Theorem drngui 20644

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 = (0g‘𝑅)
drngui.r	⊢ 𝑅 ∈ DivRing

Assertion

Ref	Expression
drngui	⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Proof of Theorem drngui

Step	Hyp	Ref	Expression
1		drngui.r	. . . 4 ⊢ 𝑅 ∈ DivRing
2		drngui.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
3		eqid 2729	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
4		drngui.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
5	2, 3, 4	isdrng 20642	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
6	1, 5	mpbi 230	. . 3 ⊢ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))
7	6	simpri 485	. 2 ⊢ (Unit‘𝑅) = (𝐵 ∖ { 0 })
8	7	eqcomi 2738	1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911  {csn 4589  ‘cfv 6511  Basecbs 17179  0_gc0g 17402  Ringcrg 20142  Unitcui 20264  DivRingcdr 20638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-drng 20640

This theorem is referenced by:  cnflddiv  21312  cnflddivOLD  21313  cnfldinv  21314  cnsubdrglem  21335  cnmgpabl  21345  cnmsubglem  21347  gzrngunit  21350  zringunit  21376  expghm  21385  psgninv  21491  zrhpsgnmhm  21493  amgmlem  26900  dchrghm  27167  dchrabs  27171  sum2dchr  27185  lgseisenlem4  27289  qrngdiv  27535  proot1ex  43185  amgmwlem  49791  amgmlemALT  49792