Proof of Theorem flt4lem5e
Step | Hyp | Ref
| Expression |
1 | | flt4lem5a.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℕ) |
2 | 1 | nnsqcld 13959 |
. . . . . 6
⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | | flt4lem5a.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℕ) |
4 | 3 | nnsqcld 13959 |
. . . . . 6
⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | | flt4lem5a.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ ℕ) |
6 | | flt4lem5a.1 |
. . . . . . 7
⊢ (𝜑 → ¬ 2 ∥ 𝐴) |
7 | | 2prm 16397 |
. . . . . . . 8
⊢ 2 ∈
ℙ |
8 | 1 | nnzd 12425 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | | prmdvdssq 16423 |
. . . . . . . 8
⊢ ((2
∈ ℙ ∧ 𝐴
∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
10 | 7, 8, 9 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
11 | 6, 10 | mtbid 324 |
. . . . . 6
⊢ (𝜑 → ¬ 2 ∥ (𝐴↑2)) |
12 | | flt4lem5a.2 |
. . . . . . 7
⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) |
13 | | 2nn 12046 |
. . . . . . . . 9
⊢ 2 ∈
ℕ |
14 | 13 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℕ) |
15 | | rplpwr 16267 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈
ℕ) → ((𝐴 gcd
𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
16 | 1, 5, 14, 15 | syl3anc 1370 |
. . . . . . 7
⊢ (𝜑 → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
17 | 12, 16 | mpd 15 |
. . . . . 6
⊢ (𝜑 → ((𝐴↑2) gcd 𝐶) = 1) |
18 | 1 | nncnd 11989 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 40482 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 11989 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 40482 |
. . . . . . . 8
⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7293 |
. . . . . . 7
⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (((𝐴↑2)↑2) + ((𝐵↑2)↑2))) |
23 | | flt4lem5a.3 |
. . . . . . 7
⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2)) |
24 | 22, 23 | eqtr3d 2780 |
. . . . . 6
⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
25 | 2, 4, 5, 11, 17, 24 | flt4lem1 40483 |
. . . . 5
⊢ (𝜑 → (((𝐴↑2) ∈ ℕ ∧ (𝐵↑2) ∈ ℕ ∧
𝐶 ∈ ℕ) ∧
(((𝐴↑2)↑2) +
((𝐵↑2)↑2)) =
(𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2)))) |
26 | | flt4lem5a.n |
. . . . . 6
⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) |
27 | 26 | pythagtriplem13 16528 |
. . . . 5
⊢ ((((𝐴↑2) ∈ ℕ ∧
(𝐵↑2) ∈ ℕ
∧ 𝐶 ∈ ℕ)
∧ (((𝐴↑2)↑2)
+ ((𝐵↑2)↑2)) =
(𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2))) → 𝑁 ∈
ℕ) |
28 | 25, 27 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑁 ∈ ℕ) |
29 | | flt4lem5a.m |
. . . . . 6
⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2) |
30 | 29 | pythagtriplem11 16526 |
. . . . 5
⊢ ((((𝐴↑2) ∈ ℕ ∧
(𝐵↑2) ∈ ℕ
∧ 𝐶 ∈ ℕ)
∧ (((𝐴↑2)↑2)
+ ((𝐵↑2)↑2)) =
(𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2))) → 𝑀 ∈
ℕ) |
31 | 25, 30 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℕ) |
32 | | flt4lem5a.r |
. . . . 5
⊢ 𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2) |
33 | | flt4lem5a.s |
. . . . 5
⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2) |
34 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5a 40489 |
. . . 4
⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
