Theorem zrtdvds

Description: A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023)

	Ref	Expression
Assertion	zrtdvds	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∥ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ → ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℤ )
2	1	3ad2ant3	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℤ )
3		simp2	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → 𝑁 ∈ ℕ )
4		iddvdsexp	⊢ ( ( ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∥ ( ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ↑ 𝑁 ) )
5	2 3 4	syl2anc	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∥ ( ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ↑ 𝑁 ) )
6		zcn	⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℂ )
7	6	3ad2ant1	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → 𝐴 ∈ ℂ )
8		cxproot	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ↑ 𝑁 ) = 𝐴 )
9	7 3 8	syl2anc	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → ( ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ↑ 𝑁 ) = 𝐴 )
10	5 9	breqtrd	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∈ ℕ ) → ( 𝐴 ↑𝑐 ( 1 / 𝑁 ) ) ∥ 𝐴 )