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Theorem rngogrpo 36264
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 36262 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3 ablogrpo 29287 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6491  1st c1st 7909  GrpOpcgr 29229  AbelOpcablo 29284  RingOpscrngo 36248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7662
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-ov 7352  df-1st 7911  df-2nd 7912  df-ablo 29285  df-rngo 36249
This theorem is referenced by:  rngone0  36265  rngogcl  36266  rngoaass  36268  rngorcan  36271  rngolcan  36272  rngo0cl  36273  rngo0rid  36274  rngo0lid  36275  rngolz  36276  rngorz  36277  rngosn3  36278  rngonegcl  36281  rngoaddneg1  36282  rngoaddneg2  36283  rngosub  36284  rngodm1dm2  36286  rngorn1  36287  rngonegmn1l  36295  rngonegmn1r  36296  rngogrphom  36325  rngohom0  36326  rngohomsub  36327  rngokerinj  36329  keridl  36386  dmncan1  36430
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