Theorem orngring 29928

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 2651	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2651	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2651	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2651	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 29927	. 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 1096	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030  ∀wral 2941   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  ._rcmulr 15989  lecple 15995  0_gc0g 16147  Ringcrg 18593  oGrpcogrp 29826  oRingcorng 29923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-orng 29925

This theorem is referenced by:  orngsqr  29932  ornglmulle  29933  orngrmulle  29934  ornglmullt  29935  orngrmullt  29936  orngmullt  29937  orng0le1  29940  suborng  29943  isarchiofld  29945