Theorem rngogrpo 37958

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 37956	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 30527	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  1^st c1st 7919  GrpOpcgr 30469  AbelOpcablo 30524  RingOpscrngo 37942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-1st 7921  df-2nd 7922  df-ablo 30525  df-rngo 37943

This theorem is referenced by:  rngone0  37959  rngogcl  37960  rngoaass  37962  rngorcan  37965  rngolcan  37966  rngo0cl  37967  rngo0rid  37968  rngo0lid  37969  rngolz  37970  rngorz  37971  rngosn3  37972  rngonegcl  37975  rngoaddneg1  37976  rngoaddneg2  37977  rngosub  37978  rngodm1dm2  37980  rngorn1  37981  rngonegmn1l  37989  rngonegmn1r  37990  rngogrphom  38019  rngohom0  38020  rngohomsub  38021  rngokerinj  38023  keridl  38080  dmncan1  38124