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Theorem orngrng 27453
Description: An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Assertion
Ref Expression
orngrng  |-  ( R  e. oRing  ->  R  e.  Ring )

Proof of Theorem orngrng
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
3 eqid 2467 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
4 eqid 2467 . . 3  |-  ( le
`  R )  =  ( le `  R
)
51, 2, 3, 4isorng 27452 . 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 ) ) ) )
65simp1bi 1011 1  |-  ( R  e. oRing  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2814   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   .rcmulr 14552   lecple 14558   0gc0g 14691   Ringcrg 16986  oGrpcogrp 27350  oRingcorng 27448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-orng 27450
This theorem is referenced by:  orngsqr  27457  ornglmulle  27458  orngrmulle  27459  ornglmullt  27460  orngrmullt  27461  orngmullt  27462  orng0le1  27465  suborng  27468  isarchiofld  27470
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