Theorem 2sqreultb

Description: There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers iff P == 1 (mod 4). (Contributed by AV, 10-Jun-2023) The prime needs not be odd, as observed by WL. (Revised by AV, 25-Jun-2023)

	Ref	Expression
Hypothesis	2sqreult.1	⊢ ( 𝜑 ↔ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) )
Assertion	2sqreultb	⊢ ( 𝑃 ∈ ℙ → ( ( 𝑃 mod 4 ) = 1 ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2sqreult.1	⊢ ( 𝜑 ↔ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) )
2		2sqreultblem	⊢ ( 𝑃 ∈ ℙ → ( ( 𝑃 mod 4 ) = 1 ↔ ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) )
3	1	bicomi	⊢ ( ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ 𝜑 )
4	3	reubii	⊢ ( ∃! 𝑏 ∈ ℕ0 ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ0 𝜑 )
5	4	reubii	⊢ ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 𝜑 )
6	5	a1i	⊢ ( 𝑃 ∈ ℙ → ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 𝜑 ) )
7	1	2sqreulem4	⊢ ∀ 𝑎 ∈ ℕ0 ∃* 𝑏 ∈ ℕ0 𝜑
8		2reu1	⊢ ( ∀ 𝑎 ∈ ℕ0 ∃* 𝑏 ∈ ℕ0 𝜑 → ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 𝜑 ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )
9	7 8	mp1i	⊢ ( 𝑃 ∈ ℙ → ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 𝜑 ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )
10	2 6 9	3bitrd	⊢ ( 𝑃 ∈ ℙ → ( ( 𝑃 mod 4 ) = 1 ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )