Proof of Theorem pythagtriplem12
Step | Hyp | Ref
| Expression |
1 | | pythagtriplem11.1 |
. . 3
⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) |
2 | 1 | oveq1i 6559 |
. 2
⊢ (𝑀↑2) =
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) |
3 | | nncn 10905 |
. . . . . . . . 9
⊢ (𝐶 ∈ ℕ → 𝐶 ∈
ℂ) |
4 | | nncn 10905 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℂ) |
5 | | addcl 9897 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 + 𝐵) ∈ ℂ) |
6 | 3, 4, 5 | syl2anr 494 |
. . . . . . . 8
⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ) |
7 | 6 | 3adant1 1072 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℂ) |
8 | 7 | sqrtcld 14024 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) →
(√‘(𝐶 + 𝐵)) ∈
ℂ) |
9 | | subcl 10159 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶 − 𝐵) ∈ ℂ) |
10 | 3, 4, 9 | syl2anr 494 |
. . . . . . . 8
⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℂ) |
11 | 10 | 3adant1 1072 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℂ) |
12 | 11 | sqrtcld 14024 |
. . . . . 6
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) →
(√‘(𝐶 −
𝐵)) ∈
ℂ) |
13 | 8, 12 | addcld 9938 |
. . . . 5
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) →
((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ) |
14 | 13 | 3ad2ant1 1075 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℂ) |
15 | | 2cn 10968 |
. . . . . 6
⊢ 2 ∈
ℂ |
16 | | 2ne0 10990 |
. . . . . 6
⊢ 2 ≠
0 |
17 | | sqdiv 12790 |
. . . . . 6
⊢
((((√‘(𝐶
+ 𝐵)) +
(√‘(𝐶 −
𝐵))) ∈ ℂ ∧ 2
∈ ℂ ∧ 2 ≠ 0) → ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2↑2))) |
18 | 15, 16, 17 | mp3an23 1408 |
. . . . 5
⊢
(((√‘(𝐶
+ 𝐵)) +
(√‘(𝐶 −
𝐵))) ∈ ℂ →
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2↑2))) |
19 | 15 | sqvali 12805 |
. . . . . 6
⊢
(2↑2) = (2 · 2) |
20 | 19 | oveq2i 6560 |
. . . . 5
⊢
((((√‘(𝐶
+ 𝐵)) +
(√‘(𝐶 −
𝐵)))↑2) / (2↑2))
= ((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 ·
2)) |
21 | 18, 20 | syl6eq 2660 |
. . . 4
⊢
(((√‘(𝐶
+ 𝐵)) +
(√‘(𝐶 −
𝐵))) ∈ ℂ →
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 ·
2))) |
22 | 14, 21 | syl 17 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 ·
2))) |
23 | 8 | 3ad2ant1 1075 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℂ) |
24 | 12 | 3ad2ant1 1075 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℂ) |
25 | | binom2 12841 |
. . . . . . 7
⊢
(((√‘(𝐶
+ 𝐵)) ∈ ℂ ∧
(√‘(𝐶 −
𝐵)) ∈ ℂ) →
(((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵))))) +
((√‘(𝐶 −
𝐵))↑2))) |
26 | 23, 24, 25 | syl2anc 691 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
(((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) = ((((√‘(𝐶 + 𝐵))↑2) + (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵))))) +
((√‘(𝐶 −
𝐵))↑2))) |
27 | | nnre 10904 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ ℕ → 𝐶 ∈
ℝ) |
28 | | nnre 10904 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℝ) |
29 | | readdcl 9898 |
. . . . . . . . . . . 12
⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + 𝐵) ∈ ℝ) |
30 | 27, 28, 29 | syl2anr 494 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ) |
31 | 30 | 3adant1 1072 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℝ) |
32 | 31 | 3ad2ant1 1075 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℝ) |
33 | 27 | 3ad2ant3 1077 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈
ℝ) |
34 | 28 | 3ad2ant2 1076 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈
ℝ) |
35 | | nngt0 10926 |
. . . . . . . . . . . . 13
⊢ (𝐶 ∈ ℕ → 0 <
𝐶) |
36 | 35 | 3ad2ant3 1077 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 <
𝐶) |
37 | | nngt0 10926 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ ℕ → 0 <
𝐵) |
38 | 37 | 3ad2ant2 1076 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 <
𝐵) |
39 | 33, 34, 36, 38 | addgt0d 10481 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 <
(𝐶 + 𝐵)) |
40 | 39 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 + 𝐵)) |
41 | | 0re 9919 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
42 | | ltle 10005 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ (𝐶 +
𝐵) ∈ ℝ) →
(0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵))) |
43 | 41, 42 | mpan 702 |
. . . . . . . . . 10
⊢ ((𝐶 + 𝐵) ∈ ℝ → (0 < (𝐶 + 𝐵) → 0 ≤ (𝐶 + 𝐵))) |
