Theorem drngui 18959

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 2771	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
4		drngui.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
5	2, 3, 4	isdrng 18957	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })))
6	1, 5	mpbi 220	. . 3 ⊢ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))
7	6	simpri 473	. 2 ⊢ (Unit‘𝑅) = (𝐵 ∖ { 0 })
8	7	eqcomi 2780	1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Syntax hints:   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ∖ cdif 3720  {csn 4316  ‘cfv 6029  Basecbs 16060  0_gc0g 16304  Ringcrg 18751  Unitcui 18843  DivRingcdr 18953

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 5992  df-fv 6037  df-drng 18955

This theorem is referenced by:  cnflddiv  19987  cnfldinv  19988  cnsubdrglem  20008  cnmgpabl  20018  cnmsubglem  20020  gzrngunit  20023  zringunit  20047  expghm  20055  psgninv  20139  zrhpsgnmhm  20141  amgmlem  24933  dchrghm  25198  dchrabs  25202  sum2dchr  25216  lgseisenlem4  25320  qrngdiv  25530  proot1ex  38302  amgmwlem  43076  amgmlemALT  43077