Proof of Theorem pythagtriplem13
Step | Hyp | Ref
| Expression |
1 | | pythagtriplem13.1 |
. 2
⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2) |
2 | | pythagtriplem9 15367 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℕ) |
3 | 2 | nnzd 11357 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℤ) |
4 | | simp3r 1083 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴) |
5 | | simp3 1056 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈
ℕ) |
6 | | simp2 1055 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈
ℕ) |
7 | 5, 6 | nnaddcld 10944 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℕ) |
8 | 7 | nnzd 11357 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℤ) |
9 | 8 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ) |
10 | | nnz 11276 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℤ) |
11 | 10 | 3ad2ant1 1075 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈
ℤ) |
12 | 11 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ) |
13 | | 2z 11286 |
. . . . . . . . . . 11
⊢ 2 ∈
ℤ |
14 | | dvdsgcdb 15100 |
. . . . . . . . . . 11
⊢ ((2
∈ ℤ ∧ (𝐶 +
𝐵) ∈ ℤ ∧
𝐴 ∈ ℤ) →
((2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴))) |
15 | 13, 14 | mp3an1 1403 |
. . . . . . . . . 10
⊢ (((𝐶 + 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴))) |
16 | 9, 12, 15 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴))) |
17 | 16 | biimpar 501 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)) → (2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴)) |
18 | 17 | simprd 478 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)) → 2 ∥ 𝐴) |
19 | 4, 18 | mtand 689 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥
((𝐶 + 𝐵) gcd 𝐴)) |
20 | | pythagtriplem7 15365 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) = ((𝐶 + 𝐵) gcd 𝐴)) |
21 | 20 | breq2d 4595 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ∥
(√‘(𝐶 + 𝐵)) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴))) |
22 | 19, 21 | mtbird 314 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥
(√‘(𝐶 + 𝐵))) |
23 | | pythagtriplem8 15366 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℕ) |
24 | 23 | nnzd 11357 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℤ) |
25 | | nnz 11276 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ ℕ → 𝐶 ∈
ℤ) |
26 | 25 | 3ad2ant3 1077 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈
ℤ) |
27 | | nnz 11276 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℤ) |
28 | 27 | 3ad2ant2 1076 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈
ℤ) |
29 | 26, 28 | zsubcld 11363 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ) |
30 | 29 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℤ) |
31 | | dvdsgcdb 15100 |
. . . . . . . . . . 11
⊢ ((2
∈ ℤ ∧ (𝐶
− 𝐵) ∈ ℤ
∧ 𝐴 ∈ ℤ)
→ ((2 ∥ (𝐶
− 𝐵) ∧ 2 ∥
𝐴) ↔ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴))) |
32 | 13, 31 | mp3an1 1403 |
. . . . . . . . . 10
⊢ (((𝐶 − 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((2 ∥ (𝐶 − 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴))) |
33 | 30, 12, 32 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 ∥ (𝐶 − 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴))) |
34 | 33 | biimpar 501 |
. . . . . . . 8
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)) → (2 ∥ (𝐶 − 𝐵) ∧ 2 ∥ 𝐴)) |
35 | 34 | simprd 478 |
. . . . . . 7
⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)) → 2 ∥ 𝐴) |
36 | 4, 35 | mtand 689 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥
((𝐶 − 𝐵) gcd 𝐴)) |
37 | | pythagtriplem6 15364 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) = ((𝐶 − 𝐵) gcd 𝐴)) |
38 | 37 | breq2d 4595 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ∥
(√‘(𝐶 −
𝐵)) ↔ 2 ∥
((𝐶 − 𝐵) gcd 𝐴))) |
39 | 36, 38 | mtbird 314 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥
(√‘(𝐶 −
𝐵))) |
40 | | omoe 14926 |
. . . . 5
⊢
((((√‘(𝐶
+ 𝐵)) ∈ ℤ ∧
¬ 2 ∥ (√‘(𝐶 + 𝐵))) ∧ ((√‘(𝐶 − 𝐵)) ∈ ℤ ∧ ¬ 2 ∥
(√‘(𝐶 −
𝐵)))) → 2 ∥
((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))) |
41 | 3, 22, 24, 39, 40 | syl22anc 1319 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 2 ∥
((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))) |
42 | 29 | zred 11358 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ) |
43 | 42 | 3ad2ant1 1075 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℝ) |
44 | | simp13 1086 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℕ) |
45 | 44 | nnred 10912 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℝ) |
46 | 7 | nnred 10912 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ) |
47 | 46 | 3ad2ant1 1075 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ) |
48 | | nnrp 11718 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℝ+) |
49 | 48 | 3ad2ant2 1076 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈
ℝ+) |
