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Theorem rngoablo 37414
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 2728 . . 3 (2nd β€˜π‘…) = (2nd β€˜π‘…)
3 eqid 2728 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3rngoi 37405 . 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 766 1 (𝑅 ∈ RingOps β†’ 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067   Γ— cxp 5680  ran crn 5683  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  AbelOpcablo 30374  RingOpscrngo 37400
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-1st 7999  df-2nd 8000  df-rngo 37401
This theorem is referenced by:  rngoablo2  37415  rngogrpo  37416  rngocom  37419  rngoa32  37421  rngoa4  37422
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