Theorem orngogrp 29801

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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2622	. . 3 |- ( 0g ` R ) = ( 0g ` R )
3		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
4		eqid 2622	. . 3 |- ( le ` R ) = ( le ` R )
5	1, 2, 3, 4	isorng 29799	. 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 ) ) ) )
6	5	simp2bi 1077	1 |- ( R e. oRing -> R e. oGrp )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   lecple 15948   0gc0g 16100   Ringcrg 18547  oGrpcogrp 29698  oRingcorng 29795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-orng 29797

This theorem is referenced by:  orngsqr  29804  ornglmulle  29805  orngrmulle  29806  ofldtos  29811  ofldchr  29814  suborng  29815  isarchiofld  29817  nn0omnd  29841