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Theorem istdrg2 21886
Description: A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istdrg2.m 𝑀 = (mulGrp‘𝑅)
istdrg2.b 𝐵 = (Base‘𝑅)
istdrg2.z 0 = (0g𝑅)
Assertion
Ref Expression
istdrg2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))

Proof of Theorem istdrg2
StepHypRef Expression
1 istdrg2.m . . 3 𝑀 = (mulGrp‘𝑅)
2 eqid 2626 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 21874 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp))
4 istdrg2.b . . . . . . . . 9 𝐵 = (Base‘𝑅)
5 istdrg2.z . . . . . . . . 9 0 = (0g𝑅)
64, 2, 5isdrng 18667 . . . . . . . 8 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
76simprbi 480 . . . . . . 7 (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
87adantl 482 . . . . . 6 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
98oveq2d 6621 . . . . 5 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ { 0 })))
109eleq1d 2688 . . . 4 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1110pm5.32i 668 . . 3 (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
12 df-3an 1038 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp))
13 df-3an 1038 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1411, 12, 133bitr4i 292 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
153, 14bitri 264 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  cdif 3557  {csn 4153  cfv 5850  (class class class)co 6605  Basecbs 15776  s cress 15777  0gc0g 16016  mulGrpcmgp 18405  Ringcrg 18463  Unitcui 18555  DivRingcdr 18663  TopGrpctgp 21780  TopRingctrg 21864  TopDRingctdrg 21865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-ov 6608  df-drng 18665  df-tdrg 21869
This theorem is referenced by: (None)
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