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Theorem orngring 30906
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 2824 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2824 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2824 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2824 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 30905 . 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 1142 1 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3133   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  .rcmulr 16566  lecple 16572  0gc0g 16713  Ringcrg 19297  oGrpcogrp 30731  oRingcorng 30901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-orng 30903
This theorem is referenced by:  orngsqr  30910  ornglmulle  30911  orngrmulle  30912  ornglmullt  30913  orngrmullt  30914  orngmullt  30915  orng0le1  30918  suborng  30921  isarchiofld  30923
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