Theorem drngunit 20702

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 20701	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
5	4	simprbi 497	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
6	5	eleq2d 2823	. 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 })))
7		eldifsn 4730	. 2 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
8	6, 7	bitrdi 287	1 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∖ cdif 3887  {csn 4568  ‘cfv 6492  Basecbs 17170  0_gc0g 17393  Ringcrg 20205  Unitcui 20326  DivRingcdr 20697

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-drng 20699

This theorem is referenced by:  drngunz  20715  drnginvrcl  20721  drnginvrn0  20722  drnginvrl  20724  drnginvrr  20725  issubdrg  20748  sdrgunit  20764  abvdiv  20797  ornglmullt  20837  orngrmullt  20838  qsssubdrg  21416  redvr  21607  drnguc1p  26149  lgseisenlem3  27354  fxpsdrg  33251  isarchiofld  33275  sdrgdvcl  33375  sdrginvcl  33376  1arithufd  33623  ply1asclunit  33649  ply1dg1rt  33655  qqhval2lem  34141  qqhf  34146  matunitlindf  37953  fldhmf1  42543  lincreslvec3  48970  isldepslvec2  48973