Theorem orngogrp 20838

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 2741	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2741	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2741	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2741	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 20836	. 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 1153	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2121  ∀wral 3055   class class class wbr 5074  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  ._rcmulr 17216  lecple 17222  0_gc0g 17397  oGrpcogrp 20089  Ringcrg 20208  oRingcorng 20832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5230

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3725  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362  df-orng 20834

This theorem is referenced by:  orngsqr  20841  ornglmulle  20842  orngrmulle  20843  ofldtos  20848  suborng  20851  ofldchr  21554  isarchiofld  33282  nn0omnd  33429