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Theorem tdrgunit 24026
Description: The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istrg.1 𝑀 = (mulGrpβ€˜π‘…)
istdrg.1 π‘ˆ = (Unitβ€˜π‘…)
Assertion
Ref Expression
tdrgunit (𝑅 ∈ TopDRing β†’ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp)

Proof of Theorem tdrgunit
StepHypRef Expression
1 istrg.1 . . 3 𝑀 = (mulGrpβ€˜π‘…)
2 istdrg.1 . . 3 π‘ˆ = (Unitβ€˜π‘…)
31, 2istdrg 24025 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
43simp3bi 1144 1 (𝑅 ∈ TopDRing β†’ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6537  (class class class)co 7405   β†Ύs cress 17182  mulGrpcmgp 20039  Unitcui 20257  DivRingcdr 20587  TopGrpctgp 23930  TopRingctrg 24015  TopDRingctdrg 24016
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 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-tdrg 24020
This theorem is referenced by:  invrcn2  24039
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