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Theorem orngogrp 30882
Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngogrp (𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Proof of Theorem orngogrp
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2821 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2821 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2821 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 30880 . 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))))
65simp2bi 1143 1 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3126   class class class wbr 5039  cfv 6328  (class class class)co 7130  Basecbs 16462  .rcmulr 16545  lecple 16551  0gc0g 16692  Ringcrg 19276  oGrpcogrp 30707  oRingcorng 30876
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 2178  ax-ext 2793  ax-nul 5183
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-orng 30878
This theorem is referenced by:  orngsqr  30885  ornglmulle  30886  orngrmulle  30887  ofldtos  30892  ofldchr  30895  suborng  30896  isarchiofld  30898  nn0omnd  30922
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