Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  orngogrp Structured version   Visualization version   GIF version

Theorem orngogrp 29132
 Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngogrp (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Proof of Theorem orngogrp
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2610 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2610 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2610 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 29130 . 2 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)(((0g𝑅)(le‘𝑅)𝑎 ∧ (0g𝑅)(le‘𝑅)𝑏) → (0g𝑅)(le‘𝑅)(𝑎(.r𝑅)𝑏))))
65simp2bi 1070 1 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  lecple 15775  0gc0g 15923  Ringcrg 18370  oGrpcogrp 29029  oRingcorng 29126 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  ax-nul 4717 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-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-ov 6552  df-orng 29128 This theorem is referenced by:  orngsqr  29135  ornglmulle  29136  orngrmulle  29137  ofldtos  29142  ofldchr  29145  suborng  29146  isarchiofld  29148  nn0omnd  29172
 Copyright terms: Public domain W3C validator