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Mirrors > Home > MPE Home > Th. List > rnggrp | Structured version Visualization version GIF version |
Description: A non-unital ring is a (additive) group. (Contributed by AV, 16-Feb-2025.) |
Ref | Expression |
---|---|
rnggrp | ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngabl 20089 | . 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
2 | 1 | ablgrpd 19735 | 1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Grpcgrp 18884 Rngcrng 20086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-nul 5301 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-iota 6495 df-fv 6551 df-ov 7418 df-abl 19732 df-rng 20087 |
This theorem is referenced by: rngacl 20096 rng0cl 20097 rngrz 20100 rngmneg1 20101 rngmneg2 20102 rngm2neg 20103 rngsubdi 20105 rngsubdir 20106 prdsrngd 20110 subrngsubg 20483 cntzsubrng 20498 rnglidlmcl 21106 rnglidl0 21119 rnglidl1 21122 2idlcpblrng 21159 rngqiprngimfolem 21174 rngqiprngimf1lem 21178 rngqiprngghm 21183 rngqiprngimf1 21184 rngqiprngimfo 21185 rngqiprngfulem3 21197 rngqiprngfulem4 21198 rngqiprngfulem5 21199 pzriprnglem4 21404 pzriprnglem10 21410 |
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