44 | 32, 40, 43 | sylc 63 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 + 𝐵)) |
45 | | resqrtth 13844 |
. . . . . . . . 9
⊢ (((𝐶 + 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 + 𝐵)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵)) |
46 | 32, 44, 45 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵))↑2) = (𝐶 + 𝐵)) |
47 | 46 | oveq1d 6564 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
(((√‘(𝐶 + 𝐵))↑2) + (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵))))) = ((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))))) |
48 | | resubcl 10224 |
. . . . . . . . . . 11
⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) |
49 | 27, 28, 48 | syl2anr 494 |
. . . . . . . . . 10
⊢ ((𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ) |
50 | 49 | 3adant1 1072 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℝ) |
51 | 50 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℝ) |
52 | | pythagtriplem10 15363 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) → 0 < (𝐶 − 𝐵)) |
53 | 52 | 3adant3 1074 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (𝐶 − 𝐵)) |
54 | | ltle 10005 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (𝐶
− 𝐵) ∈ ℝ)
→ (0 < (𝐶 −
𝐵) → 0 ≤ (𝐶 − 𝐵))) |
55 | 41, 54 | mpan 702 |
. . . . . . . . 9
⊢ ((𝐶 − 𝐵) ∈ ℝ → (0 < (𝐶 − 𝐵) → 0 ≤ (𝐶 − 𝐵))) |
56 | 51, 53, 55 | sylc 63 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ (𝐶 − 𝐵)) |
57 | | resqrtth 13844 |
. . . . . . . 8
⊢ (((𝐶 − 𝐵) ∈ ℝ ∧ 0 ≤ (𝐶 − 𝐵)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵)) |
58 | 51, 56, 57 | syl2anc 691 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 − 𝐵))↑2) = (𝐶 − 𝐵)) |
59 | 47, 58 | oveq12d 6567 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
((((√‘(𝐶 +
𝐵))↑2) + (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵))))) +
((√‘(𝐶 −
𝐵))↑2)) = (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵))) |
60 | 7 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℂ) |
61 | 8, 12 | mulcld 9939 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) →
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵))) ∈
ℂ) |
62 | | mulcl 9899 |
. . . . . . . . . 10
⊢ ((2
∈ ℂ ∧ ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) ∈ ℂ) → (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵)))) ∈
ℂ) |
63 | 15, 61, 62 | sylancr 694 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (2
· ((√‘(𝐶
+ 𝐵)) ·
(√‘(𝐶 −
𝐵)))) ∈
ℂ) |
64 | 63 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵)))) ∈
ℂ) |
65 | 11 | 3ad2ant1 1075 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℂ) |
66 | 60, 64, 65 | add32d 10142 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)) = (((𝐶 + 𝐵) + (𝐶 − 𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))))) |
67 | 3 | 3ad2ant3 1077 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈
ℂ) |
68 | 67 | 3ad2ant1 1075 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐶 ∈ ℂ) |
69 | | nncn 10905 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℂ) |
70 | 69 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈
ℂ) |
71 | 70 | 3ad2ant1 1075 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℂ) |
72 | | adddi 9904 |
. . . . . . . . . 10
⊢ ((2
∈ ℂ ∧ 𝐶
∈ ℂ ∧ 𝐴
∈ ℂ) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴))) |
73 | 15, 72 | mp3an1 1403 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (2
· (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴))) |
74 | 68, 71, 73 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) = ((2 · 𝐶) + (2 · 𝐴))) |
75 | 4 | 3ad2ant2 1076 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈
ℂ) |
76 | 75 | 3ad2ant1 1075 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐵 ∈ ℂ) |
77 | 68, 76, 68 | ppncand 10311 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (𝐶 + 𝐶)) |
78 | 68 | 2timesd 11152 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · 𝐶) = (𝐶 + 𝐶)) |
79 | 77, 78 | eqtr4d 2647 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) + (𝐶 − 𝐵)) = (2 · 𝐶)) |
80 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2))) |
81 | 80 | 3ad2ant2 1076 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = ((𝐶↑2) − (𝐵↑2))) |
82 | 71 | sqcld 12868 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐴↑2) ∈ ℂ) |
83 | 76 | sqcld 12868 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐵↑2) ∈ ℂ) |
