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Theorem 2sqreunnlem1 25939
Description: Lemma 1 for 2sqreunn 25947. (Contributed by AV, 11-Jun-2023.)
Assertion
Ref Expression
2sqreunnlem1 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
Distinct variable group:   𝑃,𝑎,𝑏

Proof of Theorem 2sqreunnlem1
Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2sqnn 25929 . . 3 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)))
2 simpll 763 . . . . . . . . 9 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ)
32adantl 482 . . . . . . . 8 ((𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ)
4 breq1 5065 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑎𝑏𝑥𝑏))
5 oveq1 7158 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2))
65oveq1d 7166 . . . . . . . . . . . 12 (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2)))
76eqeq1d 2826 . . . . . . . . . . 11 (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
84, 7anbi12d 630 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
98reubidv 3394 . . . . . . . . 9 (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
109adantl 482 . . . . . . . 8 (((𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑥) → (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)))
11 simpr 485 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℕ)
1211adantr 481 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → 𝑦 ∈ ℕ)
13 breq2 5066 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → (𝑥𝑏𝑥𝑦))
14 oveq1 7158 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2))
1514oveq2d 7167 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))
1615eqeq1d 2826 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))
1713, 16anbi12d 630 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑦 → ((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))))
18 equequ1 2025 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑦 → (𝑏 = 𝑐𝑦 = 𝑐))
1918imbi2d 342 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → (((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
2019ralbidv 3201 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑦 → (∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
2117, 20anbi12d 630 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦 → (((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))))
2221adantl 482 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑏 = 𝑦) → (((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))))
23 simpr 485 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → 𝑥𝑦)
24 eqidd 2825 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))
25 nnre 11637 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ ℕ → 𝑐 ∈ ℝ)
2625resqcld 13604 . . . . . . . . . . . . . . . . . . . 20 (𝑐 ∈ ℕ → (𝑐↑2) ∈ ℝ)
2726adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
28 nnre 11637 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
2928resqcld 13604 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℝ)
3029adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
3130ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
32 nnre 11637 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
3332resqcld 13604 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℝ)
3433adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
3534ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
36 readdcan 10806 . . . . . . . . . . . . . . . . . . 19 (((𝑐↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
3727, 31, 35, 36syl3anc 1365 . . . . . . . . . . . . . . . . . 18 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2)))
3828ad4antlr 729 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ)
3925ad2antlr 723 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ)
40 nnnn0 11896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0)
4140nn0ge0d 11950 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ℕ → 0 ≤ 𝑦)
4241ad4antlr 729 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦)
43 nnnn0 11896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
4443nn0ge0d 11950 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ ℕ → 0 ≤ 𝑐)
4544ad2antlr 723 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐)
46 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2))
4746eqcomd 2830 . . . . . . . . . . . . . . . . . . . 20 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2))
4838, 39, 42, 45, 47sq11d 13614 . . . . . . . . . . . . . . . . . . 19 (((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐)
4948ex 413 . . . . . . . . . . . . . . . . . 18 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐))
5037, 49sylbid 241 . . . . . . . . . . . . . . . . 17 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐))
5150adantld 491 . . . . . . . . . . . . . . . 16 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) ∧ 𝑐 ∈ ℕ) → ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
5251ralrimiva 3186 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))
5323, 24, 52jca31 515 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → ((𝑥𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))
5412, 22, 53rspcedvd 3629 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → ∃𝑏 ∈ ℕ ((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
55 breq2 5066 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (𝑥𝑏𝑥𝑐))
56 oveq1 7158 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2))
5756oveq2d 7167 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2)))
5857eqeq1d 2826 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
5955, 58anbi12d 630 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → ((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))))
6059reu8 3727 . . . . . . . . . . . . 13 (∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ ((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑥𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
6154, 60sylibr 235 . . . . . . . . . . . 12 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑥𝑦) → ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
6261ex 413 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥𝑦 → ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
6362adantr 481 . . . . . . . . . 10 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (𝑥𝑦 → ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
6463impcom 408 . . . . . . . . 9 ((𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
