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

Theorem orngogrp 30801
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 2818 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2818 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2818 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2818 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 30799 . 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 1138 1 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  lecple 16560  0gc0g 16701  Ringcrg 19226  oGrpcogrp 30626  oRingcorng 30795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-orng 30797
This theorem is referenced by:  orngsqr  30804  ornglmulle  30805  orngrmulle  30806  ofldtos  30811  ofldchr  30814  suborng  30815  isarchiofld  30817  nn0omnd  30841
  Copyright terms: Public domain W3C validator