84 | 82, 83 | pncand 10272 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐴↑2) + (𝐵↑2)) − (𝐵↑2)) = (𝐴↑2)) |
85 | | subsq 12834 |
. . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵))) |
86 | 68, 76, 85 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶↑2) − (𝐵↑2)) = ((𝐶 + 𝐵) · (𝐶 − 𝐵))) |
87 | 81, 84, 86 | 3eqtr3rd 2653 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((𝐶 + 𝐵) · (𝐶 − 𝐵)) = (𝐴↑2)) |
88 | 87 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘((𝐶 + 𝐵) · (𝐶 − 𝐵))) = (√‘(𝐴↑2))) |
89 | 32, 44, 51, 56 | sqrtmuld 14011 |
. . . . . . . . . . 11
⊢ (((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)))) |
90 | | nnre 10904 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℝ) |
91 | 90 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈
ℝ) |
92 | 91 | 3ad2ant1 1075 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℝ) |
93 | | nnnn0 11176 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℕ0) |
94 | 93 | nn0ge0d 11231 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℕ → 0 ≤
𝐴) |
95 | 94 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 0 ≤
𝐴) |
96 | 95 | 3ad2ant1 1075 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 ≤ 𝐴) |
97 | 92, 96 | sqrtsqd 14006 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐴↑2)) = 𝐴) |
98 | 88, 89, 97 | 3eqtr3d 2652 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))) = 𝐴) |
99 | 98 | oveq2d 6565 |
. . . . . . . . 9
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ·
((√‘(𝐶 + 𝐵)) ·
(√‘(𝐶 −
𝐵)))) = (2 · 𝐴)) |
100 | 79, 99 | oveq12d 6567 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (𝐶 − 𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) = ((2 · 𝐶) + (2 · 𝐴))) |
101 | 74, 100 | eqtr4d 2647 |
. . . . . . 7
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) = (((𝐶 + 𝐵) + (𝐶 − 𝐵)) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵)))))) |
102 | 66, 101 | eqtr4d 2647 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((𝐶 + 𝐵) + (2 · ((√‘(𝐶 + 𝐵)) · (√‘(𝐶 − 𝐵))))) + (𝐶 − 𝐵)) = (2 · (𝐶 + 𝐴))) |
103 | 26, 59, 102 | 3eqtrd 2648 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
(((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵)))↑2) = (2 · (𝐶 + 𝐴))) |
104 | 103 | oveq1d 6564 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)) = ((2 ·
(𝐶 + 𝐴)) / (2 · 2))) |
105 | | addcl 9897 |
. . . . . . . . 9
⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐶 + 𝐴) ∈ ℂ) |
106 | 3, 69, 105 | syl2anr 494 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ) |
107 | 106 | 3adant2 1073 |
. . . . . . 7
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐴) ∈ ℂ) |
108 | 107 | 3ad2ant1 1075 |
. . . . . 6
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐴) ∈ ℂ) |
109 | | mulcl 9899 |
. . . . . 6
⊢ ((2
∈ ℂ ∧ (𝐶 +
𝐴) ∈ ℂ) →
(2 · (𝐶 + 𝐴)) ∈
ℂ) |
110 | 15, 108, 109 | sylancr 694 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 · (𝐶 + 𝐴)) ∈ ℂ) |
111 | | 2cnne0 11119 |
. . . . . 6
⊢ (2 ∈
ℂ ∧ 2 ≠ 0) |
112 | | divdiv1 10615 |
. . . . . 6
⊢ (((2
· (𝐶 + 𝐴)) ∈ ℂ ∧ (2
∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) →
(((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 ·
(𝐶 + 𝐴)) / (2 · 2))) |
113 | 111, 111,
112 | mp3an23 1408 |
. . . . 5
⊢ ((2
· (𝐶 + 𝐴)) ∈ ℂ → (((2
· (𝐶 + 𝐴)) / 2) / 2) = ((2 ·
(𝐶 + 𝐴)) / (2 · 2))) |
114 | 110, 113 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((2 · (𝐶 + 𝐴)) / (2 · 2))) |
115 | 104, 114 | eqtr4d 2647 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵)))↑2) / (2 · 2)) = (((2
· (𝐶 + 𝐴)) / 2) / 2)) |
116 | | divcan3 10590 |
. . . . . 6
⊢ (((𝐶 + 𝐴) ∈ ℂ ∧ 2 ∈ ℂ
∧ 2 ≠ 0) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴)) |
117 | 15, 16, 116 | mp3an23 1408 |
. . . . 5
⊢ ((𝐶 + 𝐴) ∈ ℂ → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴)) |
118 | 108, 117 | syl 17 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 · (𝐶 + 𝐴)) / 2) = (𝐶 + 𝐴)) |
119 | 118 | oveq1d 6564 |
. . 3
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((2 · (𝐶 + 𝐴)) / 2) / 2) = ((𝐶 + 𝐴) / 2)) |
120 | 22, 115, 119 | 3eqtrd 2648 |
. 2
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) →
((((√‘(𝐶 +
𝐵)) + (√‘(𝐶 − 𝐵))) / 2)↑2) = ((𝐶 + 𝐴) / 2)) |
121 | 2, 120 | syl5eq 2656 |
1
⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀↑2) = ((𝐶 + 𝐴) / 2)) |