Theorem rngoablo 38951

Description: Obsolete theorem, use ringabl 20519 instead. 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.) (Proof modification 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 2760	. . 3 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2760	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	rngoi 38942	. 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 780	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   × cxp 5649  ran crn 5652  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004  AbelOpcablo 31155  RingOpscrngo 38937

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7427  df-1st 8005  df-2nd 8006  df-rngo 38938

This theorem is used by:  rngoablo2  38952  rngogrpo  38953  rngocom  38956  rngoa32  38958  rngoa4  38959