Theorem orngring 30552

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 2772	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2772	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2772	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2772	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 30551	. 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 1125	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 387   ∈ wcel 2050  ∀wral 3082   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970  Basecbs 16333  ._rcmulr 16416  lecple 16422  0_gc0g 16563  Ringcrg 19014  oGrpcogrp 30417  oRingcorng 30547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744  ax-nul 5061

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-iota 6146  df-fv 6190  df-ov 6973  df-orng 30549

This theorem is referenced by:  orngsqr  30556  ornglmulle  30557  orngrmulle  30558  ornglmullt  30559  orngrmullt  30560  orngmullt  30561  orng0le1  30564  suborng  30567  isarchiofld  30569