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| Mirrors > Home > MPE Home > Th. List > cmnmndd | Structured version Visualization version GIF version | ||
| Description: A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.) |
| Ref | Expression |
|---|---|
| cmnmndd.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
| Ref | Expression |
|---|---|
| cmnmndd | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cmnmndd.1 | . 2 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
| 2 | cmnmnd 19815 | . 2 ⊢ (𝐺 ∈ CMnd → 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 CMndccmn 19798 |
| 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-cmn 19800 |
| This theorem is referenced by: psrbagev1 22101 evlslem1 22106 psdadd 22167 evls1fpws 22373 mdetrsca 22609 cmn246135 33038 cmn145236 33039 gsummptres2 33056 gsumfs2d 33058 gsumtp 33061 gsumhashmul 33064 gsumwun 33068 elrgspnlem1 33246 elrgspnlem2 33247 elrgspnsubrunlem1 33251 elrgspnsubrunlem2 33252 elrspunidl 33456 elrspunsn 33457 rprmdvdsprod 33562 dfufd2lem 33577 fldextrspunlsplem 33723 fldextrspunlsp 33724 isprimroot2 42095 primrootsunit1 42098 primrootscoprmpow 42100 primrootscoprbij 42103 aks6d1c1p3 42111 aks6d1c1p4 42112 aks6d1c1p5 42113 aks6d1c1p7 42114 aks6d1c1p6 42115 aks6d1c1 42117 aks6d1c2lem4 42128 aks6d1c5lem0 42136 aks6d1c5lem2 42139 aks6d1c5 42140 aks5lem3a 42190 unitscyglem5 42200 pwsgprod 42554 evlsvvval 42573 selvvvval 42595 |
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