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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoi | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ringi.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringi.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
ringi.3 | ⊢ 𝑋 = ran 𝐺 |
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)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringi.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | ringi.2 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | 1, 2 | opeq12i 4902 | . . . 4 ⊢ 〈𝐺, 𝐻〉 = 〈(1st ‘𝑅), (2nd ‘𝑅)〉 |
4 | relrngo 37805 | . . . . 5 ⊢ Rel RingOps | |
5 | 1st2nd 8076 | . . . . 5 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = 〈(1st ‘𝑅), (2nd ‘𝑅)〉) | |
6 | 4, 5 | mpan 689 | . . . 4 ⊢ (𝑅 ∈ RingOps → 𝑅 = 〈(1st ‘𝑅), (2nd ‘𝑅)〉) |
7 | 3, 6 | eqtr4id 2793 | . . 3 ⊢ (𝑅 ∈ RingOps → 〈𝐺, 𝐻〉 = 𝑅) |
8 | id 22 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑅 ∈ RingOps) | |
9 | 7, 8 | eqeltrd 2838 | . 2 ⊢ (𝑅 ∈ RingOps → 〈𝐺, 𝐻〉 ∈ RingOps) |
10 | 2 | fvexi 6933 | . . 3 ⊢ 𝐻 ∈ V |
11 | ringi.3 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
12 | 11 | isrngo 37806 | . . 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)))) |
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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