Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isabl2 | Structured version Visualization version GIF version |
Description: The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
iscmn.b | ⊢ 𝐵 = (Base‘𝐺) |
iscmn.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
isabl2 | ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isabl 18841 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
2 | grpmnd 18050 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
3 | iscmn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | iscmn.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
5 | 3, 4 | iscmn 18845 | . . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
6 | 5 | baib 536 | . . . 4 ⊢ (𝐺 ∈ Mnd → (𝐺 ∈ CMnd ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ CMnd ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
8 | 7 | pm5.32i 575 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd) ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
9 | 1, 8 | bitri 276 | 1 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 +gcplusg 16555 Mndcmnd 17901 Grpcgrp 18043 CMndccmn 18837 Abelcabl 18838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7148 df-grp 18046 df-cmn 18839 df-abl 18840 |
This theorem is referenced by: isabli 18852 invghm 18885 qusabl 18916 abl1 18917 archiabllem1 30750 archiabllem2 30754 |
Copyright terms: Public domain | W3C validator |