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Theorem rngogrpo 25096
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 25095 . 2  |-  ( R  e.  RingOps  ->  G  e.  AbelOp )
3 ablogrpo 24990 . 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 1379    e. wcel 1767   ` cfv 5588   1stc1st 6782   GrpOpcgr 24892   AbelOpcablo 24987   RingOpscrngo 25081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-1st 6784  df-2nd 6785  df-ablo 24988  df-rngo 25082
This theorem is referenced by:  rngogcl  25097  rngoaass  25099  rngorcan  25102  rngolcan  25103  rngo0cl  25104  rngo0rid  25105  rngo0lid  25106  rngolz  25107  rngorz  25108  rngon0  25122  rngodm1dm2  25124  rngorn1  25125  rngosn3  25132  rngonegcl  29979  rngoaddneg1  29980  rngoaddneg2  29981  rngosub  29982  rngonegmn1l  29983  rngonegmn1r  29984  rngogrphom  30005  rngohom0  30006  rngohomsub  30007  rngokerinj  30009  keridl  30060  dmncan1  30104
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