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Theorem istdrg2 22789
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 2824 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 22777 . 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 19506 . . . . . . . 8 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
76simprbi 500 . . . . . . 7 (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
87adantl 485 . . . . . 6 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
98oveq2d 7165 . . . . 5 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ { 0 })))
109eleq1d 2900 . . . 4 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1110pm5.32i 578 . . 3 (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
12 df-3an 1086 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp))
13 df-3an 1086 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1411, 12, 133bitr4i 306 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
153, 14bitri 278 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cdif 3916  {csn 4550  cfv 6343  (class class class)co 7149  Basecbs 16483  s cress 16484  0gc0g 16713  mulGrpcmgp 19239  Ringcrg 19297  Unitcui 19392  DivRingcdr 19502  TopGrpctgp 22682  TopRingctrg 22767  TopDRingctdrg 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-drng 19504  df-tdrg 22772
This theorem is referenced by: (None)
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