| Step | Hyp | Ref
| Expression |
| 1 | | 2sqnn0 27482 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ0
∃𝑦 ∈
ℕ0 𝑃 =
((𝑥↑2) + (𝑦↑2))) |
| 2 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ0) |
| 3 | 2 | adantl 481 |
. . . . . . . 8
⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ0) |
| 4 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏)) |
| 5 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2)) |
| 6 | 5 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2))) |
| 7 | 6 | eqeq1d 2739 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃)) |
| 8 | 4, 7 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))) |
| 9 | 8 | reubidv 3398 |
. . . . . . . . 9
⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))) |
| 10 | 9 | adantl 481 |
. . . . . . . 8
⊢ (((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑥) → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))) |
| 11 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → 𝑦 ∈ ℕ0) |
| 12 | 11 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ0) |
| 13 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦)) |
| 14 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2)) |
| 15 | 14 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) |
| 16 | 15 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 17 | 13, 16 | anbi12d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑦 → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 18 | | equequ1 2024 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑦 → (𝑏 = 𝑐 ↔ 𝑦 = 𝑐)) |
| 19 | 18 | imbi2d 340 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑦 → (((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))) |
| 20 | 19 | ralbidv 3178 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑦 → (∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))) |
| 21 | 17, 20 | anbi12d 632 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑦 → (((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))) |
| 22 | 21 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑏 = 𝑦) → (((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)))) |
| 23 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦) |
| 24 | | eqidd 2738 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) |
| 25 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 ∈ ℕ0
→ 𝑐 ∈
ℝ) |
| 26 | 25 | resqcld 14165 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 ∈ ℕ0
→ (𝑐↑2) ∈
ℝ) |
| 27 | 26 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈
ℝ) |
| 28 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℝ) |
| 29 | 28 | resqcld 14165 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ (𝑦↑2) ∈
ℝ) |
| 30 | 29 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦↑2) ∈ ℝ) |
| 31 | 30 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈
ℝ) |
| 32 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℕ0
→ 𝑥 ∈
ℝ) |
| 33 | 32 | resqcld 14165 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ℕ0
→ (𝑥↑2) ∈
ℝ) |
| 34 | 33 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥↑2) ∈ ℝ) |
| 35 | 34 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈
ℝ) |
| 36 | | readdcan 11435 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑐↑2) ∈ ℝ ∧
(𝑦↑2) ∈ ℝ
∧ (𝑥↑2) ∈
ℝ) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2))) |
| 37 | 27, 31, 35, 36 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2))) |
| 38 | 28 | ad4antlr 733 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ) |
| 39 | 25 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ) |
| 40 | | nn0ge0 12551 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ 0 ≤ 𝑦) |
| 41 | 40 | ad4antlr 733 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦) |
| 42 | | nn0ge0 12551 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 ∈ ℕ0
→ 0 ≤ 𝑐) |
| 43 | 42 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐) |
| 44 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2)) |
| 45 | 44 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2)) |
| 46 | 38, 39, 41, 43, 45 | sq11d 14297 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐) |
| 47 | 46 | ex 412 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐)) |
| 48 | 37, 47 | sylbid 240 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐)) |
| 49 | 48 | adantld 490 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)) |
| 50 | 49 | ralrimiva 3146 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)) |
| 51 | 23, 24, 50 | jca31 514 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥 ≤ 𝑦 ∧ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐))) |
| 52 | 12, 22, 51 | rspcedvd 3624 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃𝑏 ∈ ℕ0 ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐))) |
| 53 | | breq2 5147 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐)) |
| 54 | | oveq1 7438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2)) |
| 55 | 54 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2))) |
| 56 | 55 | eqeq1d 2739 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 57 | 53, 56 | anbi12d 632 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑐 → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 58 | 57 | reu8 3739 |
. . . . . . . . . . . . 13
⊢
(∃!𝑏 ∈
ℕ0 (𝑥 ≤
𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ0 ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐))) |
| 59 | 52, 58 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 60 | 59 | ex 412 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 61 | 60 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 62 | 61 | impcom 407 |
. . . . . . . . 9
⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 63 | | eqeq2 2749 |
. . . . . . . . . . . . 13
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 64 | 63 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 65 | 64 | reubidv 3398 |
. . . . . . . . . . 11
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 66 | 65 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 67 | 66 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ0
(𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 68 | 62, 67 | mpbird 257 |
. . . . . . . 8
⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃)) |
| 69 | 3, 10, 68 | rspcedvd 3624 |
. . . . . . 7
⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 70 | 11 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ0) |
| 71 | 70 | adantl 481 |
. . . . . . . 8
⊢ ((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ0) |
| 72 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏)) |
| 73 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2)) |
| 74 | 73 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2))) |
| 75 | 74 | eqeq1d 2739 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃)) |
| 76 | 72, 75 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))) |
| 77 | 76 | reubidv 3398 |
. . . . . . . . 9
⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))) |
| 78 | 77 | adantl 481 |
. . . . . . . 8
⊢ (((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) ∧ 𝑎 = 𝑦) → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))) |
| 79 | | simpll 767 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ0) |
| 80 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥)) |
| 81 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2)) |
| 82 | 81 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2))) |
| 83 | 82 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 84 | 80, 83 | anbi12d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑥 → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 85 | | equequ1 2024 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐)) |
| 86 | 85 | imbi2d 340 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑥 → (((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))) |
| 87 | 86 | ralbidv 3178 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑥 → (∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐) ↔ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))) |
| 88 | 84, 87 | anbi12d 632 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑥 → (((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))) |
| 89 | 88 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) ∧ 𝑏 = 𝑥) → (((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐)) ↔ ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)))) |
| 90 | | ltnle 11340 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) |
| 91 | 28, 32, 90 | syl2anr 597 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦)) |
| 92 | 28 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ) |
| 93 | 32 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ) |
| 94 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥) |
| 95 | 92, 93, 94 | ltled 11409 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥) |
| 96 | 95 | ex 412 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥)) |
| 97 | 91, 96 | sylbird 260 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥)) |
| 98 | 97 | imp 406 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥) |
| 99 | 29 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℕ0
→ (𝑦↑2) ∈
ℂ) |
| 100 | 99 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦↑2) ∈ ℂ) |
| 101 | 33 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℕ0
→ (𝑥↑2) ∈
ℂ) |
| 102 | 101 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥↑2) ∈ ℂ) |
| 103 | 100, 102 | addcomd 11463 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) |
| 104 | 103 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) |
| 105 | 34 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥↑2) ∈ ℂ) |
| 106 | 105 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈
ℂ) |
| 107 | 30 | recnd 11289 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦↑2) ∈ ℂ) |
| 108 | 107 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈
ℂ) |
| 109 | 106, 108 | addcomd 11463 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2))) |
| 110 | 109 | eqeq2d 2748 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)))) |
| 111 | 26 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈
ℝ) |
| 112 | 33 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈
ℝ) |
| 113 | 29 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈
ℝ) |
| 114 | | readdcan 11435 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑐↑2) ∈ ℝ ∧
(𝑥↑2) ∈ ℝ
∧ (𝑦↑2) ∈
ℝ) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2))) |
| 115 | 111, 112,
113, 114 | syl3anc 1373 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2))) |
| 116 | 110, 115 | bitrd 279 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2))) |
| 117 | 25 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ) |
| 118 | 32 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → 𝑥 ∈ ℝ) |
| 119 | 118 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ) |
| 120 | 42 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐) |
| 121 | | nn0ge0 12551 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ ℕ0
→ 0 ≤ 𝑥) |
| 122 | 121 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → 0 ≤ 𝑥) |
| 123 | 122 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥) |
| 124 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2)) |
| 125 | 117, 119,
120, 123, 124 | sq11d 14297 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥) |
| 126 | 125 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐) |
| 127 | 126 | ex 412 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐)) |
| 128 | 116, 127 | sylbid 240 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐)) |
| 129 | 128 | adantld 490 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)) |
| 130 | 129 | ralrimiva 3146 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)) |
| 131 | 130 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)) |
| 132 | 98, 104, 131 | jca31 514 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦 ≤ 𝑥 ∧ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐))) |
| 133 | 79, 89, 132 | rspcedvd 3624 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃𝑏 ∈ ℕ0 ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐))) |
| 134 | | breq2 5147 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐)) |
| 135 | 54 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2))) |
| 136 | 135 | eqeq1d 2739 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 137 | 134, 136 | anbi12d 632 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑐 → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ (𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 138 | 137 | reu8 3739 |
. . . . . . . . . . . . 13
⊢
(∃!𝑏 ∈
ℕ0 (𝑦 ≤
𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑏 ∈ ℕ0 ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) ∧ ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑏 = 𝑐))) |
| 139 | 133, 138 | sylibr 234 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 140 | 139 | ex 412 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 141 | 140 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 142 | 141 | impcom 407 |
. . . . . . . . 9
⊢ ((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 143 | | eqeq2 2749 |
. . . . . . . . . . . . 13
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
| 144 | 143 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ((𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 145 | 144 | reubidv 3398 |
. . . . . . . . . . 11
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 146 | 145 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → (∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 147 | 146 | adantl 481 |
. . . . . . . . 9
⊢ ((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → (∃!𝑏 ∈ ℕ0
(𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
| 148 | 142, 147 | mpbird 257 |
. . . . . . . 8
⊢ ((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃)) |
| 149 | 71, 78, 148 | rspcedvd 3624 |
. . . . . . 7
⊢ ((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 150 | 69, 149 | pm2.61ian 812 |
. . . . . 6
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 151 | 150 | ex 412 |
. . . . 5
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 152 | 151 | adantl 481 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ (𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0)) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 153 | 152 | rexlimdvva 3213 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) →
(∃𝑥 ∈
ℕ0 ∃𝑦 ∈ ℕ0 𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 154 | 1, 153 | mpd 15 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑎 ∈ ℕ0
∃!𝑏 ∈
ℕ0 (𝑎 ≤
𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 155 | | reurex 3384 |
. . . . 5
⊢
(∃!𝑏 ∈
ℕ0 (𝑎 ≤
𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 156 | 155 | a1i 11 |
. . . 4
⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) ∧ 𝑎 ∈ ℕ0)
→ (∃!𝑏 ∈
ℕ0 (𝑎 ≤
𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 157 | 156 | ralrimiva 3146 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) →
∀𝑎 ∈
ℕ0 (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 158 | | 2sqmo 27481 |
. . . 4
⊢ (𝑃 ∈ ℙ →
∃*𝑎 ∈
ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 159 | 158 | adantr 480 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) →
∃*𝑎 ∈
ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 160 | | rmoim 3746 |
. . 3
⊢
(∀𝑎 ∈
ℕ0 (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) → (∃*𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) → ∃*𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 161 | 157, 159,
160 | sylc 65 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) →
∃*𝑎 ∈
ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
| 162 | | reu5 3382 |
. 2
⊢
(∃!𝑎 ∈
ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ∃*𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
| 163 | 154, 161,
162 | sylanbrc 583 |
1
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) →
∃!𝑎 ∈
ℕ0 ∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |