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Theorem rngoablo 35709
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 2738 . . 3 (2nd𝑅) = (2nd𝑅)
3 eqid 2738 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 35700 . 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 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wrex 3054   × cxp 5523  ran crn 5526  wf 6335  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  AbelOpcablo 28479  RingOpscrngo 35695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-1st 7714  df-2nd 7715  df-rngo 35696
This theorem is referenced by:  rngoablo2  35710  rngogrpo  35711  rngocom  35714  rngoa32  35716  rngoa4  35717
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