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Mirrors > Home > MPE Home > Th. List > isabld | Structured version Visualization version GIF version |
Description: Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.) |
Ref | Expression |
---|---|
isabld.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
isabld.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
isabld.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
isabld.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
Ref | Expression |
---|---|
isabld | ⊢ (𝜑 → 𝐺 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isabld.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | isabld.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
3 | isabld.p | . . 3 ⊢ (𝜑 → + = (+g‘𝐺)) | |
4 | grpmnd 18048 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
6 | isabld.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
7 | 2, 3, 5, 6 | iscmnd 18848 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
8 | isabl 18839 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
9 | 1, 7, 8 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mndcmnd 17899 Grpcgrp 18041 CMndccmn 18835 Abelcabl 18836 |
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 |
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-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 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-grp 18044 df-cmn 18837 df-abl 18838 |
This theorem is referenced by: subgabl 18885 gex2abl 18900 cygabl 18939 cygablOLD 18940 ringabl 19259 lmodabl 19610 dchrabl 25757 tgrpabl 37767 erngdvlem2N 38005 erngdvlem2-rN 38013 |
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