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Theorem drngunit 18954
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 = (0g𝑅)
Assertion
Ref Expression
drngunit (𝑅 ∈ DivRing → (𝑋𝑈 ↔ (𝑋𝐵𝑋0 )))

Proof of Theorem drngunit
StepHypRef Expression
1 isdrng.b . . . . 5 𝐵 = (Base‘𝑅)
2 isdrng.u . . . . 5 𝑈 = (Unit‘𝑅)
3 isdrng.z . . . . 5 0 = (0g𝑅)
41, 2, 3isdrng 18953 . . . 4 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
54simprbi 483 . . 3 (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
65eleq2d 2825 . 2 (𝑅 ∈ DivRing → (𝑋𝑈𝑋 ∈ (𝐵 ∖ { 0 })))
7 eldifsn 4462 . 2 (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋𝐵𝑋0 ))
86, 7syl6bb 276 1 (𝑅 ∈ DivRing → (𝑋𝑈 ↔ (𝑋𝐵𝑋0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  cdif 3712  {csn 4321  cfv 6049  Basecbs 16059  0gc0g 16302  Ringcrg 18747  Unitcui 18839  DivRingcdr 18949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-drng 18951
This theorem is referenced by:  drngunz  18964  drnginvrcl  18966  drnginvrn0  18967  drnginvrl  18968  drnginvrr  18969  issubdrg  19007  abvdiv  19039  qsssubdrg  20007  redvr  20165  drnguc1p  24129  lgseisenlem3  25301  ornglmullt  30116  orngrmullt  30117  isarchiofld  30126  qqhval2lem  30334  qqhf  30339  matunitlindf  33720  lincreslvec3  42781  isldepslvec2  42784
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