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Theorem rngogrpo 21970
Description: A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
ringgrp.1  |-  G  =  ( 1st `  R
)
Assertion
Ref Expression
rngogrpo  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )

Proof of Theorem rngogrpo
StepHypRef Expression
1 ringgrp.1 . . 3  |-  G  =  ( 1st `  R
)
21rngoablo 21969 . 2  |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
3 ablogrpo 21864 . 2  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
42, 3syl 16 1  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446   1stc1st 6339   GrpOpcgr 21766   AbelOpcablo 21861   RingOpscrngo 21955
This theorem is referenced by:  rngogcl  21971  rngoaass  21973  rngorcan  21976  rngolcan  21977  rngo0cl  21978  rngo0rid  21979  rngo0lid  21980  rngolz  21981  rngorz  21982  rngon0  21996  rngodm1dm2  21998  rngorn1  21999  rngosn3  22006  rngonegcl  26542  rngoaddneg1  26543  rngoaddneg2  26544  rngosub  26545  rngonegmn1l  26546  rngonegmn1r  26547  rngogrphom  26568  rngohom0  26569  rngohomsub  26570  rngokerinj  26572  keridl  26623  dmncan1  26667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-ablo 21862  df-rngo 21956
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