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Theorem rngoi 37808
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 = (1st𝑅)
ringi.2 𝐻 = (2nd𝑅)
ringi.3 𝑋 = ran 𝐺
Assertion
Ref Expression
rngoi (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((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
StepHypRef Expression
1 ringi.1 . . . . 5 𝐺 = (1st𝑅)
2 ringi.2 . . . . 5 𝐻 = (2nd𝑅)
31, 2opeq12i 4902 . . . 4 𝐺, 𝐻⟩ = ⟨(1st𝑅), (2nd𝑅)⟩
4 relrngo 37805 . . . . 5 Rel RingOps
5 1st2nd 8076 . . . . 5 ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
64, 5mpan 689 . . . 4 (𝑅 ∈ RingOps → 𝑅 = ⟨(1st𝑅), (2nd𝑅)⟩)
73, 6eqtr4id 2793 . . 3 (𝑅 ∈ RingOps → ⟨𝐺, 𝐻⟩ = 𝑅)
8 id 22 . . 3 (𝑅 ∈ RingOps → 𝑅 ∈ RingOps)
97, 8eqeltrd 2838 . 2 (𝑅 ∈ RingOps → ⟨𝐺, 𝐻⟩ ∈ RingOps)
102fvexi 6933 . . 3 𝐻 ∈ V
11 ringi.3 . . . 4 𝑋 = ran 𝐺
1211isrngo 37806 . . 3 (𝐻 ∈ V → (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))))
1310, 12ax-mp 5 . 2 (⟨𝐺, 𝐻⟩ ∈ RingOps ↔ ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
149, 13sylib 218 1 (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  wrex 3072  Vcvv 3482  cop 4654   × cxp 5697  ran crn 5700  Rel wrel 5704  wf 6568  cfv 6572  (class class class)co 7445  1st c1st 8024  2nd c2nd 8025  AbelOpcablo 30567  RingOpscrngo 37803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fv 6580  df-ov 7448  df-1st 8026  df-2nd 8027  df-rngo 37804
This theorem is referenced by:  rngosm  37809  rngoid  37811  rngoideu  37812  rngodi  37813  rngodir  37814  rngoass  37815  rngoablo  37817  rngorn1eq  37843  rngomndo  37844
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