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Theorem trgtgp 21781
 Description: A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trgtgp (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Proof of Theorem trgtgp
StepHypRef Expression
1 eqid 2610 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
21istrg 21777 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
32simp1bi 1069 1 (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ‘cfv 5804  mulGrpcmgp 18312  Ringcrg 18370  TopMndctmd 21684  TopGrpctgp 21685  TopRingctrg 21769 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-trg 21773 This theorem is referenced by:  trgtmd2  21782  trgtps  21783  pl1cn  29329
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