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Theorem tdrgtrg 24095
Description: A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tdrgtrg (𝑅 ∈ TopDRing β†’ 𝑅 ∈ TopRing)

Proof of Theorem tdrgtrg
StepHypRef Expression
1 eqid 2727 . . 3 (mulGrpβ€˜π‘…) = (mulGrpβ€˜π‘…)
2 eqid 2727 . . 3 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
31, 2istdrg 24088 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrpβ€˜π‘…) β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp))
43simp1bi 1142 1 (𝑅 ∈ TopDRing β†’ 𝑅 ∈ TopRing)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  β€˜cfv 6551  (class class class)co 7424   β†Ύs cress 17214  mulGrpcmgp 20079  Unitcui 20299  DivRingcdr 20629  TopGrpctgp 23993  TopRingctrg 24078  TopDRingctdrg 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-ov 7427  df-tdrg 24083
This theorem is referenced by:  tdrgring  24097  tdrgtmd  24098  tdrgtps  24099  dvrcn  24106
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