Metamath Proof Explorer

Theorem mgm0

Description: Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021)

Ref Expression
Assertion mgm0 
|- ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Mgm )

Proof

Step Hyp Ref Expression
1 rzal 
 |-  ( ( Base ` M ) = (/) -> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) )
2 1 adantl 
 |-  ( ( M e. V /\ ( Base ` M ) = (/) ) -> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) )
3 eqid 
 |-  ( Base ` M ) = ( Base ` M )
4 eqid 
 |-  ( +g ` M ) = ( +g ` M )
5 3 4 ismgm 
 |-  ( M e. V -> ( M e. Mgm <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
6 5 adantr 
 |-  ( ( M e. V /\ ( Base ` M ) = (/) ) -> ( M e. Mgm <-> A. x e. ( Base ` M ) A. y e. ( Base ` M ) ( x ( +g ` M ) y ) e. ( Base ` M ) ) )
7 2 6 mpbird 
 |-  ( ( M e. V /\ ( Base ` M ) = (/) ) -> M e. Mgm )