Theorem mulgnn0cld

Description: Closure of the group multiple (exponentiation) operation for a nonnegative multiplier in a monoid. Deduction associated with mulgnn0cl . (Contributed by SN, 1-Feb-2025)

	Ref	Expression
Hypotheses	mulgnn0cld.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mulgnn0cld.t	⊢ · = ( .g ‘ 𝐺 )
	mulgnn0cld.m	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
	mulgnn0cld.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	mulgnn0cld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	mulgnn0cld	⊢ ( 𝜑 → ( 𝑁 · 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mulgnn0cld.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mulgnn0cld.t	⊢ · = ( .g ‘ 𝐺 )
3		mulgnn0cld.m	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
4		mulgnn0cld.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
5		mulgnn0cld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6	1 2	mulgnn0cl	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑁 · 𝑋 ) ∈ 𝐵 )