Theorem orngring 20976

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 20975	. 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 1162	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  ∀wral 3078   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  ._rcmulr 17317  lecple 17323  0_gc0g 17498  oGrpcogrp 20196  Ringcrg 20321  oRingcorng 20971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-orng 20973

This theorem is used by:  orngsqr  20980  ornglmulle  20981  orngrmulle  20982  ornglmullt  20983  orngrmullt  20984  orngmullt  20985  orng0le1  20988  suborng  20990  isarchiofld  33528