35 | 28 | nnzd 12425 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | 8, 35 | gcdcomd 16221 |
. . . . 5
⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴)) |
37 | 31 | nnzd 12425 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
38 | 35, 37 | gcdcomd 16221 |
. . . . . . 7
⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
39 | 29, 26 | flt4lem5 40487 |
. . . . . . . 8
⊢ ((((𝐴↑2) ∈ ℕ ∧
(𝐵↑2) ∈ ℕ
∧ 𝐶 ∈ ℕ)
∧ (((𝐴↑2)↑2)
+ ((𝐵↑2)↑2)) =
(𝐶↑2) ∧ (((𝐴↑2) gcd (𝐵↑2)) = 1 ∧ ¬ 2 ∥ (𝐴↑2))) → (𝑀 gcd 𝑁) = 1) |
40 | 25, 39 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
41 | 38, 40 | eqtrd 2778 |
. . . . . 6
⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
42 | 28 | nnsqcld 13959 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
43 | 42 | nncnd 11989 |
. . . . . . . 8
⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
44 | 2 | nncnd 11989 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
45 | 43, 44 | addcomd 11177 |
. . . . . . 7
⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝑁↑2))) |
46 | 45, 34 | eqtrd 2778 |
. . . . . 6
⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = (𝑀↑2)) |
47 | 28, 1, 31, 41, 46 | fltabcoprm 40479 |
. . . . 5
⊢ (𝜑 → (𝑁 gcd 𝐴) = 1) |
48 | 36, 47 | eqtrd 2778 |
. . . 4
⊢ (𝜑 → (𝐴 gcd 𝑁) = 1) |
49 | 32, 33 | flt4lem5 40487 |
. . . 4
⊢ (((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2) ∧ ((𝐴 gcd 𝑁) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑅 gcd 𝑆) = 1) |
50 | 1, 28, 31, 34, 48, 6, 49 | syl312anc 1390 |
. . 3
⊢ (𝜑 → (𝑅 gcd 𝑆) = 1) |
51 | 32 | pythagtriplem11 16526 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2) ∧ ((𝐴 gcd 𝑁) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑅 ∈ ℕ) |
52 | 1, 28, 31, 34, 48, 6, 51 | syl312anc 1390 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ ℕ) |
53 | 33 | pythagtriplem13 16528 |
. . . . 5
⊢ (((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2) ∧ ((𝐴 gcd 𝑁) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑆 ∈ ℕ) |
54 | 1, 28, 31, 34, 48, 6, 53 | syl312anc 1390 |
. . . 4
⊢ (𝜑 → 𝑆 ∈ ℕ) |
55 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5d 40492 |
. . . 4
⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2))) |
56 | 31, 52, 54, 55, 50 | flt4lem5elem 40488 |
. . 3
⊢ (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1)) |
57 | | 3anass 1094 |
. . 3
⊢ (((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1) ↔ ((𝑅 gcd 𝑆) = 1 ∧ ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))) |
58 | 50, 56, 57 | sylanbrc 583 |
. 2
⊢ (𝜑 → ((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1)) |
59 | 52, 54, 31 | 3jca 1127 |
. 2
⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ)) |
60 | | sq2 13914 |
. . . . . . 7
⊢
(2↑2) = 4 |
61 | | 4cn 12058 |
. . . . . . 7
⊢ 4 ∈
ℂ |
62 | 60, 61 | eqeltri 2835 |
. . . . . 6
⊢
(2↑2) ∈ ℂ |
63 | 62 | a1i 11 |
. . . . 5
⊢ (𝜑 → (2↑2) ∈
ℂ) |
64 | 52, 54 | nnmulcld 12026 |
. . . . . . 7
⊢ (𝜑 → (𝑅 · 𝑆) ∈ ℕ) |
65 | 31, 64 | nnmulcld 12026 |
. . . . . 6
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℕ) |
66 | 65 | nncnd 11989 |
. . . . 5
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℂ) |
67 | | 4ne0 12081 |
. . . . . . 7
⊢ 4 ≠
0 |
68 | 60, 67 | eqnetri 3014 |
