Theorem rngoablo2 35219

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

Step	Hyp	Ref	Expression
1		df-br 5053	. . 3 ⊢ (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2		relrngo 35206	. . . . 5 ⊢ Rel RingOps
3	2	brrelex12i 5593	. . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V))
4		op1stg 7687	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
5	3, 4	syl 17	. . 3 ⊢ (𝐺RingOps𝐻 → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
6	1, 5	sylbir 237	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
7		eqid 2821	. . 3 ⊢ (1st ‘⟨𝐺, 𝐻⟩) = (1st ‘⟨𝐺, 𝐻⟩)
8	7	rngoablo 35218	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) ∈ AbelOp)
9	6, 8	eqeltrrd 2914	1 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3486  ⟨cop 4559   class class class wbr 5052  ‘cfv 6341  1^st c1st 7673  AbelOpcablo 28305  RingOpscrngo 35204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-fv 6349  df-ov 7145  df-1st 7675  df-2nd 7676  df-rngo 35205

This theorem is referenced by:  isdivrngo  35260