Theorem cmnmndd 19779

Description: A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)

Hypothesis

Ref	Expression
cmnmndd.1	⊢ (𝜑 → 𝐺 ∈ CMnd)

Assertion

Ref	Expression
cmnmndd	⊢ (𝜑 → 𝐺 ∈ Mnd)

Proof of Theorem cmnmndd

Step	Hyp	Ref	Expression
1		cmnmndd.1	. 2 ⊢ (𝜑 → 𝐺 ∈ CMnd)
2		cmnmnd 19772	. 2 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2114  Mndcmnd 18702  CMndccmn 19755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-cmn 19757

This theorem is referenced by:  pwsgprod  20309  psrbagev1  22055  evlslem1  22060  evlsvvval  22071  psdadd  22129  evls1fpws  22334  mdetrsca  22568  cmn246135  33093  cmn145236  33094  gsummptres2  33114  gsummptfzsplitra  33119  gsummptfzsplitla  33120  gsumfs2d  33122  gsumtp  33125  gsumhashmul  33128  gsumwun  33137  elrgspnlem1  33303  elrgspnlem2  33304  elrgspnsubrunlem1  33308  elrgspnsubrunlem2  33309  elrspunidl  33488  elrspunsn  33489  rprmdvdsprod  33594  dfufd2lem  33609  evlextv  33686  esplyfvaln  33718  vietalem  33723  fldextrspunlsplem  33817  fldextrspunlsp  33818  extdgfialglem2  33837  isprimroot2  42533  primrootsunit1  42536  primrootscoprmpow  42538  primrootscoprbij  42541  aks6d1c1p3  42549  aks6d1c1p4  42550  aks6d1c1p5  42551  aks6d1c1p7  42552  aks6d1c1p6  42553  aks6d1c1  42555  aks6d1c2lem4  42566  aks6d1c5lem0  42574  aks6d1c5lem2  42577  aks6d1c5  42578  aks5lem3a  42628  unitscyglem5  42638  selvvvval  43018