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Theorem rngoablo 32877
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 2610 . . 3 (2nd𝑅) = (2nd𝑅)
3 eqid 2610 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 32868 . 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 787 1 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  wrex 2897   × cxp 5036  ran crn 5039  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  AbelOpcablo 26782  RingOpscrngo 32863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-1st 7059  df-2nd 7060  df-rngo 32864
This theorem is referenced by:  rngoablo2  32878  rngogrpo  32879  rngocom  32882  rngoa32  32884  rngoa4  32885
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