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Theorem rngogrpo 21827
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 21826 . 2  |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
3 ablogrpo 21721 . 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 1649    e. wcel 1717   ` cfv 5395   1stc1st 6287   GrpOpcgr 21623   AbelOpcablo 21718   RingOpscrngo 21812
This theorem is referenced by:  rngogcl  21828  rngoaass  21830  rngorcan  21833  rngolcan  21834  rngo0cl  21835  rngo0rid  21836  rngo0lid  21837  rngolz  21838  rngorz  21839  rngon0  21853  rngodm1dm2  21855  rngorn1  21856  rngosn3  21863  rngonegcl  26253  rngoaddneg1  26254  rngoaddneg2  26255  rngosub  26256  rngonegmn1l  26257  rngonegmn1r  26258  rngogrphom  26279  rngohom0  26280  rngohomsub  26281  rngokerinj  26283  keridl  26334  dmncan1  26378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-1st 6289  df-2nd 6290  df-ablo 21719  df-rngo 21813
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