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Theorem orngogrp 33048
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 2728 . . 3 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2 eqid 2728 . . 3 (0gβ€˜π‘…) = (0gβ€˜π‘…)
3 eqid 2728 . . 3 (.rβ€˜π‘…) = (.rβ€˜π‘…)
4 eqid 2728 . . 3 (leβ€˜π‘…) = (leβ€˜π‘…)
51, 2, 3, 4isorng 33046 . 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 394   ∈ wcel 2098  βˆ€wral 3058   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  .rcmulr 17243  lecple 17249  0gc0g 17430  Ringcrg 20187  oGrpcogrp 32807  oRingcorng 33042
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 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-orng 33044
This theorem is referenced by:  orngsqr  33051  ornglmulle  33052  orngrmulle  33053  ofldtos  33058  ofldchr  33061  suborng  33062  isarchiofld  33064  nn0omnd  33089
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