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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 18549 | . 2 ⊢ Abel = (Grp ∩ CMnd) | |
2 | inss2 4058 | . 2 ⊢ (Grp ∩ CMnd) ⊆ CMnd | |
3 | 1, 2 | eqsstri 3860 | 1 ⊢ Abel ⊆ CMnd |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3797 ⊆ wss 3798 Grpcgrp 17776 CMndccmn 18546 Abelcabl 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-in 3805 df-ss 3812 df-abl 18549 |
This theorem is referenced by: bj-ablsscmnel 33686 bj-rrvecsscmn 33697 |
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