50 | 49 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈
ℝ+) |
51 | 45, 50 | ltsubrpd 11780 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) < 𝐶) |
52 | | nngt0 10926 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ ℕ → 0 <
𝐵) |
53 | 52 | 3ad2ant2 1076 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 <
𝐵) |
54 | 53 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < 𝐵) |
55 | | simp12 1085 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℕ) |
56 | 55 | nnred 10912 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℝ) |
57 | 56, 45 | ltaddposd 10490 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (0 < 𝐵 ↔ 𝐶 < (𝐶 + 𝐵))) |
58 | 54, 57 | mpbid 221 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 < (𝐶 + 𝐵)) |
59 | 43, 45, 47, 51, 58 | lttrd 10077 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) < (𝐶 + 𝐵)) |
60 | | pythagtriplem10 15363 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 − 𝐵)) |
61 | 60 | 3adant3 1074 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 − 𝐵)) |
62 | | 0re 9919 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
63 | | ltle 10005 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ (𝐶
− 𝐵) ∈ ℝ)
→ (0 < (𝐶 −
𝐵) → 0 ≤ (𝐶 − 𝐵))) |
64 | 62, 63 | mpan 702 |
. . . . . . . . . 10
⊢ ((𝐶 − 𝐵) ∈ ℝ → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵))) |
65 | 43, 61, 64 | sylc 63 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 − 𝐵)) |
66 | | nngt0 10926 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ ℕ → 0 <
𝐶) |
67 | 66 | 3ad2ant3 1077 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 <
𝐶) |
68 | 67 | 3ad2ant1 1075 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < 𝐶) |
69 | 45, 56, 68, 54 | addgt0d 10481 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 + 𝐵)) |
70 | | ltle 10005 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ (𝐶 +
𝐵) ∈ ℝ) →
(0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵))) |
71 | 62, 70 | mpan 702 |
. . . . . . . . . 10
⊢ ((𝐶 + 𝐵) ∈ ℝ → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵))) |
72 | 47, 69, 71 | sylc 63 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 + 𝐵)) |
73 | 43, 65, 47, 72 | sqrtltd 14014 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 − 𝐵) < (𝐶 + 𝐵) ↔ (√‘(𝐶 − 𝐵)) < (√‘(𝐶 + 𝐵)))) |
74 | 59, 73 | mpbid 221 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) < (√‘(𝐶 + 𝐵))) |
75 | | nnsub 10936 |
. . . . . . . 8
⊢
(((√‘(𝐶
− 𝐵)) ∈ ℕ
∧ (√‘(𝐶 +
𝐵)) ∈ ℕ) →
((√‘(𝐶 −
𝐵)) <
(√‘(𝐶 + 𝐵)) ↔ ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℕ)) |
76 | 23, 2, 75 | syl2anc 691 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 − 𝐵)) < (√‘(𝐶 + 𝐵)) ↔ ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℕ)) |
77 | 74, 76 | mpbid 221 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℕ) |
78 | 77 | nnzd 11357 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℤ) |
79 | | 2ne0 10990 |
. . . . . 6
⊢ 2 ≠
0 |
80 | | dvdsval2 14824 |
. . . . . 6
⊢ ((2
∈ ℤ ∧ 2 ≠ 0 ∧ ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℤ) → (2 ∥
((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ↔ (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ)) |
81 | 13, 79, 80 | mp3an12 1406 |
. . . . 5
⊢
(((√‘(𝐶
+ 𝐵)) −
(√‘(𝐶 −
𝐵))) ∈ ℤ →
(2 ∥ ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ↔ (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ)) |
82 | 78, 81 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ∥
((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ↔ (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ)) |
83 | 41, 82 | mpbid 221 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
(((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ) |
84 | 77 | nngt0d 10941 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 <
((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵)))) |
85 | 77 | nnred 10912 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ∈ ℝ) |
86 | | halfpos2 11138 |
. . . . 5
⊢
(((√‘(𝐶
+ 𝐵)) −
(√‘(𝐶 −
𝐵))) ∈ ℝ →
(0 < ((√‘(𝐶
+ 𝐵)) −
(√‘(𝐶 −
𝐵))) ↔ 0 <
(((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2))) |
87 | 85, 86 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (0 <
((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) ↔ 0 < (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2))) |
88 | 84, 87 | mpbid 221 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 <
(((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)) |
89 | | elnnz 11264 |
. . 3
⊢
((((√‘(𝐶
+ 𝐵)) −
(√‘(𝐶 −
𝐵))) / 2) ∈ ℕ
↔ ((((√‘(𝐶
+ 𝐵)) −
(√‘(𝐶 −
𝐵))) / 2) ∈ ℤ
∧ 0 < (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2))) |
90 | 83, 88, 89 | sylanbrc 695 |
. 2
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
(((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2) ∈ ℕ) |
91 | 1, 90 | syl5eqel 2692 |
1
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑁 ∈ ℕ) |