Theorem cmnmndd 19837

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 19830	. 2 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∈ wcel 2106  Mndcmnd 18760  CMndccmn 19813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-cmn 19815

This theorem is referenced by:  psrbagev1  22119  evlslem1  22124  psdadd  22185  evls1fpws  22389  mdetrsca  22625  cmn246135  33021  cmn145236  33022  gsummptres2  33039  gsumfs2d  33041  gsumtp  33044  gsumhashmul  33047  gsumwun  33051  elrgspnlem1  33232  elrgspnlem2  33233  elrspunidl  33436  elrspunsn  33437  rprmdvdsprod  33542  dfufd2lem  33557  isprimroot2  42076  primrootsunit1  42079  primrootscoprmpow  42081  primrootscoprbij  42084  aks6d1c1p3  42092  aks6d1c1p4  42093  aks6d1c1p5  42094  aks6d1c1p7  42095  aks6d1c1p6  42096  aks6d1c1  42098  aks6d1c2lem4  42109  aks6d1c5lem0  42117  aks6d1c5lem2  42120  aks6d1c5  42121  aks5lem3a  42171  unitscyglem5  42181  pwsgprod  42531  evlsvvval  42550  selvvvval  42572