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Theorem tdrgdrng 22777
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 2822 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2 eqid 2822 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 22769 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
43simp2bi 1143 1 (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  cfv 6334  (class class class)co 7140  s cress 16475  mulGrpcmgp 19230  Unitcui 19383  DivRingcdr 19493  TopGrpctgp 22674  TopRingctrg 22759  TopDRingctdrg 22760
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
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 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-tdrg 22764
This theorem is referenced by:  tvclvec  22802
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