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Theorem orngring 32418
Description: An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngring (𝑅 ∈ oRing β†’ 𝑅 ∈ Ring)

Proof of Theorem orngring
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2 eqid 2733 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
3 eqid 2733 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
4 eqid 2733 . . 3 (leβ€˜π‘…) = (leβ€˜π‘…)
51, 2, 3, 4isorng 32417 . 2 (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ βˆ€π‘Ž ∈ (Baseβ€˜π‘…)βˆ€π‘ ∈ (Baseβ€˜π‘…)(((0gβ€˜π‘…)(leβ€˜π‘…)π‘Ž ∧ (0gβ€˜π‘…)(leβ€˜π‘…)𝑏) β†’ (0gβ€˜π‘…)(leβ€˜π‘…)(π‘Ž(.rβ€˜π‘…)𝑏))))
65simp1bi 1146 1 (𝑅 ∈ oRing β†’ 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∈ wcel 2107  βˆ€wral 3062   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  lecple 17204  0gc0g 17385  Ringcrg 20056  oGrpcogrp 32216  oRingcorng 32413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-orng 32415
This theorem is referenced by:  orngsqr  32422  ornglmulle  32423  orngrmulle  32424  ornglmullt  32425  orngrmullt  32426  orngmullt  32427  orng0le1  32430  suborng  32433  isarchiofld  32435
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