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Theorem rngoablo2 35295
Description: In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)
Assertion
Ref Expression
rngoablo2 (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)

Proof of Theorem rngoablo2
StepHypRef Expression
1 df-br 5053 . . 3 (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2 relrngo 35282 . . . . 5 Rel RingOps
32brrelex12i 5594 . . . 4 (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V))
4 op1stg 7696 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
53, 4syl 17 . . 3 (𝐺RingOps𝐻 → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
61, 5sylbir 238 . 2 (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
7 eqid 2824 . . 3 (1st ‘⟨𝐺, 𝐻⟩) = (1st ‘⟨𝐺, 𝐻⟩)
87rngoablo 35294 . 2 (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) ∈ AbelOp)
96, 8eqeltrrd 2917 1 (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cop 4556   class class class wbr 5052  cfv 6343  1st c1st 7682  AbelOpcablo 28333  RingOpscrngo 35280
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-1st 7684  df-2nd 7685  df-rngo 35281
This theorem is referenced by:  isdivrngo  35336
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