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Theorem rngogrpo 34034
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 = (1st𝑅)
Assertion
Ref Expression
rngogrpo (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Proof of Theorem rngogrpo
StepHypRef Expression
1 ringgrp.1 . . 3 𝐺 = (1st𝑅)
21rngoablo 34032 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3 ablogrpo 27735 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  cfv 6031  1st c1st 7312  GrpOpcgr 27677  AbelOpcablo 27732  RingOpscrngo 34018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-1st 7314  df-2nd 7315  df-ablo 27733  df-rngo 34019
This theorem is referenced by:  rngone0  34035  rngogcl  34036  rngoaass  34038  rngorcan  34041  rngolcan  34042  rngo0cl  34043  rngo0rid  34044  rngo0lid  34045  rngolz  34046  rngorz  34047  rngosn3  34048  rngonegcl  34051  rngoaddneg1  34052  rngoaddneg2  34053  rngosub  34054  rngodm1dm2  34056  rngorn1  34057  rngonegmn1l  34065  rngonegmn1r  34066  rngogrphom  34095  rngohom0  34096  rngohomsub  34097  rngokerinj  34099  keridl  34156  dmncan1  34200
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