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| Mirrors > Home > MPE Home > Th. List > grpmndd | Structured version Visualization version GIF version | ||
| Description: A group is a monoid. (Contributed by SN, 1-Jun-2024.) |
| Ref | Expression |
|---|---|
| grpmndd.1 | ⊢ (𝜑 → 𝐺 ∈ Grp) |
| Ref | Expression |
|---|---|
| grpmndd | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grpmndd.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
| 2 | grpmnd 18958 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Mndcmnd 18747 Grpcgrp 18951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-grp 18954 |
| This theorem is referenced by: grpmgmd 18979 hashfingrpnn 18990 xpsinv 19078 ghmgrp 19084 mulgdirlem 19123 ghmmhm 19244 gsumccatsymgsn 19444 symggen 19488 symgtrinv 19490 psgnunilem2 19513 psgneldm2 19522 psgnfitr 19535 lsmass 19687 frgpmhm 19783 frgpuplem 19790 frgpupf 19791 frgpup1 19793 isabld 19813 gsumzinv 19963 telgsumfzslem 20006 telgsumfzs 20007 dprdssv 20036 dprdfadd 20040 pgpfac1lem3a 20096 prdsrngd 20173 ringmnd 20240 unitabl 20384 unitsubm 20386 lmodvsmmulgdi 20895 rngqiprngimf1 21310 psgnghm 21598 psdmul 22170 psdmvr 22173 ply1chr 22310 rhmcomulmpl 22386 clmmulg 25134 dchrptlem3 27310 abliso 33041 gsummulgc2 33063 cyc3genpmlem 33171 elrgspnsubrunlem2 33252 quslsm 33433 evl1deg1 33601 evl1deg2 33602 evl1deg3 33603 r1pquslmic 33631 lvecendof1f1o 33684 algextdeglem4 33761 algextdeglem5 33762 rtelextdg2lem 33767 aks6d1c6lem5 42178 rhmcomulpsr 42561 evlsbagval 42576 selvvvval 42595 evlselv 42597 gicabl 43111 mendring 43200 lmodvsmdi 48295 lincvalsng 48333 lincvalsc0 48338 linc0scn0 48340 linc1 48342 lincsum 48346 lincsumcl 48348 snlindsntor 48388 grptcmon 49190 grptcepi 49191 |
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