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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 2818 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2818 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2818 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2818 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isrng 44075 | . 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 1137 | 1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Smgrpcsgrp 17888 Abelcabl 18836 mulGrpcmgp 19168 Rngcrng 44073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-rng0 44074 |
This theorem is referenced by: isringrng 44080 rnglz 44083 isrnghm 44091 isrnghmd 44101 idrnghm 44107 c0rnghm 44112 zrrnghm 44116 |
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