Theorem orngring 30873

Description: An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)

Assertion

Ref	Expression
orngring	⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Proof of Theorem orngring

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2821	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2821	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2821	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2821	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 30872	. 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	simp1bi 1141	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∀wral 3138   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  ._rcmulr 16566  lecple 16572  0_gc0g 16713  Ringcrg 19297  oGrpcogrp 30699  oRingcorng 30868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159  df-orng 30870

This theorem is referenced by:  orngsqr  30877  ornglmulle  30878  orngrmulle  30879  ornglmullt  30880  orngrmullt  30881  orngmullt  30882  orng0le1  30885  suborng  30888  isarchiofld  30890