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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 18964 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 5 | isabld.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
| 6 | 2, 3, 4, 5 | iscmnd 19812 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
| 7 | isabl 19802 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
| 8 | 1, 6, 7 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐺 ∈ Abel) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 CMndccmn 19798 Abelcabl 19799 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-grp 18954 df-cmn 19800 df-abl 19801 |
| This theorem is referenced by: subgabl 19854 gex2abl 19869 cygabl 19909 ringabl 20278 lmodabl 20907 dchrabl 27298 tgrpabl 40753 erngdvlem2N 40991 erngdvlem2-rN 40999 |
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