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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoablo | Structured version Visualization version GIF version |
Description: A ring's addition operation is an Abelian group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringabl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
Ref | Expression |
---|---|
rngoablo | ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringabl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | eqid 2735 | . . 3 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2735 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
4 | 1, 2, 3 | rngoi 37886 | . 2 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ (2nd ‘𝑅):(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥(2nd ‘𝑅)𝑦)(2nd ‘𝑅)𝑧) = (𝑥(2nd ‘𝑅)(𝑦(2nd ‘𝑅)𝑧)) ∧ (𝑥(2nd ‘𝑅)(𝑦𝐺𝑧)) = ((𝑥(2nd ‘𝑅)𝑦)𝐺(𝑥(2nd ‘𝑅)𝑧)) ∧ ((𝑥𝐺𝑦)(2nd ‘𝑅)𝑧) = ((𝑥(2nd ‘𝑅)𝑧)𝐺(𝑦(2nd ‘𝑅)𝑧))) ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥(2nd ‘𝑅)𝑦) = 𝑦 ∧ (𝑦(2nd ‘𝑅)𝑥) = 𝑦)))) |
5 | 4 | simplld 768 | 1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 × cxp 5687 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 AbelOpcablo 30573 RingOpscrngo 37881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-1st 8013 df-2nd 8014 df-rngo 37882 |
This theorem is referenced by: rngoablo2 37896 rngogrpo 37897 rngocom 37900 rngoa32 37902 rngoa4 37903 |
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