Theorem cmnmndd 19835

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 19828	. 2 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2141  Mndcmnd 18759  CMndccmn 19811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394  df-cmn 19813

This theorem is referenced by:  pwsgprod  20365  psrbagev1  22118  evlslem1  22123  evlsvvval  22134  selvvvval  22183  psdadd  22216  evls1fpws  22420  mdetrsca  22651  cmn246135  33172  cmn145236  33173  gsummptres2  33194  gsummptfzsplitra  33199  gsummptfzsplitla  33200  gsumfs2d  33202  gsumtp  33205  gsumhashmul  33208  gsumwun  33217  elrgspnlem1  33384  elrgspnlem2  33385  elrgspnsubrunlem1  33389  elrgspnsubrunlem2  33390  elrspunidl  33575  elrspunsn  33576  rprmdvdsprod  33691  dfufd2lem  33706  evlextv  33800  esplyfvaln  33832  vietalem  33837  fldextrspunlsplem  33931  fldextrspunlsp  33932  extdgfialglem2  33951  isprimroot2  42672  primrootsunit1  42675  primrootscoprmpow  42677  primrootscoprbij  42680  aks6d1c1p3  42688  aks6d1c1p4  42689  aks6d1c1p5  42690  aks6d1c1p7  42691  aks6d1c1p6  42692  aks6d1c1  42694  aks6d1c2lem4  42705  aks6d1c5lem0  42713  aks6d1c5lem2  42716  aks6d1c5  42717  aks5lem3a  42767  unitscyglem5  42777