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Theorem rngoablo 33339
 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 2621 . . 3 (2nd𝑅) = (2nd𝑅)
3 eqid 2621 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 33330 . 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 790 1 (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908   × cxp 5072  ran crn 5075  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  AbelOpcablo 27247  RingOpscrngo 33325 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-rngo 33326 This theorem is referenced by:  rngoablo2  33340  rngogrpo  33341  rngocom  33344  rngoa32  33346  rngoa4  33347
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