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Theorem istdrg2 23545
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 = (mulGrpβ€˜π‘…)
istdrg2.b 𝐡 = (Baseβ€˜π‘…)
istdrg2.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
istdrg2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp))

Proof of Theorem istdrg2
StepHypRef Expression
1 istdrg2.m . . 3 𝑀 = (mulGrpβ€˜π‘…)
2 eqid 2733 . . 3 (Unitβ€˜π‘…) = (Unitβ€˜π‘…)
31, 2istdrg 23533 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp))
4 istdrg2.b . . . . . . . . 9 𝐡 = (Baseβ€˜π‘…)
5 istdrg2.z . . . . . . . . 9 0 = (0gβ€˜π‘…)
64, 2, 5isdrng 20201 . . . . . . . 8 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unitβ€˜π‘…) = (𝐡 βˆ– { 0 })))
76simprbi 498 . . . . . . 7 (𝑅 ∈ DivRing β†’ (Unitβ€˜π‘…) = (𝐡 βˆ– { 0 }))
87adantl 483 . . . . . 6 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) β†’ (Unitβ€˜π‘…) = (𝐡 βˆ– { 0 }))
98oveq2d 7374 . . . . 5 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) β†’ (𝑀 β†Ύs (Unitβ€˜π‘…)) = (𝑀 β†Ύs (𝐡 βˆ– { 0 })))
109eleq1d 2819 . . . 4 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) β†’ ((𝑀 β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp ↔ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp))
1110pm5.32i 576 . . 3 (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp))
12 df-3an 1090 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp))
13 df-3an 1090 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp))
1411, 12, 133bitr4i 303 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (Unitβ€˜π‘…)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp))
153, 14bitri 275 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs (𝐡 βˆ– { 0 })) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3908  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088   β†Ύs cress 17117  0gc0g 17326  mulGrpcmgp 19901  Ringcrg 19969  Unitcui 20073  DivRingcdr 20197  TopGrpctgp 23438  TopRingctrg 23523  TopDRingctdrg 23524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-drng 20199  df-tdrg 23528
This theorem is referenced by: (None)
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