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Theorem rngoablo 37288
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 β€˜π‘…)
Assertion
Ref Expression
rngoablo (𝑅 ∈ RingOps β†’ 𝐺 ∈ AbelOp)

Proof of Theorem rngoablo
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringabl.1 . . 3 𝐺 = (1st β€˜π‘…)
2 eqid 2726 . . 3 (2nd β€˜π‘…) = (2nd β€˜π‘…)
3 eqid 2726 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 37279 . 2 (𝑅 ∈ RingOps β†’ ((𝐺 ∈ AbelOp ∧ (2nd β€˜π‘…):(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) ∧ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯(2nd β€˜π‘…)𝑦)(2nd β€˜π‘…)𝑧) = (π‘₯(2nd β€˜π‘…)(𝑦(2nd β€˜π‘…)𝑧)) ∧ (π‘₯(2nd β€˜π‘…)(𝑦𝐺𝑧)) = ((π‘₯(2nd β€˜π‘…)𝑦)𝐺(π‘₯(2nd β€˜π‘…)𝑧)) ∧ ((π‘₯𝐺𝑦)(2nd β€˜π‘…)𝑧) = ((π‘₯(2nd β€˜π‘…)𝑧)𝐺(𝑦(2nd β€˜π‘…)𝑧))) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯(2nd β€˜π‘…)𝑦) = 𝑦 ∧ (𝑦(2nd β€˜π‘…)π‘₯) = 𝑦))))
54simplld 765 1 (𝑅 ∈ RingOps β†’ 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064   Γ— cxp 5667  ran crn 5670  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  AbelOpcablo 30301  RingOpscrngo 37274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-1st 7971  df-2nd 7972  df-rngo 37275
This theorem is referenced by:  rngoablo2  37289  rngogrpo  37290  rngocom  37293  rngoa32  37295  rngoa4  37296
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