Theorem drngunit 20230

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 20229	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
5	4	simprbi 497	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
6	5	eleq2d 2818	. 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 })))
7		eldifsn 4752	. 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 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939   ∖ cdif 3910  {csn 4591  ‘cfv 6501  Basecbs 17094  0_gc0g 17335  Ringcrg 19978  Unitcui 20082  DivRingcdr 20225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-drng 20227

This theorem is referenced by:  drngunz  20243  drnginvrcl  20246  drnginvrn0  20247  drnginvrl  20249  drnginvrr  20250  issubdrg  20296  sdrgunit  20319  abvdiv  20352  qsssubdrg  20893  redvr  21058  drnguc1p  25572  lgseisenlem3  26762  sdrgdvcl  32145  sdrginvcl  32146  ornglmullt  32173  orngrmullt  32174  isarchiofld  32183  qqhval2lem  32651  qqhf  32656  matunitlindf  36149  fldhmf1  40620  lincreslvec3  46683  isldepslvec2  46686