Theorem rngoablo 37875

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 = (1st ‘𝑅)

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.

Step	Hyp	Ref	Expression
1		ringabl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2		eqid 2729	. . 3 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2729	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	rngoi 37866	. 2 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ (2nd ‘𝑅):(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥(2nd ‘𝑅)𝑦)(2nd ‘𝑅)𝑧) = (𝑥(2nd ‘𝑅)(𝑦(2nd ‘𝑅)𝑧)) ∧ (𝑥(2nd ‘𝑅)(𝑦𝐺𝑧)) = ((𝑥(2nd ‘𝑅)𝑦)𝐺(𝑥(2nd ‘𝑅)𝑧)) ∧ ((𝑥𝐺𝑦)(2nd ‘𝑅)𝑧) = ((𝑥(2nd ‘𝑅)𝑧)𝐺(𝑦(2nd ‘𝑅)𝑧))) ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥(2nd ‘𝑅)𝑦) = 𝑦 ∧ (𝑦(2nd ‘𝑅)𝑥) = 𝑦))))
5	4	simplld 767	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   × cxp 5629  ran crn 5632  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  AbelOpcablo 30446  RingOpscrngo 37861

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-1st 7947  df-2nd 7948  df-rngo 37862

This theorem is referenced by:  rngoablo2  37876  rngogrpo  37877  rngocom  37880  rngoa32  37882  rngoa4  37883