. . . . . 6
⊢
(2↑2) ≠ 0 |
69 | 68 | a1i 11 |
. . . . 5
⊢ (𝜑 → (2↑2) ≠
0) |
70 | | 2cn 12048 |
. . . . . . . 8
⊢ 2 ∈
ℂ |
71 | 70 | sqvali 13897 |
. . . . . . 7
⊢
(2↑2) = (2 · 2) |
72 | 71 | oveq1i 7285 |
. . . . . 6
⊢
((2↑2) · (𝑀 · (𝑅 · 𝑆))) = ((2 · 2) · (𝑀 · (𝑅 · 𝑆))) |
73 | | 2cnd 12051 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℂ) |
74 | 31 | nncnd 11989 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℂ) |
75 | 64 | nncnd 11989 |
. . . . . . . 8
⊢ (𝜑 → (𝑅 · 𝑆) ∈ ℂ) |
76 | 73, 73, 74, 75 | mul4d 11187 |
. . . . . . 7
⊢ (𝜑 → ((2 · 2) ·
(𝑀 · (𝑅 · 𝑆))) = ((2 · 𝑀) · (2 · (𝑅 · 𝑆)))) |
77 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5c 40491 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆))) |
78 | 77, 28 | eqeltrrd 2840 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) ∈ ℕ) |
79 | 78 | nncnd 11989 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) ∈ ℂ) |
80 | 73, 74, 79 | mulassd 10998 |
. . . . . . . 8
⊢ (𝜑 → ((2 · 𝑀) · (2 · (𝑅 · 𝑆))) = (2 · (𝑀 · (2 · (𝑅 · 𝑆))))) |
81 | 77 | eqcomd 2744 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) = 𝑁) |
82 | 81 | oveq2d 7291 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 · (2 · (𝑅 · 𝑆))) = (𝑀 · 𝑁)) |
83 | 82 | oveq2d 7291 |
. . . . . . . 8
⊢ (𝜑 → (2 · (𝑀 · (2 · (𝑅 · 𝑆)))) = (2 · (𝑀 · 𝑁))) |
84 | 80, 83 | eqtrd 2778 |
. . . . . . 7
⊢ (𝜑 → ((2 · 𝑀) · (2 · (𝑅 · 𝑆))) = (2 · (𝑀 · 𝑁))) |
85 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5b 40490 |
. . . . . . 7
⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2)) |
86 | 76, 84, 85 | 3eqtrd 2782 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
(𝑀 · (𝑅 · 𝑆))) = (𝐵↑2)) |
87 | 72, 86 | eqtrid 2790 |
. . . . 5
⊢ (𝜑 → ((2↑2) ·
(𝑀 · (𝑅 · 𝑆))) = (𝐵↑2)) |
88 | 63, 66, 69, 87 | mvllmuld 11807 |
. . . 4
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) = ((𝐵↑2) / (2↑2))) |
89 | | 2ne0 12077 |
. . . . . 6
⊢ 2 ≠
0 |
90 | 89 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ≠ 0) |
91 | 20, 73, 90 | sqdivd 13877 |
. . . 4
⊢ (𝜑 → ((𝐵 / 2)↑2) = ((𝐵↑2) / (2↑2))) |
92 | 88, 91 | eqtr4d 2781 |
. . 3
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2)) |
93 | 65 | nnzd 12425 |
. . . . 5
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℤ) |
94 | 92, 93 | eqeltrrd 2840 |
. . . 4
⊢ (𝜑 → ((𝐵 / 2)↑2) ∈
ℤ) |
95 | 3 | nnzd 12425 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℤ) |
96 | | znq 12692 |
. . . . 5
⊢ ((𝐵 ∈ ℤ ∧ 2 ∈
ℕ) → (𝐵 / 2)
∈ ℚ) |
97 | 95, 13, 96 | sylancl 586 |
. . . 4
⊢ (𝜑 → (𝐵 / 2) ∈ ℚ) |
98 | 3 | nngt0d 12022 |
. . . . 5
⊢ (𝜑 → 0 < 𝐵) |
99 | 3 | nnred 11988 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
100 | | halfpos2 12202 |
. . . . . 6
⊢ (𝐵 ∈ ℝ → (0 <
𝐵 ↔ 0 < (𝐵 / 2))) |
101 | 99, 100 | syl 17 |
. . . . 5
⊢ (𝜑 → (0 < 𝐵 ↔ 0 < (𝐵 / 2))) |
102 | 98, 101 | mpbid 231 |
. . . 4
⊢ (𝜑 → 0 < (𝐵 / 2)) |
103 | 94, 97, 102 | posqsqznn 40343 |
. . 3
⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
104 | 92, 103 | jca 512 |
. 2
⊢ (𝜑 → ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ)) |
105 | 58, 59, 104 | 3jca 1127 |
1
⊢ (𝜑 → (((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1) ∧ (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ))) |