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Theorem tdrgdrng 23509
Description: A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tdrgdrng (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

Proof of Theorem tdrgdrng
StepHypRef Expression
1 eqid 2736 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2 eqid 2736 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 23501 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
43simp2bi 1146 1 (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cfv 6493  (class class class)co 7353  s cress 17104  mulGrpcmgp 19887  Unitcui 20053  DivRingcdr 20170  TopGrpctgp 23406  TopRingctrg 23491  TopDRingctdrg 23492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7356  df-tdrg 23496
This theorem is referenced by:  tvclvec  23534
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