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Theorem rngogrpo 35347
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 35345 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3 ablogrpo 28334 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cfv 6328  1st c1st 7673  GrpOpcgr 28276  AbelOpcablo 28331  RingOpscrngo 35331
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-1st 7675  df-2nd 7676  df-ablo 28332  df-rngo 35332
This theorem is referenced by:  rngone0  35348  rngogcl  35349  rngoaass  35351  rngorcan  35354  rngolcan  35355  rngo0cl  35356  rngo0rid  35357  rngo0lid  35358  rngolz  35359  rngorz  35360  rngosn3  35361  rngonegcl  35364  rngoaddneg1  35365  rngoaddneg2  35366  rngosub  35367  rngodm1dm2  35369  rngorn1  35370  rngonegmn1l  35378  rngonegmn1r  35379  rngogrphom  35408  rngohom0  35409  rngohomsub  35410  rngokerinj  35412  keridl  35469  dmncan1  35513
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