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Theorem drngui 18963
Description: The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
drngui.b 𝐵 = (Base‘𝑅)
drngui.z 0 = (0g𝑅)
drngui.r 𝑅 ∈ DivRing
Assertion
Ref Expression
drngui (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Proof of Theorem drngui
StepHypRef Expression
1 drngui.r . . . 4 𝑅 ∈ DivRing
2 drngui.b . . . . 5 𝐵 = (Base‘𝑅)
3 eqid 2771 . . . . 5 (Unit‘𝑅) = (Unit‘𝑅)
4 drngui.z . . . . 5 0 = (0g𝑅)
52, 3, 4isdrng 18961 . . . 4 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
61, 5mpbi 220 . . 3 (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))
76simpri 473 . 2 (Unit‘𝑅) = (𝐵 ∖ { 0 })
87eqcomi 2780 1 (𝐵 ∖ { 0 }) = (Unit‘𝑅)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  cdif 3720  {csn 4316  cfv 6031  Basecbs 16064  0gc0g 16308  Ringcrg 18755  Unitcui 18847  DivRingcdr 18957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-drng 18959
This theorem is referenced by:  cnflddiv  19991  cnfldinv  19992  cnsubdrglem  20012  cnmgpabl  20022  cnmsubglem  20024  gzrngunit  20027  zringunit  20051  expghm  20059  psgninv  20143  zrhpsgnmhm  20145  amgmlem  24937  dchrghm  25202  dchrabs  25206  sum2dchr  25220  lgseisenlem4  25324  qrngdiv  25534  proot1ex  38305  amgmwlem  43079  amgmlemALT  43080
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