Theorem drngui 21016

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 2763	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
4		drngui.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
5	2, 3, 4	isdrng 21014	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
6	1, 5	mpbi 233	. . 3 ⊢ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))
7	6	simpri 491	. 2 ⊢ (Unit‘𝑅) = (𝐵 ∖ { 0 })
8	7	eqcomi 2772	1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  {csn 4588  ‘cfv 6558  Basecbs 17416  0_gc0g 17639  Ringcrg 20489  Unitcui 20615  DivRingcdr 21010

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-drng 21012

This theorem is used by:  cnflddiv  21738  cnfldinv  21739  cnsubdrglem  21754  cnmgpabl  21764  cnmsubglem  21766  gzrngunit  21769  zringunit  21802  expghm  21811  psgninv  21918  zrhpsgnmhm  21920  amgmlem  27347  dchrghm  27613  dchrabs  27617  sum2dchr  27631  lgseisenlem4  27735  qrngdiv  27981  proot1ex  44390  amgmwlem  51267  amgmlemALT  51268