Theorem mulgcld

Description: Deduction associated with mulgcl . (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	mulgcld.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
	mulgcld.2	⊢ · = ( .g ‘ 𝐺 )
	mulgcld.3	⊢ ( 𝜑 → 𝐺 ∈ Grp )
	mulgcld.4	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	mulgcld.5	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	mulgcld	⊢ ( 𝜑 → ( 𝑁 · 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mulgcld.1	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		mulgcld.2	⊢ · = ( .g ‘ 𝐺 )
3		mulgcld.3	⊢ ( 𝜑 → 𝐺 ∈ Grp )
4		mulgcld.4	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
5		mulgcld.5	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6	1 2	mulgcl	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 )
7	3 4 5 6	syl3anc	⊢ ( 𝜑 → ( 𝑁 · 𝑋 ) ∈ 𝐵 )