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

Theorem drngui 20752
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 2735 . . . . 5 (Unit‘𝑅) = (Unit‘𝑅)
4 drngui.z . . . . 5 0 = (0g𝑅)
52, 3, 4isdrng 20750 . . . 4 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
61, 5mpbi 230 . . 3 (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))
76simpri 485 . 2 (Unit‘𝑅) = (𝐵 ∖ { 0 })
87eqcomi 2744 1 (𝐵 ∖ { 0 }) = (Unit‘𝑅)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  cdif 3960  {csn 4631  cfv 6563  Basecbs 17245  0gc0g 17486  Ringcrg 20251  Unitcui 20372  DivRingcdr 20746
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-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-drng 20748
This theorem is referenced by:  cnflddiv  21431  cnflddivOLD  21432  cnfldinv  21433  cnsubdrglem  21454  cnmgpabl  21464  cnmsubglem  21466  gzrngunit  21469  zringunit  21495  expghm  21504  psgninv  21618  zrhpsgnmhm  21620  amgmlem  27048  dchrghm  27315  dchrabs  27319  sum2dchr  27333  lgseisenlem4  27437  qrngdiv  27683  proot1ex  43185  amgmwlem  49033  amgmlemALT  49034
  Copyright terms: Public domain W3C validator