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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 20153 | . 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
| 2 | 1 | ablgrpd 19805 | 1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Grpcgrp 18952 Rngcrng 20150 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-abl 19802 df-rng 20151 | 
| This theorem is referenced by: rngacl 20160 rng0cl 20161 rngrz 20164 rngmneg1 20165 rngmneg2 20166 rngm2neg 20167 rngsubdi 20169 rngsubdir 20170 prdsrngd 20174 subrngsubg 20553 cntzsubrng 20568 rnglidlmcl 21227 rnglidl0 21240 rnglidl1 21243 2idlcpblrng 21282 rngqiprngimfolem 21301 rngqiprngimf1lem 21305 rngqiprngghm 21310 rngqiprngimf1 21311 rngqiprngimfo 21312 rngqiprngfulem3 21324 rngqiprngfulem4 21325 rngqiprngfulem5 21326 pzriprnglem4 21496 pzriprnglem10 21502 | 
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