Step | Hyp | Ref
| Expression |
1 | | 2sqnn0 26586 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) → ∃𝑥 ∈ ℕ0
∃𝑦 ∈
ℕ0 𝑃 =
((𝑥↑2) + (𝑦↑2))) |
2 | | simpll 764 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑥 ∈ ℕ0) |
3 | 2 | adantl 482 |
. . . . . . . 8
⊢ ((𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑥 ∈ ℕ0) |
4 | | breq1 5077 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏)) |
5 | | oveq1 7282 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑥 → (𝑎↑2) = (𝑥↑2)) |
6 | 5 | oveq1d 7290 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑥 → ((𝑎↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑏↑2))) |
7 | 6 | eqeq1d 2740 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑥 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = 𝑃)) |
8 | 4, 7 | anbi12d 631 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑥 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))) |
9 | 8 | reubidv 3323 |
. . . . . . . . 9
⊢ (𝑎 = 𝑥 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = 𝑃))) |
10 | 9 | adantl 482 |
. . . . . . . 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 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → 𝑦 ∈ ℕ0) |
12 | 11 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑦 ∈ ℕ0) |
13 | | breq2 5078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑦 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦)) |
14 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑦 → (𝑏↑2) = (𝑦↑2)) |
15 | 14 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑦 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))) |
16 | 15 | eqeq1d 2740 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑦 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
17 | 13, 16 | anbi12d 631 |
. . . . . . . . . . . . . . . 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 2028 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑦 → (𝑏 = 𝑐 ↔ 𝑦 = 𝑐)) |
19 | 18 | imbi2d 341 |
. . . . . . . . . . . . . . . . 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 3112 |
. . . . . . . . . . . . . . . 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 631 |
. . . . . . . . . . . . . . 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 482 |
. . . . . . . . . . . . . 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 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝑦) |
24 | | eqidd 2739 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ((𝑥↑2) + (𝑦↑2)) = ((𝑥↑2) + (𝑦↑2))) |
25 | | nn0re 12242 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 ∈ ℕ0
→ 𝑐 ∈
ℝ) |
26 | 25 | resqcld 13965 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 ∈ ℕ0
→ (𝑐↑2) ∈
ℝ) |
27 | 26 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈
ℝ) |
28 | | nn0re 12242 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℝ) |
29 | 28 | resqcld 13965 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ (𝑦↑2) ∈
ℝ) |
30 | 29 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦↑2) ∈ ℝ) |
31 | 30 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈
ℝ) |
32 | | nn0re 12242 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℕ0
→ 𝑥 ∈
ℝ) |
33 | 32 | resqcld 13965 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ ℕ0
→ (𝑥↑2) ∈
ℝ) |
34 | 33 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥↑2) ∈ ℝ) |
35 | 34 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈
ℝ) |
36 | | readdcan 11149 |
. . . . . . . . . . . . . . . . . . 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 1370 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑦↑2))) |
38 | 28 | ad4antlr 730 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 ∈ ℝ) |
39 | 25 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑐 ∈ ℝ) |
40 | | nn0ge0 12258 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ 0 ≤ 𝑦) |
41 | 40 | ad4antlr 730 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑦) |
42 | | nn0ge0 12258 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 ∈ ℕ0
→ 0 ≤ 𝑐) |
43 | 42 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 0 ≤ 𝑐) |
44 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑐↑2) = (𝑦↑2)) |
45 | 44 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → (𝑦↑2) = (𝑐↑2)) |
46 | 38, 39, 41, 43, 45 | sq11d 13975 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝑥 ∈
ℕ0 ∧ 𝑦
∈ ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑦↑2)) → 𝑦 = 𝑐) |
47 | 46 | ex 413 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑦↑2) → 𝑦 = 𝑐)) |
48 | 37, 47 | sylbid 239 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → (((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑦 = 𝑐)) |
49 | 48 | adantld 491 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) ∧ 𝑐 ∈ ℕ0) → ((𝑥 ≤ 𝑐 ∧ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑦 = 𝑐)) |
50 | 49 | ralrimiva 3103 |
. . . . . . . . . . . . . . 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 515 |
. . . . . . . . . . . . . 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 3563 |
. . . . . . . . . . . . 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 5078 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐)) |
54 | | oveq1 7282 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2)) |
