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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 | 1 | grpmndd 18377 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
5 | isabld.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
6 | 2, 3, 4, 5 | iscmnd 19183 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
7 | isabl 19174 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
8 | 1, 6, 7 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐺 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 Grpcgrp 18365 CMndccmn 19170 Abelcabl 19171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-ov 7216 df-grp 18368 df-cmn 19172 df-abl 19173 |
This theorem is referenced by: subgabl 19221 gex2abl 19236 cygabl 19275 cygablOLD 19276 ringabl 19598 lmodabl 19946 dchrabl 26135 tgrpabl 38502 erngdvlem2N 38740 erngdvlem2-rN 38748 |
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