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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 2824 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2824 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2824 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2824 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isrng 44154 | . 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 1141 | 1 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 +gcplusg 16568 .rcmulr 16569 Smgrpcsgrp 17903 Abelcabl 18910 mulGrpcmgp 19242 Rngcrng 44152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-rng0 44153 |
This theorem is referenced by: isringrng 44159 rnglz 44162 isrnghm 44170 isrnghmd 44180 idrnghm 44186 c0rnghm 44191 zrrnghm 44195 |
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