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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 18976 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
5 | isabld.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
6 | 2, 3, 4, 5 | iscmnd 19826 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
7 | isabl 19816 | . 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 1086 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 +gcplusg 17297 Grpcgrp 18963 CMndccmn 19812 Abelcabl 19813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-ov 7433 df-grp 18966 df-cmn 19814 df-abl 19815 |
This theorem is referenced by: subgabl 19868 gex2abl 19883 cygabl 19923 ringabl 20294 lmodabl 20923 dchrabl 27312 tgrpabl 40733 erngdvlem2N 40971 erngdvlem2-rN 40979 |
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