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Theorem tdrgdrng 24198
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 2735 . . 3 (mulGrp‘𝑅) = (mulGrp‘𝑅)
2 eqid 2735 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 24190 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
43simp2bi 1145 1 (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cfv 6563  (class class class)co 7431  s cress 17274  mulGrpcmgp 20152  Unitcui 20372  DivRingcdr 20746  TopGrpctgp 24095  TopRingctrg 24180  TopDRingctdrg 24181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tdrg 24185
This theorem is referenced by:  tvclvec  24223
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