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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 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2738 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2738 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isrng 45434 | . 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 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 .rcmulr 16963 Smgrpcsgrp 18374 Abelcabl 19387 mulGrpcmgp 19720 Rngcrng 45432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-rng0 45433 |
This theorem is referenced by: isringrng 45439 rnglz 45442 isrnghm 45450 isrnghmd 45460 idrnghm 45466 c0rnghm 45471 zrrnghm 45475 |
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