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Theorem rngoablo 37915
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 2737 . . 3 (2nd𝑅) = (2nd𝑅)
3 eqid 2737 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 37906 . 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 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070   × cxp 5683  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  AbelOpcablo 30563  RingOpscrngo 37901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-rngo 37902
This theorem is referenced by:  rngoablo2  37916  rngogrpo  37917  rngocom  37920  rngoa32  37922  rngoa4  37923
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