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Theorem rngogrpo 36255
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 36253 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3 ablogrpo 29275 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6492  1st c1st 7910  GrpOpcgr 29217  AbelOpcablo 29272  RingOpscrngo 36239
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7353  df-1st 7912  df-2nd 7913  df-ablo 29273  df-rngo 36240
This theorem is referenced by:  rngone0  36256  rngogcl  36257  rngoaass  36259  rngorcan  36262  rngolcan  36263  rngo0cl  36264  rngo0rid  36265  rngo0lid  36266  rngolz  36267  rngorz  36268  rngosn3  36269  rngonegcl  36272  rngoaddneg1  36273  rngoaddneg2  36274  rngosub  36275  rngodm1dm2  36277  rngorn1  36278  rngonegmn1l  36286  rngonegmn1r  36287  rngogrphom  36316  rngohom0  36317  rngohomsub  36318  rngokerinj  36320  keridl  36377  dmncan1  36421
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