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Theorem cmnmndd 19193
Description: A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.)
Hypothesis
Ref Expression
cmnmndd.1 (𝜑𝐺 ∈ CMnd)
Assertion
Ref Expression
cmnmndd (𝜑𝐺 ∈ Mnd)

Proof of Theorem cmnmndd
StepHypRef Expression
1 cmnmndd.1 . 2 (𝜑𝐺 ∈ CMnd)
2 cmnmnd 19186 . 2 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
31, 2syl 17 1 (𝜑𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  Mndcmnd 18173  CMndccmn 19170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-cmn 19172
This theorem is referenced by:  psrbagev1  21035  psrbagev1OLD  21036  evlslem1  21042  gsummptres2  31032  gsumhashmul  31035  elrspunidl  31320  pwsgprod  39981  evlsbagval  39985
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