Theorem cmnmndd 19734

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 19727	. 2 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2109  Mndcmnd 18661  CMndccmn 19710

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-cmn 19712

This theorem is referenced by:  psrbagev1  21984  evlslem1  21989  psdadd  22050  evls1fpws  22256  mdetrsca  22490  cmn246135  32974  cmn145236  32975  gsummptres2  32993  gsumfs2d  32995  gsumtp  32998  gsumhashmul  33001  gsumwun  33005  elrgspnlem1  33193  elrgspnlem2  33194  elrgspnsubrunlem1  33198  elrgspnsubrunlem2  33199  elrspunidl  33399  elrspunsn  33400  rprmdvdsprod  33505  dfufd2lem  33520  fldextrspunlsplem  33668  fldextrspunlsp  33669  isprimroot2  42082  primrootsunit1  42085  primrootscoprmpow  42087  primrootscoprbij  42090  aks6d1c1p3  42098  aks6d1c1p4  42099  aks6d1c1p5  42100  aks6d1c1p7  42101  aks6d1c1p6  42102  aks6d1c1  42104  aks6d1c2lem4  42115  aks6d1c5lem0  42123  aks6d1c5lem2  42126  aks6d1c5  42127  aks5lem3a  42177  unitscyglem5  42187  pwsgprod  42532  evlsvvval  42551  selvvvval  42573