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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 18910 | . 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} | |
2 | 1 | ssrab3 4059 | 1 ⊢ CMnd ⊆ Mnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∀wral 3140 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Mndcmnd 17913 CMndccmn 18908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-in 3945 df-ss 3954 df-cmn 18910 |
This theorem is referenced by: bj-cmnssmndel 34557 |
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