Theorem sgrpmgm

Description: A semigroup is a magma. (Contributed by FL, 2-Nov-2009) (Revised by AV, 6-Jan-2020)

	Ref	Expression
Assertion	sgrpmgm	⊢ ( 𝑀 ∈ Smgrp → 𝑀 ∈ Mgm )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
2		eqid	⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 )
3	1 2	issgrp	⊢ ( 𝑀 ∈ Smgrp ↔ ( 𝑀 ∈ Mgm ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ∀ 𝑧 ∈ ( Base ‘ 𝑀 ) ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ( +g ‘ 𝑀 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝑀 ) ( 𝑦 ( +g ‘ 𝑀 ) 𝑧 ) ) ) )
4	3	simplbi	⊢ ( 𝑀 ∈ Smgrp → 𝑀 ∈ Mgm )