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Theorem rngogrpo 33338
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 33336 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3 ablogrpo 27247 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cfv 5847  1st c1st 7111  GrpOpcgr 27189  AbelOpcablo 27244  RingOpscrngo 33322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-1st 7113  df-2nd 7114  df-ablo 27245  df-rngo 33323
This theorem is referenced by:  rngone0  33339  rngogcl  33340  rngoaass  33342  rngorcan  33345  rngolcan  33346  rngo0cl  33347  rngo0rid  33348  rngo0lid  33349  rngolz  33350  rngorz  33351  rngosn3  33352  rngonegcl  33355  rngoaddneg1  33356  rngoaddneg2  33357  rngosub  33358  rngodm1dm2  33360  rngorn1  33361  rngonegmn1l  33369  rngonegmn1r  33370  rngogrphom  33399  rngohom0  33400  rngohomsub  33401  rngokerinj  33403  keridl  33460  dmncan1  33504
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