Metamath Proof Explorer

Theorem 2sqreunnltb

Description: There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4 k + 1 . (Contributed by AV, 11-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 2sqreunnltb ⊢ 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 2sqreunnltblem ⊢ 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 ax-mp ⊢ ∃! a ∈ ℕ ∃! b ∈ ℕ φ ↔ ∃! a ∈ ℕ ∃ b ∈ ℕ φ ∧ ∃! b ∈ ℕ ∃ a ∈ ℕ φ
9 5 8 bitri ⊢ ∃! a ∈ ℕ ∃! b ∈ ℕ a < b ∧ a 2 + b 2 = P ↔ ∃! a ∈ ℕ ∃ b ∈ ℕ φ ∧ ∃! b ∈ ℕ ∃ a ∈ ℕ φ
10 2 9 syl6bb ⊢ P ∈ ℙ → P mod 4 = 1 ↔ ∃! a ∈ ℕ ∃ b ∈ ℕ φ ∧ ∃! b ∈ ℕ ∃ a ∈ ℕ φ