![]() |
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 19800 | . 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)} | |
2 | 1 | ssrab3 4092 | 1 ⊢ CMnd ⊆ Mnd |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1535 ∀wral 3057 ⊆ wss 3963 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 +gcplusg 17287 Mndcmnd 18748 CMndccmn 19798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-rab 3433 df-ss 3980 df-cmn 19800 |
This theorem is referenced by: bj-cmnssmndel 37216 |
Copyright terms: Public domain | W3C validator |