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Mirrors > Home > ILE Home > Th. List > ablgrp | GIF version |
Description: An Abelian group is a group. (Contributed by NM, 26-Aug-2011.) |
Ref | Expression |
---|---|
ablgrp | ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isabl 13045 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
2 | 1 | simplbi 274 | 1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 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-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 df-abl 13044 |
This theorem is referenced by: ablgrpd 13047 ablinvadd 13066 ablsub2inv 13067 ablsubadd 13068 ablsub4 13069 abladdsub4 13070 abladdsub 13071 ablpncan2 13072 ablpncan3 13073 ablsubsub 13074 ablsubsub4 13075 ablpnpcan 13076 ablnncan 13077 ablnnncan 13079 ablnnncan1 13080 ablsubsub23 13081 |
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