Theorem rngogrpo 33709

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

Step	Hyp	Ref	Expression
1		ringgrp.1	. . 3 |- G = ( 1st ` R )
2	1	rngoablo 33707	. 2 |- ( R e. RingOps -> G e. AbelOp )
3		ablogrpo 27401	. 2 |- ( G e. AbelOp -> G e. GrpOp )
4	2, 3	syl 17	1 |- ( R e. RingOps -> G e. GrpOp )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   1stc1st 7166   GrpOpcgr 27343   AbelOpcablo 27398   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-ablo 27399  df-rngo 33694

This theorem is referenced by:  rngone0  33710  rngogcl  33711  rngoaass  33713  rngorcan  33716  rngolcan  33717  rngo0cl  33718  rngo0rid  33719  rngo0lid  33720  rngolz  33721  rngorz  33722  rngosn3  33723  rngonegcl  33726  rngoaddneg1  33727  rngoaddneg2  33728  rngosub  33729  rngodm1dm2  33731  rngorn1  33732  rngonegmn1l  33740  rngonegmn1r  33741  rngogrphom  33770  rngohom0  33771  rngohomsub  33772  rngokerinj  33774  keridl  33831  dmncan1  33875