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Theorem orngogrp 27442
Description: An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngogrp  |-  ( R  e. oRing  ->  R  e. oGrp )

Proof of Theorem orngogrp
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2462 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2462 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
3 eqid 2462 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2462 . . 3  |-  ( le
`  R )  =  ( le `  R
)
51, 2, 3, 4isorng 27440 . 2  |-  ( R  e. oRing 
<->  ( R  e.  Ring  /\  R  e. oGrp  /\  A. a  e.  ( Base `  R ) A. b  e.  ( Base `  R
) ( ( ( 0g `  R ) ( le `  R
) a  /\  ( 0g `  R ) ( le `  R ) b )  ->  ( 0g `  R ) ( le `  R ) ( a ( .r
`  R ) b ) ) ) )
65simp2bi 1007 1  |-  ( R  e. oRing  ->  R  e. oGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   A.wral 2809   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   .rcmulr 14547   lecple 14553   0gc0g 14686   Ringcrg 16981  oGrpcogrp 27338  oRingcorng 27436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-orng 27438
This theorem is referenced by:  orngsqr  27445  ornglmulle  27446  orngrmulle  27447  ofldtos  27452  ofldchr  27455  suborng  27456  isarchiofld  27458  nn0omnd  27482
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