Theorem orngogrp 31500

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 2738	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2738	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2738	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2738	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 31498	. 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 1145	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3064   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  ._rcmulr 16963  lecple 16969  0_gc0g 17150  Ringcrg 19783  oGrpcogrp 31324  oRingcorng 31494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-orng 31496

This theorem is referenced by:  orngsqr  31503  ornglmulle  31504  orngrmulle  31505  ofldtos  31510  ofldchr  31513  suborng  31514  isarchiofld  31516  nn0omnd  31545