Theorem orngring 20884

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 2756	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2756	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		eqid 2756	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2756	. . 3 ⊢ (le‘𝑅) = (le‘𝑅)
5	1, 2, 3, 4	isorng 20883	. 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 1154	1 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2136  ∀wral 3070   class class class wbr 5094  ‘cfv 6510  (class class class)co 7385  Basecbs 17221  ._rcmulr 17263  lecple 17269  0_gc0g 17444  oGrpcogrp 20136  Ringcrg 20255  oRingcorng 20879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-nul 5250

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-iota 6466  df-fv 6518  df-ov 7388  df-orng 20881

This theorem is referenced by:  orngsqr  20888  ornglmulle  20889  orngrmulle  20890  ornglmullt  20891  orngrmullt  20892  orngmullt  20893  orng0le1  20896  suborng  20898  isarchiofld  33333