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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 2818 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2818 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | iscmn 18843 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
4 | 3 | simplbi 498 | 1 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Mndcmnd 17899 CMndccmn 18835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-cmn 18837 |
This theorem is referenced by: cmn32 18854 cmn4 18855 cmn12 18856 rinvmod 18858 mulgnn0di 18875 mulgmhm 18877 ghmcmn 18881 prdscmnd 18910 gsumres 18962 gsumcl2 18963 gsumf1o 18965 gsumsubmcl 18968 gsumadd 18972 gsumsplit 18977 gsummhm 18987 gsummulglem 18990 gsuminv 18995 gsumpr 19004 gsumunsnfd 19006 gsumdifsnd 19010 gsum2d 19021 prdsgsum 19030 srgmnd 19188 gsumvsmul 19627 psrbagev1 20218 evlslem3 20221 evlslem1 20223 frlmgsum 20844 frlmup2 20871 islindf4 20910 mdetdiagid 21137 mdetrlin 21139 mdetrsca 21140 gsummatr01lem3 21194 gsummatr01 21196 chpscmat 21378 chp0mat 21382 chpidmat 21383 tmdgsum 22631 tmdgsum2 22632 tsms0 22677 tsmsmhm 22681 tsmsadd 22682 tgptsmscls 22685 tsmssplit 22687 tsmsxplem1 22688 tsmsxplem2 22689 imasdsf1olem 22910 lgseisenlem4 25881 xrge00 30600 gsumvsmul1 30616 gsummptres 30617 xrge0omnd 30639 gsumle 30652 slmdmnd 30761 lbsdiflsp0 30921 xrge0iifmhm 31081 xrge0tmdALT 31088 esum0 31207 esumsnf 31222 esumcocn 31238 gsumge0cl 42530 sge0tsms 42539 gsumdifsndf 43965 |
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