65 eqeq2 2836 . . . . . . . . . . . . 13 (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
6665anbi2d 628 . . . . . . . . . . . 12 (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
6766reubidv 3394 . . . . . . . . . . 11 (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
6867adantl 482 . . . . . . . . . 10 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
6968adantl 482 . . . . . . . . 9 ((𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
7064, 69mpbird 258 . . . . . . . 8 ((𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑥𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))
713, 10, 70rspcedvd 3629 . . . . . . 7 ((𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
7211adantr 481 . . . . . . . . 9 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ)
7372adantl 482 . . . . . . . 8 ((¬ 𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ)
74 breq1 5065 . . . . . . . . . . 11 (𝑎 = 𝑦 → (𝑎𝑏𝑦𝑏))
75 oveq1 7158 . . . . . . . . . . . . 13 (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2))
7675oveq1d 7166 . . . . . . . . . . . 12 (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2)))
7776eqeq1d 2826 . . . . . . . . . . 11 (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
7874, 77anbi12d 630 . . . . . . . . . 10 (𝑎 = 𝑦 → ((𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
7978reubidv 3394 . . . . . . . . 9 (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
8079adantl 482 . . . . . . . 8 (((¬ 𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑦) → (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)))
81 simpll 763 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → 𝑥 ∈ ℕ)
82 breq2 5066 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑥 → (𝑦𝑏𝑦𝑥))
83 oveq1 7158 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2))
8483oveq2d 7167 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2)))
8584eqeq1d 2826 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))
8682, 85anbi12d 630 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑥 → ((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))))
87 equequ1 2025 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑥 → (𝑏 = 𝑐𝑥 = 𝑐))
8887imbi2d 342 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑥 → (((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
8988ralbidv 3201 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑥 → (∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
9086, 89anbi12d 630 . . . . . . . . . . . . . . 15 (𝑏 = 𝑥 → (((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))))
9190adantl 482 . . . . . . . . . . . . . 14 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) ∧ 𝑏 = 𝑥) → (((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))))
92 ltnle 10712 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥𝑦))
9328, 32, 92syl2anr 596 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < 𝑥 ↔ ¬ 𝑥𝑦))
9428ad2antlr 723 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ)
9532ad2antrr 722 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ)
96 simpr 485 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥)
9794, 95, 96ltled 10780 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑦 < 𝑥) → 𝑦𝑥)
9897ex 413 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦 < 𝑥𝑦𝑥))
9993, 98sylbird 261 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥𝑦𝑦𝑥))
10099imp 407 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → 𝑦𝑥)
10129recnd 10661 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ℕ → (𝑦↑2) ∈ ℂ)
102101adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
10333recnd 10661 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℕ → (𝑥↑2) ∈ ℂ)
104103adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
105102, 104addcomd 10834 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
106105adantr 481 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))
10734recnd 10661 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
108107adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℂ)
10930recnd 10661 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
110109adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℂ)
111108, 110addcomd 10834 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2)))
112111eqeq2d 2835 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2))))
11326adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑐↑2) ∈ ℝ)
11433ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑥↑2) ∈ ℝ)
11529ad2antlr 723 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (𝑦↑2) ∈ ℝ)
116 readdcan 10806 . . . . . . . . . . . . . . . . . . . . 21 (((𝑐↑2) ∈ ℝ ∧ (𝑥↑2) ∈ ℝ ∧ (𝑦↑2) ∈ ℝ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
117113, 114, 115, 116syl3anc 1365 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
118112, 117bitrd 280 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2)))
11925ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ)
12032adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℝ)
121120ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ)
12244ad2antlr 723 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐)
123 nnnn0 11896 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
124123nn0ge0d 11950 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℕ → 0 ≤ 𝑥)
125124adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → 0 ≤ 𝑥)
126125ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥)
127 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2))
128119, 121, 122, 126, 127sq11d 13614 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥)
129128eqcomd 2830 . . . . . . . . . . . . . . . . . . . 20 ((((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐)
130129ex 413 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐))
131118, 130sylbid 241 . . . . . . . . . . . . . . . . . 18 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐))
132131adantld 491 . . . . . . . . . . . . . . . . 17 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑐 ∈ ℕ) → ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
133132ralrimiva 3186 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
134133adantr 481 . . . . . . . . . . . . . . 15 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))
135100, 106, 134jca31 515 . . . . . . . . . . . . . 14 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → ((𝑦𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))
