Theorem 2sqreunnlt

Description: There exists a unique decomposition of a prime of the form 4 k + 1 as a sum of squares of two different positive integers. (Contributed by AV, 4-Jun-2023) Specialization to different integers, proposed by GL. (Revised by AV, 25-Jun-2023)

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

Proof

Step	Hyp	Ref	Expression
1		2sqreult.1	⊢ ( 𝜑 ↔ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) )
2		2sqreunnltlem	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) )
3	1	bicomi	⊢ ( ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ 𝜑 )
4	3	reubii	⊢ ( ∃! 𝑏 ∈ ℕ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ 𝜑 )
5	4	reubii	⊢ ( ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ 𝜑 )
6	1	2sqreunnlem2	⊢ ∀ 𝑎 ∈ ℕ ∃* 𝑏 ∈ ℕ 𝜑
7		2reu1	⊢ ( ∀ 𝑎 ∈ ℕ ∃* 𝑏 ∈ ℕ 𝜑 → ( ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ 𝜑 ↔ ( ∃! 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ 𝜑 ∧ ∃! 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ 𝜑 ) ) )
8	6 7	mp1i	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ 𝜑 ↔ ( ∃! 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ 𝜑 ∧ ∃! 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ 𝜑 ) ) )
9	5 8	syl5bb	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 < 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( ∃! 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ 𝜑 ∧ ∃! 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ 𝜑 ) ) )
10	2 9	mpbid	⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃! 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ 𝜑 ∧ ∃! 𝑏 ∈ ℕ ∃ 𝑎 ∈ ℕ 𝜑 ) )