Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  orngring Structured version   Visualization version   GIF version

Theorem orngring 30552
 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 2772 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2772 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2772 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2772 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 30551 . 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 1125 1 (𝑅 ∈ oRing → 𝑅 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∈ wcel 2050  ∀wral 3082   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970  Basecbs 16333  .rcmulr 16416  lecple 16422  0gc0g 16563  Ringcrg 19014  oGrpcogrp 30417  oRingcorng 30547 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744  ax-nul 5061 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-iota 6146  df-fv 6190  df-ov 6973  df-orng 30549 This theorem is referenced by:  orngsqr  30556  ornglmulle  30557  orngrmulle  30558  ornglmullt  30559  orngrmullt  30560  orngmullt  30561  orng0le1  30564  suborng  30567  isarchiofld  30569
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