Theorem ablcmnd 14072

Description: An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)

Hypothesis

Ref	Expression
ablcmnd.1	|- ( ph -> G e. Abel )

Assertion

Ref	Expression
ablcmnd	|- ( ph -> G e. CMnd )

Proof of Theorem ablcmnd

Step	Hyp	Ref	Expression
1		ablcmnd.1	. 2 |- ( ph -> G e. Abel )
2		ablcmn 14071	. 2 |- ( G e. Abel -> G e. CMnd )
3	1, 2	syl 14	1 |- ( ph -> G e. CMnd )

Syntax hints:   -> wi 4   e. wcel 2209  CMndccmn 14064   Abelcabl 14065

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-abl 14067

This theorem is referenced by:  ringcmnd  14313