Theorem orngogrp 20940

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.

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2769	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2769	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2769	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 20938	. 2 ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)(((0g‘𝑅)(le‘𝑅)𝑎 ∧ (0g‘𝑅)(le‘𝑅)𝑏) → (0g‘𝑅)(le‘𝑅)(𝑎(.r‘𝑅)𝑏))))
6	5	simp2bi 1162	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  ∀wral 3085   class class class wbr 5110  ‘cfv 6533  (class class class)co 7408  Basecbs 17265  ._rcmulr 17307  lecple 17313  0_gc0g 17488  oGrpcogrp 20186  Ringcrg 20311  oRingcorng 20934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6489  df-fv 6541  df-ov 7411  df-orng 20936

This theorem is referenced by:  orngsqr  20943  ornglmulle  20944  orngrmulle  20945  ofldtos  20950  suborng  20953  ofldchr  21691  isarchiofld  33456  nn0omnd  33603