Theorem dvdsexpad

Description: Deduction associated with dvdsexpim . (Contributed by SN, 21-Aug-2024)

	Ref	Expression
Hypotheses	dvdsexpad.1	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
	dvdsexpad.2	⊢ ( 𝜑 → 𝐵 ∈ ℤ )
	dvdsexpad.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
	dvdsexpad.5	⊢ ( 𝜑 → 𝐴 ∥ 𝐵 )
Assertion	dvdsexpad	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∥ ( 𝐵 ↑ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsexpad.1	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
2		dvdsexpad.2	⊢ ( 𝜑 → 𝐵 ∈ ℤ )
3		dvdsexpad.3	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
4		dvdsexpad.5	⊢ ( 𝜑 → 𝐴 ∥ 𝐵 )
5		dvdsexpim	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ∥ 𝐵 → ( 𝐴 ↑ 𝑁 ) ∥ ( 𝐵 ↑ 𝑁 ) ) )
6	1 2 3 5	syl3anc	⊢ ( 𝜑 → ( 𝐴 ∥ 𝐵 → ( 𝐴 ↑ 𝑁 ) ∥ ( 𝐵 ↑ 𝑁 ) ) )
7	4 6	mpd	⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∥ ( 𝐵 ↑ 𝑁 ) )