Theorem mulgnncl

Description: Closure of the group multiple (exponentiation) operation for a positive multiplier in a magma. (Contributed by Mario Carneiro, 11-Dec-2014) (Revised by AV, 29-Aug-2021)

	Ref	Expression
Hypotheses	mulgnncl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mulgnncl.t	⊢ · = ( .g ‘ 𝐺 )
Assertion	mulgnncl	⊢ ( ( 𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mulgnncl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mulgnncl.t	⊢ · = ( .g ‘ 𝐺 )
3		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
4		id	⊢ ( 𝐺 ∈ Mgm → 𝐺 ∈ Mgm )
5		ssidd	⊢ ( 𝐺 ∈ Mgm → 𝐵 ⊆ 𝐵 )
6	1 3	mgmcl	⊢ ( ( 𝐺 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 )
7	1 2 3 4 5 6	mulgnnsubcl	⊢ ( ( 𝐺 ∈ Mgm ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 )