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Theorem orngogrp 28125
 Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngogrp oRing oGrp

Proof of Theorem orngogrp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2400 . . 3
2 eqid 2400 . . 3
3 eqid 2400 . . 3
4 eqid 2400 . . 3
51, 2, 3, 4isorng 28123 . 2 oRing oGrp
65simp2bi 1011 1 oRing oGrp
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wcel 1840  wral 2751   class class class wbr 4392  cfv 5523  (class class class)co 6232  cbs 14731  cmulr 14800  cple 14806  c0g 14944  crg 17408  oGrpcogrp 28021  oRingcorng 28119 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-nul 4522 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-iota 5487  df-fv 5531  df-ov 6235  df-orng 28121 This theorem is referenced by:  orngsqr  28128  ornglmulle  28129  orngrmulle  28130  ofldtos  28135  ofldchr  28138  suborng  28139  isarchiofld  28141  nn0omnd  28165
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