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Theorem istdrg2 23357
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 2733 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 23345 . 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 20023 . . . . . . . 8 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
76simprbi 496 . . . . . . 7 (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
87adantl 481 . . . . . 6 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
98oveq2d 7311 . . . . 5 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ { 0 })))
109eleq1d 2818 . . . 4 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1110pm5.32i 574 . . 3 (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
12 df-3an 1087 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp))
13 df-3an 1087 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1411, 12, 133bitr4i 302 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
153, 14bitri 274 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1085   = wceq 1537  wcel 2101  cdif 3886  {csn 4564  cfv 6447  (class class class)co 7295  Basecbs 16940  s cress 16969  0gc0g 17178  mulGrpcmgp 19748  Ringcrg 19811  Unitcui 19909  DivRingcdr 20019  TopGrpctgp 23250  TopRingctrg 23335  TopDRingctdrg 23336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-ov 7298  df-drng 20021  df-tdrg 23340
This theorem is referenced by: (None)
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