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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cmnssmnd | Structured version Visualization version GIF version |
Description: Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cmnssmnd | ⊢ CMnd ⊆ Mnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cmn 19739 | . 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} | |
2 | 1 | ssrab3 4072 | 1 ⊢ CMnd ⊆ Mnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∀wral 3051 ⊆ wss 3940 ‘cfv 6542 (class class class)co 7415 Basecbs 17177 +gcplusg 17230 Mndcmnd 18691 CMndccmn 19737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-ss 3957 df-cmn 19739 |
This theorem is referenced by: bj-cmnssmndel 36808 |
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