Theorem drngui 20668

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 2736	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
4		drngui.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
5	2, 3, 4	isdrng 20666	. . . 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 2745	1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898  {csn 4580  ‘cfv 6492  Basecbs 17136  0_gc0g 17359  Ringcrg 20168  Unitcui 20291  DivRingcdr 20662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-drng 20664

This theorem is referenced by:  cnflddiv  21355  cnflddivOLD  21356  cnfldinv  21357  cnsubdrglem  21373  cnmgpabl  21383  cnmsubglem  21385  gzrngunit  21388  zringunit  21421  expghm  21430  psgninv  21537  zrhpsgnmhm  21539  amgmlem  26956  dchrghm  27223  dchrabs  27227  sum2dchr  27241  lgseisenlem4  27345  qrngdiv  27591  proot1ex  43434  amgmwlem  50043  amgmlemALT  50044