Theorem drngui 20966

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 2761	. . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
4		drngui.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
5	2, 3, 4	isdrng 20964	. . . 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 2770	1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896  {csn 4584  ‘cfv 6531  Basecbs 17367  0_gc0g 17590  Ringcrg 20439  Unitcui 20565  DivRingcdr 20960

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-drng 20962

This theorem is used by:  cnflddiv  21688  cnfldinv  21689  cnsubdrglem  21704  cnmgpabl  21714  cnmsubglem  21716  gzrngunit  21719  zringunit  21752  expghm  21761  psgninv  21868  zrhpsgnmhm  21870  amgmlem  27299  dchrghm  27565  dchrabs  27569  sum2dchr  27583  lgseisenlem4  27687  qrngdiv  27933  proot1ex  44156  amgmwlem  50931  amgmlemALT  50932