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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ablsscmn | Structured version Visualization version GIF version |
Description: Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ablsscmn | ⊢ Abel ⊆ CMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-abl 18019 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
2 | inss2 3796 | . 2 ⊢ (Grp ∩ CMnd) ⊆ CMnd | |
3 | 1, 2 | eqsstri 3598 | 1 ⊢ Abel ⊆ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3539 ⊆ wss 3540 Grpcgrp 17245 CMndccmn 18016 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-ss 3554 df-abl 18019 |
This theorem is referenced by: bj-ablsscmnel 32318 bj-rrvecsscmn 32329 |
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