Theorem orngogrp 20966

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 2763	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2763	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2763	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2763	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 20964	. 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 1164	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  ∀wral 3079   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  ._rcmulr 17306  lecple 17312  0_gc0g 17487  oGrpcogrp 20185  Ringcrg 20310  oRingcorng 20960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-orng 20962

This theorem is referenced by:  orngsqr  20969  ornglmulle  20970  orngrmulle  20971  ofldtos  20976  suborng  20979  ofldchr  21726  isarchiofld  33519  nn0omnd  33664