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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 3203 | . 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 18915 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
7 | 1, 3, 6 | mpbir2an 709 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Grpcgrp 18103 Abelcabl 18907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-grp 18106 df-cmn 18908 df-abl 18909 |
This theorem is referenced by: cnaddablx 18988 cnaddabl 18989 zaddablx 18992 |
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