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Theorem rngoablo 37895
Description: A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ringabl.1 𝐺 = (1st𝑅)
Assertion
Ref Expression
rngoablo (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)

Proof of Theorem rngoablo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringabl.1 . . 3 𝐺 = (1st𝑅)
2 eqid 2735 . . 3 (2nd𝑅) = (2nd𝑅)
3 eqid 2735 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 37886 . 2 (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ (2nd𝑅):(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥(2nd𝑅)𝑦)(2nd𝑅)𝑧) = (𝑥(2nd𝑅)(𝑦(2nd𝑅)𝑧)) ∧ (𝑥(2nd𝑅)(𝑦𝐺𝑧)) = ((𝑥(2nd𝑅)𝑦)𝐺(𝑥(2nd𝑅)𝑧)) ∧ ((𝑥𝐺𝑦)(2nd𝑅)𝑧) = ((𝑥(2nd𝑅)𝑧)𝐺(𝑦(2nd𝑅)𝑧))) ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥(2nd𝑅)𝑦) = 𝑦 ∧ (𝑦(2nd𝑅)𝑥) = 𝑦))))
54simplld 768 1 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068   × cxp 5687  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  AbelOpcablo 30573  RingOpscrngo 37881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014  df-rngo 37882
This theorem is referenced by:  rngoablo2  37896  rngogrpo  37897  rngocom  37900  rngoa32  37902  rngoa4  37903
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