MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  drngunit Structured version   Visualization version   GIF version

Theorem drngunit 18575
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 18574 . . . 4 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
54simprbi 479 . . 3 (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 }))
65eleq2d 2673 . 2 (𝑅 ∈ DivRing → (𝑋𝑈𝑋 ∈ (𝐵 ∖ { 0 })))
7 eldifsn 4260 . 2 (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋𝐵𝑋0 ))
86, 7syl6bb 275 1 (𝑅 ∈ DivRing → (𝑋𝑈 ↔ (𝑋𝐵𝑋0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  cdif 3537  {csn 4125  cfv 5804  Basecbs 15695  0gc0g 15923  Ringcrg 18370  Unitcui 18462  DivRingcdr 18570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-drng 18572
This theorem is referenced by:  drngunz  18585  drnginvrcl  18587  drnginvrn0  18588  drnginvrl  18589  drnginvrr  18590  issubdrg  18628  abvdiv  18660  qsssubdrg  19624  redvr  19782  drnguc1p  23734  lgseisenlem3  24902  ornglmullt  29138  orngrmullt  29139  isarchiofld  29148  qqhval2lem  29353  qqhf  29358  matunitlindf  32577  lincreslvec3  42065  isldepslvec2  42068
  Copyright terms: Public domain W3C validator