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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 2738 | . . 3 ⊢ (2nd ‘𝑅) = (2nd ‘𝑅) | |
3 | eqid 2738 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
4 | 1, 2, 3 | rngoi 35700 | . 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 × cxp 5523 ran crn 5526 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 1st c1st 7712 2nd c2nd 7713 AbelOpcablo 28479 RingOpscrngo 35695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-1st 7714 df-2nd 7715 df-rngo 35696 |
This theorem is referenced by: rngoablo2 35710 rngogrpo 35711 rngocom 35714 rngoa32 35716 rngoa4 35717 |
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