Theorem rngoablo2 38230

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 5087	. . 3 ⊢ (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2		relrngo 38217	. . . . 5 ⊢ Rel RingOps
3	2	brrelex12i 5686	. . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V))
4		op1stg 7954	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
5	3, 4	syl 17	. . 3 ⊢ (𝐺RingOps𝐻 → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
6	1, 5	sylbir 235	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) = 𝐺)
7		eqid 2737	. . 3 ⊢ (1st ‘⟨𝐺, 𝐻⟩) = (1st ‘⟨𝐺, 𝐻⟩)
8	7	rngoablo 38229	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) ∈ AbelOp)
9	6, 8	eqeltrrd 2838	1 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   class class class wbr 5086  ‘cfv 6499  1^st c1st 7940  AbelOpcablo 30615  RingOpscrngo 38215

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 5232  ax-nul 5242  ax-pr 5376  ax-un 7689

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7370  df-1st 7942  df-2nd 7943  df-rngo 38216

This theorem is referenced by:  isdivrngo  38271