Theorem orngring 21032

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 2762	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2762	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2762	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2762	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 21031	. 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 1163	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  ._rcmulr 17347  lecple 17353  0_gc0g 17528  oGrpcogrp 20251  Ringcrg 20376  oRingcorng 21027

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-orng 21029

This theorem is used by:  orngsqr  21036  ornglmulle  21037  orngrmulle  21038  ornglmullt  21039  orngrmullt  21040  orngmullt  21041  orng0le1  21044  suborng  21046  isarchiofld  33641