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Theorem orngring 29928
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.
StepHypRef Expression
1 eqid 2651 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2651 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2651 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2651 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 29927 . 2 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ (Base‘𝑅)∀𝑏 ∈ (Base‘𝑅)(((0g𝑅)(le‘𝑅)𝑎 ∧ (0g𝑅)(le‘𝑅)𝑏) → (0g𝑅)(le‘𝑅)(𝑎(.r𝑅)𝑏))))
65simp1bi 1096 1 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  lecple 15995  0gc0g 16147  Ringcrg 18593  oGrpcogrp 29826  oRingcorng 29923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-orng 29925
This theorem is referenced by:  orngsqr  29932  ornglmulle  29933  orngrmulle  29934  ornglmullt  29935  orngrmullt  29936  orngmullt  29937  orng0le1  29940  suborng  29943  isarchiofld  29945
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