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Theorem rngogrpo 21073
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 21072 . 2  |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
3 ablogrpo 20967 . 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 1632    e. wcel 1696   ` cfv 5271   1stc1st 6136   GrpOpcgr 20869   AbelOpcablo 20964   RingOpscrngo 21058
This theorem is referenced by:  rngogcl  21074  rngoaass  21076  rngorcan  21079  rngolcan  21080  rngo0cl  21081  rngo0rid  21082  rngo0lid  21083  rngolz  21084  rngorz  21085  rngon0  21099  rngodm1dm2  21101  rngorn1  21102  rngosn3  21109  rngodmeqrn  25522  multinv  25525  multinvb  25526  mult2inv  25527  mulinvsca  25583  muldisc  25584  svli2  25587  rngonegcl  26679  rngoaddneg1  26680  rngoaddneg2  26681  rngosub  26682  rngonegmn1l  26683  rngonegmn1r  26684  rngogrphom  26705  rngohom0  26706  rngohomsub  26707  rngokerinj  26709  keridl  26760  dmncan1  26804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-ablo 20965  df-rngo 21059
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