55 | 54 | oveq2d 7291 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑐 → ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑐↑2))) |
56 | 55 | eqeq1d 2740 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑥↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
57 | 53, 56 | anbi12d 631 |
. . . . . . . . . . . . . 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 3668 |
. . . . . . . . . . . . 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 233 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
60 | 59 | ex 413 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑥 ≤ 𝑏 ∧ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
61 | 60 | adantr 481 |
. . . . . . . . . 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 408 |
. . . . . . . . 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 2750 |
. . . . . . . . . . . . 13
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑥↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑥↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
64 | 63 | anbi2d 629 |
. . . . . . . . . . . 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 3323 |
. . . . . . . . . . 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 482 |
. . . . . . . . . 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 482 |
. . . . . . . . 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 256 |
. . . . . . . 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 3563 |
. . . . . . 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 481 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → 𝑦 ∈ ℕ0) |
71 | 70 | adantl 482 |
. . . . . . . 8
⊢ ((¬
𝑥 ≤ 𝑦 ∧ ((𝑥 ∈ ℕ0 ∧ 𝑦 ∈ ℕ0)
∧ 𝑃 = ((𝑥↑2) + (𝑦↑2)))) → 𝑦 ∈ ℕ0) |
72 | | breq1 5077 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑦 → (𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏)) |
73 | | oveq1 7282 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑦 → (𝑎↑2) = (𝑦↑2)) |
74 | 73 | oveq1d 7290 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑦 → ((𝑎↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑏↑2))) |
75 | 74 | eqeq1d 2740 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑦 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = 𝑃)) |
76 | 72, 75 | anbi12d 631 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑦 → ((𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))) |
77 | 76 | reubidv 3323 |
. . . . . . . . 9
⊢ (𝑎 = 𝑦 → (∃!𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = 𝑃))) |
78 | 77 | adantl 482 |
. . . . . . . 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 764 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑥 ∈ ℕ0) |
80 | | breq2 5078 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑥 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥)) |
81 | | oveq1 7282 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑥 → (𝑏↑2) = (𝑥↑2)) |
82 | 81 | oveq2d 7291 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑥 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑥↑2))) |
83 | 82 | eqeq1d 2740 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑥 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
84 | 80, 83 | anbi12d 631 |
. . . . . . . . . . . . . . . 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 2028 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑥 → (𝑏 = 𝑐 ↔ 𝑥 = 𝑐)) |
86 | 85 | imbi2d 341 |
. . . . . . . . . . . . . . . . 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 3112 |
. . . . . . . . . . . . . . . 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 631 |
. . . . . . . . . . . . . . 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 482 |
. . . . . . . . . . . . . 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 11054 |
. . . . . . . . . . . . . . . . . 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 724 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ∈ ℝ) |
93 | 32 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑥 ∈ ℝ) |
94 | | simpr 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 < 𝑥) |
95 | 92, 93, 94 | ltled 11123 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑦 < 𝑥) → 𝑦 ≤ 𝑥) |
96 | 95 | ex 413 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥)) |
97 | 91, 96 | sylbird 259 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥)) |
98 | 97 | imp 407 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → 𝑦 ≤ 𝑥) |
99 | 29 | recnd 11003 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈ ℕ0
→ (𝑦↑2) ∈
ℂ) |
100 | 99 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦↑2) ∈ ℂ) |
101 | 33 | recnd 11003 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℕ0
→ (𝑥↑2) ∈
ℂ) |
102 | 101 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥↑2) ∈ ℂ) |
103 | 100, 102 | addcomd 11177 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) |
104 | 103 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ((𝑦↑2) + (𝑥↑2)) = ((𝑥↑2) + (𝑦↑2))) |
105 | 34 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑥↑2) ∈ ℂ) |
106 | 105 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈
ℂ) |
107 | 30 | recnd 11003 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑦↑2) ∈ ℂ) |
