Theorem cmnmndd 19822

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 19815	. 2 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

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