Theorem orngring 20934

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 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 20933	. 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 1161	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ∀wral 3079   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  ._rcmulr 17301  lecple 17307  0_gc0g 17482  oGrpcogrp 20181  Ringcrg 20306  oRingcorng 20929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-orng 20931

This theorem is referenced by:  orngsqr  20938  ornglmulle  20939  orngrmulle  20940  ornglmullt  20941  orngrmullt  20942  orngmullt  20943  orng0le1  20946  suborng  20948  isarchiofld  33432