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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 18585 | . 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} | |
2 | ssrab2 3908 | . 2 ⊢ {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} ⊆ Mnd | |
3 | 1, 2 | eqsstri 3854 | 1 ⊢ CMnd ⊆ Mnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∀wral 3090 {crab 3094 ⊆ wss 3792 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 +gcplusg 16342 Mndcmnd 17684 CMndccmn 18583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rab 3099 df-in 3799 df-ss 3806 df-cmn 18585 |
This theorem is referenced by: bj-cmnssmndel 33742 |
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