Theorem rngoablo2 35189

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 5069	. . 3 ⊢ (𝐺RingOps𝐻 ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps)
2		relrngo 35176	. . . . 5 ⊢ Rel RingOps
3	2	brrelex12i 5609	. . . 4 ⊢ (𝐺RingOps𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V))
4		op1stg 7703	. . . 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 2823	. . 3 ⊢ (1st ‘⟨𝐺, 𝐻⟩) = (1st ‘⟨𝐺, 𝐻⟩)
8	7	rngoablo 35188	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → (1st ‘⟨𝐺, 𝐻⟩) ∈ AbelOp)
9	6, 8	eqeltrrd 2916	1 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575   class class class wbr 5068  ‘cfv 6357  1^st c1st 7689  AbelOpcablo 28323  RingOpscrngo 35174

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 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-1st 7691  df-2nd 7692  df-rngo 35175

This theorem is referenced by:  isdivrngo  35230