MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istdrg2 Structured version   Visualization version   GIF version

Theorem istdrg2 22260
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 2765 . . 3 (Unit‘𝑅) = (Unit‘𝑅)
31, 2istdrg 22248 . 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 19020 . . . . . . . 8 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
76simprbi 490 . . . . . . 7 (𝑅 ∈ DivRing → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
87adantl 473 . . . . . 6 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (Unit‘𝑅) = (𝐵 ∖ { 0 }))
98oveq2d 6858 . . . . 5 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → (𝑀s (Unit‘𝑅)) = (𝑀s (𝐵 ∖ { 0 })))
109eleq1d 2829 . . . 4 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) → ((𝑀s (Unit‘𝑅)) ∈ TopGrp ↔ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1110pm5.32i 570 . . 3 (((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
12 df-3an 1109 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp))
13 df-3an 1109 . . 3 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
1411, 12, 133bitr4i 294 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (Unit‘𝑅)) ∈ TopGrp) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
153, 14bitri 266 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cdif 3729  {csn 4334  cfv 6068  (class class class)co 6842  Basecbs 16130  s cress 16131  0gc0g 16366  mulGrpcmgp 18756  Ringcrg 18814  Unitcui 18906  DivRingcdr 19016  TopGrpctgp 22154  TopRingctrg 22238  TopDRingctdrg 22239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076  df-ov 6845  df-drng 19018  df-tdrg 22243
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator