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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 2735 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2735 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2735 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2735 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isrng 20172 | . 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 1144 | 1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Smgrpcsgrp 18744 Abelcabl 19814 mulGrpcmgp 20152 Rngcrng 20170 |
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-ext 2706 ax-nul 5312 |
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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 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-iota 6516 df-fv 6571 df-ov 7434 df-rng 20171 |
This theorem is referenced by: rnggrp 20176 rnglz 20183 rngansg 20188 prdsrngd 20194 imasrng 20195 isringrng 20301 opprrng 20362 isrnghm 20458 isrnghmd 20468 idrnghm 20475 c0rnghm 20552 zrrnghm 20553 subrngringnsg 20570 issubrng2 20575 rnglidlrng 21275 2idlcpblrng 21299 qus2idrng 21301 rngqiprngimf1lem 21322 rngqiprngimfo 21329 rngqiprngfulem2 21340 rngqiprngfulem4 21342 |
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