Theorem rngoablo 38159

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 2737	. . 3 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅)
3		eqid 2737	. . 3 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	rngoi 38150	. 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 768	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   × cxp 5630  ran crn 5633  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  AbelOpcablo 30632  RingOpscrngo 38145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-1st 7943  df-2nd 7944  df-rngo 38146

This theorem is referenced by:  rngoablo2  38160  rngogrpo  38161  rngocom  38164  rngoa32  38166  rngoa4  38167