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| Mirrors > Home > MPE Home > Th. List > rngabl | Structured version Visualization version GIF version | ||
| Description: A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.) | 
| Ref | Expression | 
|---|---|
| rngabl | ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 3 | eqid 2737 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 4 | eqid 2737 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 5 | 1, 2, 3, 4 | isrng 20151 | . 2 ⊢ (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ (mulGrp‘𝑅) ∈ Smgrp ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))))) | 
| 6 | 5 | simp1bi 1146 | 1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Smgrpcsgrp 18731 Abelcabl 19799 mulGrpcmgp 20137 Rngcrng 20149 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-rng 20150 | 
| This theorem is referenced by: rnggrp 20155 rnglz 20162 rngansg 20167 prdsrngd 20173 imasrng 20174 isringrng 20284 opprrng 20345 isrnghm 20441 isrnghmd 20451 idrnghm 20458 c0rnghm 20535 zrrnghm 20536 subrngringnsg 20553 issubrng2 20558 rnglidlrng 21257 2idlcpblrng 21281 qus2idrng 21283 rngqiprngimf1lem 21304 rngqiprngimfo 21311 rngqiprngfulem2 21322 rngqiprngfulem4 21324 | 
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