctrg 21769  TopDRingctdrg 21770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-tdrg 21774 This theorem is referenced by:  invrcn2  21793
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