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Theorem tdrgtrg 22353
 Description: A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tdrgtrg (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Proof of Theorem tdrgtrg
StepHypRef Expression
1 eqid 2825 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2 eqid 2825 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 22346 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
43simp1bi 1179 1 (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2164  ‘cfv 6127  (class class class)co 6910   ↾s cress 16230  mulGrpcmgp 18850  Unitcui 19000  DivRingcdr 19110  TopGrpctgp 22252  TopRingctrg 22336  TopDRingctdrg 22337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-tdrg 22341 This theorem is referenced by:  tdrgring  22355  tdrgtmd  22356  tdrgtps  22357  dvrcn  22364
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