![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cmnmnd | GIF version |
Description: A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
cmnmnd | ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2177 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | iscmn 13049 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
4 | 3 | simplbi 274 | 1 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 +gcplusg 12530 Mndcmnd 12771 CMndccmn 13041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 df-cmn 13043 |
This theorem is referenced by: cmn32 13060 cmn4 13061 cmn12 13062 cmnmndd 13064 rinvmod 13065 srgmnd 13103 |
Copyright terms: Public domain | W3C validator |