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Theorem istdrg2 24202
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 2735 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 24190 . 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 20750 . . . . . . . 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 7447 . . . . 5 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ { 0 })))
109eleq1d 2824 . . . 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 1088 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp))
13 df-3an 1088 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1411, 12, 133bitr4i 303 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
153, 14bitri 275 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  0gc0g 17486  mulGrpcmgp 20152  Ringcrg 20251  Unitcui 20372  DivRingcdr 20746  TopGrpctgp 24095  TopRingctrg 24180  TopDRingctdrg 24181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-drng 20748  df-tdrg 24185
This theorem is referenced by: (None)
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