13681, 91, 135rspcedvd 3629 . . . . . . . . . . . . 13 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → ∃𝑏 ∈ ℕ ((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
137 breq2 5066 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (𝑦𝑏𝑦𝑐))
13856oveq2d 7167 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2)))
139138eqeq1d 2826 . . . . . . . . . . . . . . 15 (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))
140137, 139anbi12d 630 . . . . . . . . . . . . . 14 (𝑏 = 𝑐 → ((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))))
141140reu8 3727 . . . . . . . . . . . . 13 (∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ ((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ ((𝑦𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)))
142136, 141sylibr 235 . . . . . . . . . . . 12 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ ¬ 𝑥𝑦) → ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
143142ex 413 . . . . . . . . . . 11 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (¬ 𝑥𝑦 → ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
144143adantr 481 . . . . . . . . . 10 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (¬ 𝑥𝑦 → ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
145144impcom 408 . . . . . . . . 9 ((¬ 𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
146 eqeq2 2836 . . . . . . . . . . . . 13 (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))
147146anbi2d 628 . . . . . . . . . . . 12 (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
148147reubidv 3394 . . . . . . . . . . 11 (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
149148adantl 482 . . . . . . . . . 10 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
150149adantl 482 . . . . . . . . 9 ((¬ 𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))))
151145, 150mpbird 258 . . . . . . . 8 ((¬ 𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ (𝑦𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))
15273, 80, 151rspcedvd 3629 . . . . . . 7 ((¬ 𝑥𝑦 ∧ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
15371, 152pm2.61ian 808 . . . . . 6 (((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
154153ex 413 . . . . 5 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
155154adantl 482 . . . 4 (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
156155rexlimdvva 3298 . . 3 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → (∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
1571, 156mpd 15 . 2 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
158 reurex 3436 . . . . 5 (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
159158a1i 11 . . . 4 (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ) → (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
160159ralrimiva 3186 . . 3 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∀𝑎 ∈ ℕ (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
161 2sqmo 25927 . . . . 5 (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ0𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
162 nnssnn0 11892 . . . . . 6 ℕ ⊆ ℕ0
163 nfcv 2981 . . . . . . 7 𝑎
164 nfcv 2981 . . . . . . 7 𝑎0
165163, 164ssrmof 4035 . . . . . 6 (ℕ ⊆ ℕ0 → (∃*𝑎 ∈ ℕ0𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
166162, 165ax-mp 5 . . . . 5 (∃*𝑎 ∈ ℕ0𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
167 ssrexv 4037 . . . . . . 7 (ℕ ⊆ ℕ0 → (∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
168162, 167ax-mp 5 . . . . . 6 (∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
169168rmoimi 3736 . . . . 5 (∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ0 (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
170161, 166, 1693syl 18 . . . 4 (𝑃 ∈ ℙ → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
171170adantr 481 . . 3 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
172 rmoim 3734 . . 3 (∀𝑎 ∈ ℕ (∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (∃*𝑎 ∈ ℕ ∃𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃*𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
173160, 171, 172sylc 65 . 2 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃*𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
174 reu5 3435 . 2 (∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ∃*𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
175157, 173, 174sylanbrc 583 1 ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃!𝑎 ∈ ℕ ∃!𝑏 ∈ ℕ (𝑎𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  wral 3142  wrex 3143  ∃!wreu 3144  ∃*wrmo 3145  wss 3939   class class class wbr 5062  (class class class)co 7151  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   < clt 10667  cle 10668  cn 11630  2c2 11684  4c4 11686  0cn0 11889   mod cmo 13230  cexp 13422  cprime 16007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-ofr 7403  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-tpos 7886  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-ec 8284  df-qs 8288  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12383  df-fz 12886  df-fzo 13027  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-dvds 15600  df-gcd 15836  df-prm 16008  df-phi 16095  df-pc 16166  df-gz 16258  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-imas 16773  df-qus 16774  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-mhm 17946  df-submnd 17947  df-grp 18038  df-minusg 18039  df-sbg 18040  df-mulg 18157  df-subg 18208  df-nsg 18209  df-eqg 18210  df-ghm 18288  df-cntz 18379  df-cmn 18830  df-abl 18831  df-mgp 19162  df-ur 19174  df-srg 19178  df-ring 19221  df-cring 19222  df-oppr 19295  df-dvdsr 19313  df-unit 19314  df-invr 19344  df-dvr 19355  df-rnghom 19389  df-drng 19426  df-field 19427  df-subrg 19455  df-lmod 19558  df-lss 19626  df-lsp 19666  df-sra 19866  df-rgmod 19867  df-lidl 19868  df-rsp 19869  df-2idl 19926  df-nzr 19952  df-rlreg 19977  df-domn 19978  df-idom 19979  df-assa 20006  df-asp 20007  df-ascl 20008  df-psr 20057  df-mvr 20058  df-mpl 20059  df-opsr 20061  df-evls 20206  df-evl 20207  df-psr1 20265  df-vr1 20266  df-ply1 20267  df-coe1 20268  df-evl1 20396  df-cnfld 20462  df-zring 20534  df-zrh 20567  df-zn 20570  df-mdeg 24564  df-deg1 24565  df-mon1 24639  df-uc1p 24640  df-q1p 24641  df-r1p 24642  df-lgs 25785
This theorem is referenced by:  2sqreunnltlem  25940  2sqreunn  25947
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