Theorem orngogrp 21171

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 21169	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  ∀wral 3080   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  ._rcmulr 17473  lecple 17479  0_gc0g 17660  oGrpcogrp 20385  Ringcrg 20510  oRingcorng 21165

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-orng 21167

This theorem is used by:  orngsqr  21174  ornglmulle  21175  orngrmulle  21176  ofldtos  21181  suborng  21184  ofldchr  21937  isarchiofld  33859  nn0omnd  34009