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Mirrors > Home > MPE Home > Th. List > Mathboxes > ablcmnd | Structured version Visualization version GIF version |
Description: An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.) |
Ref | Expression |
---|---|
ablcmnd.1 | ⊢ (𝜑 → 𝐺 ∈ Abel) |
Ref | Expression |
---|---|
ablcmnd | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablcmnd.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Abel) | |
2 | ablcmn 19174 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 CMndccmn 19167 Abelcabl 19168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3407 df-in 3870 df-abl 19170 |
This theorem is referenced by: ringcmnd 39956 |
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