Theorem rngogrpo 38448

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 38446	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 30839	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 18	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  1^st c1st 7983  GrpOpcgr 30781  AbelOpcablo 30836  RingOpscrngo 38432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-1st 7985  df-2nd 7986  df-ablo 30837  df-rngo 38433

This theorem is referenced by:  rngone0  38449  rngogcl  38450  rngoaass  38452  rngorcan  38455  rngolcan  38456  rngo0cl  38457  rngo0rid  38458  rngo0lid  38459  rngolz  38460  rngorz  38461  rngosn3  38462  rngonegcl  38465  rngoaddneg1  38466  rngoaddneg2  38467  rngosub  38468  rngodm1dm2  38470  rngorn1  38471  rngonegmn1l  38479  rngonegmn1r  38480  rngogrphom  38509  rngohom0  38510  rngohomsub  38511  rngokerinj  38513  keridl  38570  dmncan1  38614