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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 19796 | . 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} | |
| 2 | 1 | ssrab3 4081 | 1 ⊢ CMnd ⊆ Mnd |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∀wral 3060 ⊆ wss 3950 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 +gcplusg 17293 Mndcmnd 18743 CMndccmn 19794 |
| 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 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-ss 3967 df-cmn 19796 |
| This theorem is referenced by: bj-cmnssmndel 37252 |
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