Theorem drngunit 20505

Description: Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
isdrng.b	⊢ 𝐵 = (Base‘𝑅)
isdrng.u	⊢ 𝑈 = (Unit‘𝑅)
isdrng.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
drngunit	⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))

Proof of Theorem drngunit

Step	Hyp	Ref	Expression
1		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
2		isdrng.u	. . . . 5 ⊢ 𝑈 = (Unit‘𝑅)
3		isdrng.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	isdrng 20504	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
5	4	simprbi 495	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
6	5	eleq2d 2817	. 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 })))
7		eldifsn 4789	. 2 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
8	6, 7	bitrdi 286	1 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938   ∖ cdif 3944  {csn 4627  ‘cfv 6542  Basecbs 17148  0_gc0g 17389  Ringcrg 20127  Unitcui 20246  DivRingcdr 20500

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-drng 20502

This theorem is referenced by:  drngunz  20519  drnginvrcl  20522  drnginvrn0  20523  drnginvrl  20525  drnginvrr  20526  issubdrg  20544  sdrgunit  20555  abvdiv  20588  qsssubdrg  21204  redvr  21389  drnguc1p  25923  lgseisenlem3  27116  sdrgdvcl  32667  sdrginvcl  32668  ornglmullt  32695  orngrmullt  32696  isarchiofld  32705  ply1asclunit  32928  qqhval2lem  33259  qqhf  33264  matunitlindf  36789  fldhmf1  41261  lincreslvec3  47250  isldepslvec2  47253