Theorem ablcmnd 13829

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 13828	. 2 |- ( G e. Abel -> G e. CMnd )
3	1, 2	syl 14	1 |- ( ph -> G e. CMnd )

Syntax hints:   -> wi 4   e. wcel 2200  CMndccmn 13821   Abelcabl 13822

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-abl 13824

This theorem is referenced by:  ringcmnd  13998