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Theorem drngunit 18521
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 18520 . . . 4 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
54simprbi 478 . . 3 (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
65eleq2d 2672 . 2 (𝑅 ∈ DivRing → (𝑋𝑈𝑋 ∈ (𝐵 ∖ { 0 })))
7 eldifsn 4259 . 2 (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋𝐵𝑋0 ))
86, 7syl6bb 274 1 (𝑅 ∈ DivRing → (𝑋𝑈 ↔ (𝑋𝐵𝑋0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  cdif 3536  {csn 4124  cfv 5790  Basecbs 15641  0gc0g 15869  Ringcrg 18316  Unitcui 18408  DivRingcdr 18516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-drng 18518
This theorem is referenced by:  drngunz  18531  drnginvrcl  18533  drnginvrn0  18534  drnginvrl  18535  drnginvrr  18536  issubdrg  18574  abvdiv  18606  qsssubdrg  19570  redvr  19727  drnguc1p  23651  lgseisenlem3  24819  ornglmullt  28944  orngrmullt  28945  isarchiofld  28954  qqhval2lem  29159  qqhf  29164  matunitlindf  32373  lincreslvec3  42060  isldepslvec2  42063
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