108 | 107 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈
ℂ) |
109 | 106, 108 | addcomd 11177 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑥↑2) + (𝑦↑2)) = ((𝑦↑2) + (𝑥↑2))) |
110 | 109 | eqeq2d 2749 |
. . . . . . . . . . . . . . . . . . . 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 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑐↑2) ∈
ℝ) |
112 | 33 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑥↑2) ∈
ℝ) |
113 | 29 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (𝑦↑2) ∈
ℝ) |
114 | | readdcan 11149 |
. . . . . . . . . . . . . . . . . . . . 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 1370 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑦↑2) + (𝑥↑2)) ↔ (𝑐↑2) = (𝑥↑2))) |
116 | 110, 115 | bitrd 278 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ (𝑐↑2) = (𝑥↑2))) |
117 | 25 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 ∈ ℝ) |
118 | 32 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → 𝑥 ∈ ℝ) |
119 | 118 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 ∈ ℝ) |
120 | 42 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑐) |
121 | | nn0ge0 12258 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ ℕ0
→ 0 ≤ 𝑥) |
122 | 121 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → 0 ≤ 𝑥) |
123 | 122 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 0 ≤ 𝑥) |
124 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → (𝑐↑2) = (𝑥↑2)) |
125 | 117, 119,
120, 123, 124 | sq11d 13975 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑐 = 𝑥) |
126 | 125 | eqcomd 2744 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) ∧ (𝑐↑2) = (𝑥↑2)) → 𝑥 = 𝑐) |
127 | 126 | ex 413 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑐↑2) = (𝑥↑2) → 𝑥 = 𝑐)) |
128 | 116, 127 | sylbid 239 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → (((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)) → 𝑥 = 𝑐)) |
129 | 128 | adantld 491 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑐 ∈ ℕ0) → ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)) |
130 | 129 | ralrimiva 3103 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → ∀𝑐 ∈ ℕ0 ((𝑦 ≤ 𝑐 ∧ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2))) → 𝑥 = 𝑐)) |
131 | 130 | adantr 481 |
. . . . . . . . . . . . . . 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 515 |
. . . . . . . . . . . . . 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 3563 |
. . . . . . . . . . . . 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 5078 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐)) |
135 | 54 | oveq2d 7291 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑐 → ((𝑦↑2) + (𝑏↑2)) = ((𝑦↑2) + (𝑐↑2))) |
136 | 135 | eqeq1d 2740 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑐 → (((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)) ↔ ((𝑦↑2) + (𝑐↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
137 | 134, 136 | anbi12d 631 |
. . . . . . . . . . . . . 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 3668 |
. . . . . . . . . . . . 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 233 |
. . . . . . . . . . . 12
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ ¬ 𝑥 ≤ 𝑦) → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
140 | 139 | ex 413 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (¬ 𝑥 ≤ 𝑦 → ∃!𝑏 ∈ ℕ0 (𝑦 ≤ 𝑏 ∧ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2))))) |
141 | 140 | adantr 481 |
. . . . . . . . . 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 408 |
. . . . . . . . 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 2750 |
. . . . . . . . . . . . 13
⊢ (𝑃 = ((𝑥↑2) + (𝑦↑2)) → (((𝑦↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑦↑2) + (𝑏↑2)) = ((𝑥↑2) + (𝑦↑2)))) |
144 | 143 | anbi2d 629 |
. . . . . . . . . . . 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 3323 |
. . . . . . . . . . 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 482 |
. . . . . . . . . 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 482 |
. . . . . . . . 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 256 |
. . . . . . . 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 3563 |
. . . . . . 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 809 |
. . . . . 6
⊢ (((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) ∧ 𝑃 = ((𝑥↑2) + (𝑦↑2))) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
151 | 150 | ex 413 |
. . . . 5
⊢ ((𝑥 ∈ ℕ0
∧ 𝑦 ∈
ℕ0) → (𝑃 = ((𝑥↑2) + (𝑦↑2)) → ∃𝑎 ∈ ℕ0 ∃!𝑏 ∈ ℕ0
(𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃))) |
152 | 151 | adantl 482 |
. . . 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 3223 |
. . 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 3362 |
. . . . 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 3103 |
. . 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 26585 |
. . . 4
⊢ (𝑃 ∈ ℙ →
∃*𝑎 ∈
ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
159 | 158 | adantr 481 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 1) →
∃*𝑎 ∈
ℕ0 ∃𝑏 ∈ ℕ0 (𝑎 ≤ 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
160 | | rmoim 3675 |
. . 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 3361 |
. 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)) |