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Theorem orngrng 27453
 Description: An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngrng oRing

Proof of Theorem orngrng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3
2 eqid 2467 . . 3
3 eqid 2467 . . 3
4 eqid 2467 . . 3
51, 2, 3, 4isorng 27452 . 2 oRing oGrp
65simp1bi 1011 1 oRing
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1767  wral 2814   class class class wbr 4447  cfv 5586  (class class class)co 6282  cbs 14486  cmulr 14552  cple 14558  c0g 14691  crg 16986  oGrpcogrp 27350  oRingcorng 27448 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-orng 27450 This theorem is referenced by:  orngsqr  27457  ornglmulle  27458  orngrmulle  27459  ornglmullt  27460  orngrmullt  27461  orngmullt  27462  orng0le1  27465  suborng  27468  isarchiofld  27470
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