![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 18900 | . 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} | |
2 | 1 | ssrab3 4008 | 1 ⊢ CMnd ⊆ Mnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∀wral 3106 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Mndcmnd 17903 CMndccmn 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-cmn 18900 |
This theorem is referenced by: bj-cmnssmndel 34688 |
Copyright terms: Public domain | W3C validator |