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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  |-  ( 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 2400 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2400 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
3 eqid 2400 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2400 . . 3  |-  ( le
`  R )  =  ( le `  R
)
51, 2, 3, 4isorng 28123 . 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 1011 1  |-  ( R  e. oRing  ->  R  e. oGrp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1840   A.wral 2751   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   .rcmulr 14800   lecple 14806   0gc0g 14944   Ringcrg 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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