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Theorem tdrgtrg 22782
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 2801 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2 eqid 2801 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 22775 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
43simp1bi 1142 1 (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cfv 6328  (class class class)co 7139  s cress 16480  mulGrpcmgp 19236  Unitcui 19389  DivRingcdr 19499  TopGrpctgp 22680  TopRingctrg 22765  TopDRingctdrg 22766
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-tdrg 22770
This theorem is referenced by:  tdrgring  22784  tdrgtmd  22785  tdrgtps  22786  dvrcn  22793
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