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∧a2+b2=P
Assertion 2sqreunnltb ⊢P∈ℙ→Pmod4=1↔∃!a∈ℕ∃b∈ℕφ∧∃!b∈ℕ∃a∈ℕφ

Proof

Step Hyp Ref Expression
1 2sqreult.1 ⊢φ↔a<b∧a2+b2=P
2 2sqreunnltblem ⊢P∈ℙ→Pmod4=1↔∃!a∈ℕ∃!b∈ℕa<b∧a2+b2=P
3 1 bicomi ⊢a<b∧a2+b2=P↔φ
4 3 reubii ⊢∃!b∈ℕa<b∧a2+b2=P↔∃!b∈ℕφ
5 4 reubii ⊢∃!a∈ℕ∃!b∈ℕa<b∧a2+b2=P↔∃!a∈ℕ∃!b∈ℕφ
6 1 2sqreunnlem2 ⊢∀a∈ℕ∃*b∈ℕφ
7 2reu1 ⊢∀a∈ℕ∃*b∈ℕφ→∃!a∈ℕ∃!b∈ℕφ↔∃!a∈ℕ∃b∈ℕφ∧∃!b∈ℕ∃a∈ℕφ
8 6 7 ax-mp ⊢∃!a∈ℕ∃!b∈ℕφ↔∃!a∈ℕ∃b∈ℕφ∧∃!b∈ℕ∃a∈ℕφ
9 5 8 bitri ⊢∃!a∈ℕ∃!b∈ℕa<b∧a2+b2=P↔∃!a∈ℕ∃b∈ℕφ∧∃!b∈ℕ∃a∈ℕφ
10 2 9 bitrdi ⊢P∈ℙ→Pmod4=1↔∃!a∈ℕ∃b∈ℕφ∧∃!b∈ℕ∃a∈ℕφ