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Theorem cmnmndd 13367
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 13360 . 2 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
31, 2syl 14 1 (𝜑𝐺 ∈ Mnd)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2164  Mndcmnd 12987  CMndccmn 13343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5207  df-fv 5254  df-ov 5913  df-cmn 13345
This theorem is referenced by:  gsumfzreidx  13396  gsumfzmptfidmadd  13398  lgseisenlem3  15136
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