Theorem orngogrp 20996

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 2765	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2765	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2765	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2765	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 20994	. 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 2146  ∀wral 3081   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  ._rcmulr 17329  lecple 17335  0_gc0g 17510  oGrpcogrp 20214  Ringcrg 20339  oRingcorng 20990

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7419  df-orng 20992

This theorem is used by:  orngsqr  20999  ornglmulle  21000  orngrmulle  21001  ofldtos  21006  suborng  21009  ofldchr  21756  isarchiofld  33559  nn0omnd  33704