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Theorem rngogrpo 21057
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 21056 . 2  |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
3 ablogrpo 20951 . 2  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
42, 3syl 15 1  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255   1stc1st 6120   GrpOpcgr 20853   AbelOpcablo 20948   RingOpscrngo 21042
This theorem is referenced by:  rngogcl  21058  rngoaass  21060  rngorcan  21063  rngolcan  21064  rngo0cl  21065  rngo0rid  21066  rngo0lid  21067  rngolz  21068  rngorz  21069  rngon0  21083  rngodm1dm2  21085  rngorn1  21086  rngosn3  21093  rngodmeqrn  25419  multinv  25422  multinvb  25423  mult2inv  25424  mulinvsca  25480  muldisc  25481  svli2  25484  rngonegcl  26576  rngoaddneg1  26577  rngoaddneg2  26578  rngosub  26579  rngonegmn1l  26580  rngonegmn1r  26581  rngogrphom  26602  rngohom0  26603  rngohomsub  26604  rngokerinj  26606  keridl  26657  dmncan1  26701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-ablo 20949  df-rngo 21043
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