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| Mirrors > Home > MPE Home > Th. List > ablcmn | Structured version Visualization version GIF version | ||
| Description: An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
| Ref | Expression |
|---|---|
| ablcmn | ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isabl 19802 | . 2 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd)) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Grpcgrp 18951 CMndccmn 19798 Abelcabl 19799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-abl 19801 |
| This theorem is referenced by: ablcmnd 19806 ablcom 19817 abl32 19821 ablsub4 19828 mulgdi 19844 ghmabl 19850 ghmplusg 19864 ablcntzd 19875 prdsabld 19880 gsumsubgcl 19938 gsummulgz 19961 gsuminv 19964 gsumsub 19966 telgsumfzslem 20006 telgsums 20011 ringcmn 20279 lmodcmn 20908 clmsub4 25139 lgseisenlem4 27422 primrootspoweq0 42107 aks6d1c6lem4 42174 |
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