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Mirrors > Home > ILE Home > Th. List > isabld | 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 12843 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
5 | isabld.c | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
6 | 2, 3, 4, 5 | iscmnd 13054 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
7 | isabl 13045 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
8 | 1, 6, 7 | sylanbrc 417 | 1 ⊢ (𝜑 → 𝐺 ∈ Abel) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 +gcplusg 12530 Grpcgrp 12831 CMndccmn 13041 Abelcabl 13042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 df-grp 12834 df-cmn 13043 df-abl 13044 |
This theorem is referenced by: ringabl 13168 |
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