Theorem mulgnn0cl

Description: Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. (Contributed by Mario Carneiro, 11-Dec-2014)

	Ref	Expression
Hypotheses	mulgnncl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mulgnncl.t	⊢ · = ( .g ‘ 𝐺 )
Assertion	mulgnn0cl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 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 ∈ Mnd → 𝐺 ∈ Mnd )
5		ssidd	⊢ ( 𝐺 ∈ Mnd → 𝐵 ⊆ 𝐵 )
6	1 3	mndcl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 )
7		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
8	1 7	mndidcl	⊢ ( 𝐺 ∈ Mnd → ( 0g ‘ 𝐺 ) ∈ 𝐵 )
9	1 2 3 4 5 6 7 8	mulgnn0subcl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 )