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| Mirrors > Home > MPE Home > Th. List > isabli | Structured version Visualization version GIF version | ||
| Description: Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.) | 
| Ref | Expression | 
|---|---|
| isabli.g | ⊢ 𝐺 ∈ Grp | 
| isabli.b | ⊢ 𝐵 = (Base‘𝐺) | 
| isabli.p | ⊢ + = (+g‘𝐺) | 
| isabli.c | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | 
| Ref | Expression | 
|---|---|
| isabli | ⊢ 𝐺 ∈ Abel | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isabli.g | . 2 ⊢ 𝐺 ∈ Grp | |
| 2 | isabli.c | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
| 3 | 2 | rgen2 3198 | . 2 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) | 
| 4 | isabli.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 5 | isabli.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 6 | 4, 5 | isabl2 19809 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) | 
| 7 | 1, 3, 6 | mpbir2an 711 | 1 ⊢ 𝐺 ∈ Abel | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 Grpcgrp 18952 Abelcabl 19800 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-grp 18955 df-cmn 19801 df-abl 19802 | 
| This theorem is referenced by: cnaddablx 19887 cnaddabl 19888 zaddablx 19891 | 
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