Theorem rngoi 37893

Description: The properties of a unital ring. (Contributed by Steve Rodriguez, 8-Sep-2007.) (Proof shortened by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringi.1	⊢ 𝐺 = (1st ‘𝑅)
ringi.2	⊢ 𝐻 = (2nd ‘𝑅)
ringi.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
rngoi	⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝑥,𝐻,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑅

Allowed substitution hints:   𝑅(𝑦,𝑧)

Proof of Theorem rngoi

Step	Hyp	Ref	Expression
1		ringi.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		ringi.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
3	1, 2	opeq12i 4842	. . . 4 ⊢ ⟨𝐺, 𝐻⟩ = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩
4		relrngo 37890	. . . . 5 ⊢ Rel RingOps
5		1st2nd 8018	. . . . 5 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
6	4, 5	mpan 690	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
7	3, 6	eqtr4id 2783	. . 3 ⊢ (𝑅 ∈ RingOps → ⟨𝐺, 𝐻⟩ = 𝑅)
8		id 22	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑅 ∈ RingOps)
9	7, 8	eqeltrd 2828	. 2 ⊢ (𝑅 ∈ RingOps → ⟨𝐺, 𝐻⟩ ∈ RingOps)
10	2	fvexi 6872	. . 3 ⊢ 𝐻 ∈ V
11		ringi.3	. . . 4 ⊢ 𝑋 = ran 𝐺
12	11	isrngo 37891	. . 3 ⊢ (𝐻 ∈ V → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
13	10, 12	ax-mp 5	. 2 ⊢ (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
14	9, 13	sylib 218	1 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ⟨cop 4595   × cxp 5636  ran crn 5639  Rel wrel 5643  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  AbelOpcablo 30473  RingOpscrngo 37888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-1st 7968  df-2nd 7969  df-rngo 37889

This theorem is referenced by:  rngosm  37894  rngoid  37896  rngoideu  37897  rngodi  37898  rngodir  37899  rngoass  37900  rngoablo  37902  rngorn1eq  37928  rngomndo  37929