Theorem drngunit 19996

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 19995	. . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
5	4	simprbi 497	. . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
6	5	eleq2d 2824	. 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 })))
7		eldifsn 4720	. 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∖ cdif 3884  {csn 4561  ‘cfv 6433  Basecbs 16912  0_gc0g 17150  Ringcrg 19783  Unitcui 19881  DivRingcdr 19991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-drng 19993

This theorem is referenced by:  drngunz  20006  drnginvrcl  20008  drnginvrn0  20009  drnginvrl  20010  drnginvrr  20011  issubdrg  20049  abvdiv  20097  qsssubdrg  20657  redvr  20822  drnguc1p  25335  lgseisenlem3  26525  ornglmullt  31506  orngrmullt  31507  isarchiofld  31516  qqhval2lem  31931  qqhf  31936  matunitlindf  35775  lincreslvec3  45823  isldepslvec2  45826