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Theorem rngogrpo 23875
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 23874 . 2  |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
3 ablogrpo 23769 . 2  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
42, 3syl 16 1  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   ` cfv 5416   1stc1st 6573   GrpOpcgr 23671   AbelOpcablo 23766   RingOpscrngo 23860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-1st 6575  df-2nd 6576  df-ablo 23767  df-rngo 23861
This theorem is referenced by:  rngogcl  23876  rngoaass  23878  rngorcan  23881  rngolcan  23882  rngo0cl  23883  rngo0rid  23884  rngo0lid  23885  rngolz  23886  rngorz  23887  rngon0  23901  rngodm1dm2  23903  rngorn1  23904  rngosn3  23911  rngonegcl  28748  rngoaddneg1  28749  rngoaddneg2  28750  rngosub  28751  rngonegmn1l  28752  rngonegmn1r  28753  rngogrphom  28774  rngohom0  28775  rngohomsub  28776  rngokerinj  28778  keridl  28829  dmncan1  28873
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