Theorem 2sqreult

Description: There exists a unique decomposition of a prime as a sum of squares of two different nonnegative integers. (Contributed by AV, 8-Jun-2023) (Proposed by GL, 8-Jun-2023.) (Revised by AV, 25-Jun-2023)

	Ref	Expression
Hypothesis	2sqreult.1	⊢ ( 𝜑 ↔ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) )
Assertion	2sqreult	⊢ ( ( 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		2sqreultlem	⊢ ( ( 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	1	2sqreulem4	⊢ ∀ 𝑎 ∈ ℕ0 ∃* 𝑏 ∈ ℕ0 𝜑
7		2reu1	⊢ ( ∀ 𝑎 ∈ ℕ0 ∃* 𝑏 ∈ ℕ0 𝜑 → ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 𝜑 ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )
8	6 7	mp1i	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 𝜑 ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )
9	5 8	syl5bb	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃! 𝑎 ∈ ℕ0 ∃! 𝑏 ∈ ℕ0 ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) ) )
10	2 9	mpbid	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃! 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 𝜑 ∧ ∃! 𝑏 ∈ ℕ0 ∃ 𝑎 ∈ ℕ0 𝜑 ) )