Theorem smgrpmgm

Description: A semigroup is a magma. (Contributed by FL, 2-Nov-2009) (New usage is discouraged.)

	Ref	Expression
Hypothesis	smgrpmgm.1	⊢ 𝑋 = dom dom 𝐺
Assertion	smgrpmgm	⊢ ( 𝐺 ∈ SemiGrp → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		smgrpmgm.1	⊢ 𝑋 = dom dom 𝐺
2	1	issmgrpOLD	⊢ ( 𝐺 ∈ SemiGrp → ( 𝐺 ∈ SemiGrp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) ) )
3		simpl	⊢ ( ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
4	2 3	biimtrdi	⊢ ( 𝐺 ∈ SemiGrp → ( 𝐺 ∈ SemiGrp → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ) )
5	4	pm2.43i	⊢ ( 𝐺 ∈ SemiGrp → 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )