Theorem orngring 32459

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 2732	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2732	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2732	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2732	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 32458	. 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 1145	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∀wral 3061   class class class wbr 5148  ‘cfv 6543  (class class class)co 7411  Basecbs 17146  ._rcmulr 17200  lecple 17206  0_gc0g 17387  Ringcrg 20058  oGrpcogrp 32257  oRingcorng 32454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-orng 32456

This theorem is referenced by:  orngsqr  32463  ornglmulle  32464  orngrmulle  32465  ornglmullt  32466  orngrmullt  32467  orngmullt  32468  orng0le1  32471  suborng  32474  isarchiofld  32476