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	$$⊢B={\mathrm{Base}}_G$$
	mulgnn0cld.t	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
	mulgnn0cld.m	$$⊢\phi \to G\in \mathrm{Mnd}$$
	mulgnn0cld.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
	mulgnn0cld.x	$$⊢\phi \to X\in B$$
Assertion	mulgnn0cld	$$⊢\phi \to N\underset{\dot{}}{·}X\in B$$

Proof

Step	Hyp	Ref	Expression
1		mulgnn0cld.b	$$⊢B={\mathrm{Base}}_G$$
2		mulgnn0cld.t	$$⊢\underset{\dot{}}{·}={\cdot }_G$$
3		mulgnn0cld.m	$$⊢\phi \to G\in \mathrm{Mnd}$$
4		mulgnn0cld.n	$$⊢\phi \to N\in {\mathbb{N}}_0$$
5		mulgnn0cld.x	$$⊢\phi \to X\in B$$
6	1 2	mulgnn0cl	$$⊢\left(G\in \mathrm{Mnd}\wedge N\in {\mathbb{N}}_0\wedge X\in B\right)\to N\underset{\dot{}}{·}X\in B$$
7	3 4 5 6	syl3anc	$$⊢\phi \to N\underset{\dot{}}{·}X\in B$$