Theorem mgmplusgiopALT

Description: Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	mgmplusgiopALT	⊢ ( 𝑀 ∈ Mgm → ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
2		eqid	⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 )
3	1 2	mgmcl	⊢ ( ( 𝑀 ∈ Mgm ∧ 𝑥 ∈ ( Base ‘ 𝑀 ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) )
4	3	3expb	⊢ ( ( 𝑀 ∈ Mgm ∧ ( 𝑥 ∈ ( Base ‘ 𝑀 ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) )
5	4	ralrimivva	⊢ ( 𝑀 ∈ Mgm → ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) )
6		fvex	⊢ ( +g ‘ 𝑀 ) ∈ V
7		fvex	⊢ ( Base ‘ 𝑀 ) ∈ V
8	6 7	pm3.2i	⊢ ( ( +g ‘ 𝑀 ) ∈ V ∧ ( Base ‘ 𝑀 ) ∈ V )
9		iscllaw	⊢ ( ( ( +g ‘ 𝑀 ) ∈ V ∧ ( Base ‘ 𝑀 ) ∈ V ) → ( ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) )
10	8 9	mp1i	⊢ ( 𝑀 ∈ Mgm → ( ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝑀 ) ∀ 𝑦 ∈ ( Base ‘ 𝑀 ) ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ ( Base ‘ 𝑀 ) ) )
11	5 10	mpbird	⊢ ( 𝑀 ∈ Mgm → ( +g ‘ 𝑀 ) clLaw ( Base ‘ 𝑀 ) )