Theorem rngogrpo 36076

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 36074	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 28917	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ‘cfv 6426  1^st c1st 7818  GrpOpcgr 28859  AbelOpcablo 28914  RingOpscrngo 36060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350  ax-un 7578

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-fv 6434  df-ov 7270  df-1st 7820  df-2nd 7821  df-ablo 28915  df-rngo 36061

This theorem is referenced by:  rngone0  36077  rngogcl  36078  rngoaass  36080  rngorcan  36083  rngolcan  36084  rngo0cl  36085  rngo0rid  36086  rngo0lid  36087  rngolz  36088  rngorz  36089  rngosn3  36090  rngonegcl  36093  rngoaddneg1  36094  rngoaddneg2  36095  rngosub  36096  rngodm1dm2  36098  rngorn1  36099  rngonegmn1l  36107  rngonegmn1r  36108  rngogrphom  36137  rngohom0  36138  rngohomsub  36139  rngokerinj  36141  keridl  36198  dmncan1  36242