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Mirrors > Home > MPE Home > Th. List > cmnmnd | Structured version Visualization version 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 2622 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2622 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | iscmn 18200 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
4 | 3 | simplbi 476 | 1 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Mndcmnd 17294 CMndccmn 18193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-cmn 18195 |
This theorem is referenced by: cmn32 18211 cmn4 18212 cmn12 18213 mulgnn0di 18231 mulgmhm 18233 ghmcmn 18237 prdscmnd 18264 gsumres 18314 gsumcl2 18315 gsumf1o 18317 gsumsubmcl 18319 gsumadd 18323 gsumsplit 18328 gsummhm 18338 gsummulglem 18341 gsuminv 18346 gsumunsnfd 18356 gsumdifsnd 18360 gsum2d 18371 prdsgsum 18377 srgmnd 18509 gsumvsmul 18927 psrbagev1 19510 evlslem3 19514 evlslem1 19515 frlmgsum 20111 frlmup2 20138 islindf4 20177 mdetdiagid 20406 mdetrlin 20408 mdetrsca 20409 gsummatr01lem3 20463 gsummatr01 20465 chpscmat 20647 chp0mat 20651 chpidmat 20652 tmdgsum 21899 tmdgsum2 21900 tsms0 21945 tsmsmhm 21949 tsmsadd 21950 tgptsmscls 21953 tsmssplit 21955 tsmsxplem1 21956 tsmsxplem2 21957 imasdsf1olem 22178 lgseisenlem4 25103 xrge00 29686 xrge0omnd 29711 slmdmnd 29759 gsumle 29779 gsummptres 29784 xrge0iifmhm 29985 xrge0tmdOLD 29991 esum0 30111 esumsnf 30126 esumcocn 30142 gsumge0cl 40588 sge0tsms 40597 gsumpr 42139 gsumdifsndf 42144 |
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