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Metamath Proof Explorer

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 )