Theorem bj-smgrpssmgm

Description: Semigroups are magmas. (Contributed by BJ, 12-Apr-2024) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-smgrpssmgm	⊢ Smgrp ⊆ Mgm

Proof

Step	Hyp	Ref	Expression
1		df-sgrp	⊢ Smgrp = { 𝑔 ∈ Mgm ∣ [ ( Base ‘ 𝑔 ) / 𝑏 ] [ ( +g ‘ 𝑔 ) / 𝑝 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑝 𝑦 ) 𝑝 𝑧 ) = ( 𝑥 𝑝 ( 𝑦 𝑝 𝑧 ) ) }
2	1	ssrab3	⊢ Smgrp ⊆ Mgm