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Theorem orngogrp 29775
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 2620 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2620 . . 3 (0g𝑅) = (0g𝑅)
3 eqid 2620 . . 3 (.r𝑅) = (.r𝑅)
4 eqid 2620 . . 3 (le‘𝑅) = (le‘𝑅)
51, 2, 3, 4isorng 29773 . 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 1075 1 (𝑅 ∈ oRing → 𝑅 ∈ oGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1988  wral 2909   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  .rcmulr 15923  lecple 15929  0gc0g 16081  Ringcrg 18528  oGrpcogrp 29672  oRingcorng 29769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-orng 29771
This theorem is referenced by:  orngsqr  29778  ornglmulle  29779  orngrmulle  29780  ofldtos  29785  ofldchr  29788  suborng  29789  isarchiofld  29791  nn0omnd  29815
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