Theorem rngogrpo 34746

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 ∈ RingOps → 𝐺 ∈ GrpOp)

Proof of Theorem rngogrpo

Step	Hyp	Ref	Expression
1		ringgrp.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngoablo 34744	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 28020	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1522   ∈ wcel 2081  ‘cfv 6230  1^st c1st 7548  GrpOpcgr 27962  AbelOpcablo 28017  RingOpscrngo 34730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-fv 6238  df-ov 7024  df-1st 7550  df-2nd 7551  df-ablo 28018  df-rngo 34731

This theorem is referenced by:  rngone0  34747  rngogcl  34748  rngoaass  34750  rngorcan  34753  rngolcan  34754  rngo0cl  34755  rngo0rid  34756  rngo0lid  34757  rngolz  34758  rngorz  34759  rngosn3  34760  rngonegcl  34763  rngoaddneg1  34764  rngoaddneg2  34765  rngosub  34766  rngodm1dm2  34768  rngorn1  34769  rngonegmn1l  34777  rngonegmn1r  34778  rngogrphom  34807  rngohom0  34808  rngohomsub  34809  rngokerinj  34811  keridl  34868  dmncan1  34912