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Theorem orngogrp 32922
Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngogrp (𝑅 ∈ oRing β†’ 𝑅 ∈ oGrp)

Proof of Theorem orngogrp
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2726 . . 3 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2 eqid 2726 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
3 eqid 2726 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
4 eqid 2726 . . 3 (leβ€˜π‘…) = (leβ€˜π‘…)
51, 2, 3, 4isorng 32920 . 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))))
65simp2bi 1143 1 (𝑅 ∈ oRing β†’ 𝑅 ∈ oGrp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  .rcmulr 17207  lecple 17213  0gc0g 17394  Ringcrg 20138  oGrpcogrp 32722  oRingcorng 32916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-orng 32918
This theorem is referenced by:  orngsqr  32925  ornglmulle  32926  orngrmulle  32927  ofldtos  32932  ofldchr  32935  suborng  32936  isarchiofld  32938  nn0omnd  32963
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