Proof of Theorem flt4lem5e
Step | Hyp | Ref
| Expression |
1 | | flt4lem5a.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℕ) |
2 | 1 | nnsqcld 13699 |
. . . . . . 7
⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
3 | | flt4lem5a.b |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℕ) |
4 | 3 | nnsqcld 13699 |
. . . . . . 7
⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
5 | | flt4lem5a.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ ℕ) |
6 | | flt4lem5a.1 |
. . . . . . . 8
⊢ (𝜑 → ¬ 2 ∥ 𝐴) |
7 | | 2prm 16135 |
. . . . . . . . 9
⊢ 2 ∈
ℙ |
8 | 1 | nnzd 12169 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℤ) |
9 | | prmdvdssq 16161 |
. . . . . . . . 9
⊢ ((2
∈ ℙ ∧ 𝐴
∈ ℤ) → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
10 | 7, 8, 9 | sylancr 590 |
. . . . . . . 8
⊢ (𝜑 → (2 ∥ 𝐴 ↔ 2 ∥ (𝐴↑2))) |
11 | 6, 10 | mtbid 327 |
. . . . . . 7
⊢ (𝜑 → ¬ 2 ∥ (𝐴↑2)) |
12 | | flt4lem5a.2 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) |
13 | | 2nn 11791 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ |
14 | 13 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 2 ∈
ℕ) |
15 | | rplpwr 16005 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈
ℕ) → ((𝐴 gcd
𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
16 | 1, 5, 14, 15 | syl3anc 1372 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴 gcd 𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
17 | 12, 16 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → ((𝐴↑2) gcd 𝐶) = 1) |
18 | 1 | nncnd 11734 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 18 | flt4lem 40076 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
20 | 3 | nncnd 11734 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℂ) |
21 | 20 | flt4lem 40076 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
22 | 19, 21 | oveq12d 7190 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (((𝐴↑2)↑2) + ((𝐵↑2)↑2))) |
23 | | flt4lem5a.3 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2)) |
24 | 22, 23 | eqtr3d 2775 |
. . . . . . 7
⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
25 | 2, 4, 5, 11, 17, 24 | flt4lem1 40077 |
. . . . . 6
⊢ (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 |
. . . . . . 7
⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2) |
27 | 26 | pythagtriplem13 16266 |
. . . . . 6
⊢ ((((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 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℕ) |
29 | | flt4lem5a.m |
. . . . . . 7
⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2) |
30 | 29 | pythagtriplem11 16264 |
. . . . . 6
⊢ ((((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 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℕ) |
32 | | flt4lem5a.r |
. . . . . 6
⊢ 𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2) |
33 | | flt4lem5a.s |
. . . . . 6
⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2) |
34 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5a 40083 |
. . . . 5
⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
35 | 28 | nnzd 12169 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | 8, 35 | gcdcomd 15959 |
. . . . . 6
⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴)) |
37 | 31 | nnzd 12169 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℤ) |
38 | 35, 37 | gcdcomd 15959 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
39 | 29, 26 | flt4lem5 40081 |
. . . . . . . . 9
⊢ ((((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 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 gcd 𝑁) = 1) |
41 | 38, 40 | eqtrd 2773 |
. . . . . . 7
⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
42 | 28 | nnsqcld 13699 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
43 | 42 | nncnd 11734 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
44 | 2 | nncnd 11734 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
45 | 43, 44 | addcomd 10922 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝑁↑2))) |
46 | 45, 34 | eqtrd 2773 |
. . . . . . 7
⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = (𝑀↑2)) |
47 | 28, 1, 31, 41, 46 | fltabcoprm 40073 |
. . . . . 6
⊢ (𝜑 → (𝑁 gcd 𝐴) = 1) |
48 | 36, 47 | eqtrd 2773 |
. . . . 5
⊢ (𝜑 → (𝐴 gcd 𝑁) = 1) |
49 | 32, 33 | flt4lem5 40081 |
. . . . 5
⊢ (((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 1392 |
. . . 4
⊢ (𝜑 → (𝑅 gcd 𝑆) = 1) |
51 | 32 | pythagtriplem11 16264 |
. . . . . 6
⊢ (((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 1392 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ ℕ) |
53 | 33 | pythagtriplem13 16266 |
. . . . . 6
⊢ (((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 1392 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ ℕ) |
55 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5d 40086 |
. . . . 5
⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2))) |
56 | 31, 52, 54, 55, 50 | flt4lem5elem 40082 |
. . . 4
⊢ (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1)) |
57 | 50, 56 | jca 515 |
. . 3
⊢ (𝜑 → ((𝑅 gcd 𝑆) = 1 ∧ ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))) |
58 | | 3anass 1096 |
. . 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))) |
59 | 57, 58 | sylibr 237 |
. 2
⊢ (𝜑 → ((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1)) |
60 | 52, 54, 31 | 3jca 1129 |
. 2
⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ)) |
61 | | sq2 13654 |
. . . . . . 7
⊢
(2↑2) = 4 |
62 | | 4cn 11803 |
. . . . . . 7
⊢ 4 ∈
ℂ |
63 | 61, 62 | eqeltri 2829 |
. . . . . 6
⊢
(2↑2) ∈ ℂ |
64 | 63 | a1i 11 |
. . . . 5
⊢ (𝜑 → (2↑2) ∈
ℂ) |
65 | 52, 54 | nnmulcld 11771 |
. . . . . . 7
⊢ (𝜑 → (𝑅 · 𝑆) ∈ ℕ) |
66 | 31, 65 | nnmulcld 11771 |
. . . . . 6
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℕ) |
67 | 66 | nncnd 11734 |
. . . . 5
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℂ) |
68 | | 4ne0 11826 |
. . . . . . 7
⊢ 4 ≠
0 |
69 | 61, 68 | eqnetri 3004 |
. . . . . 6
⊢
(2↑2) ≠ 0 |
70 | 69 | a1i 11 |
. . . . 5
⊢ (𝜑 → (2↑2) ≠
0) |
71 | | 2cn 11793 |
. . . . . . . 8
⊢ 2 ∈
ℂ |
72 | 71 | sqvali 13637 |
. . . . . . 7
⊢
(2↑2) = (2 · 2) |
73 | 72 | oveq1i 7182 |
. . . . . 6
⊢
((2↑2) · (𝑀 · (𝑅 · 𝑆))) = ((2 · 2) · (𝑀 · (𝑅 · 𝑆))) |
74 | | 2cnd 11796 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℂ) |
75 | 31 | nncnd 11734 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℂ) |
76 | 65 | nncnd 11734 |
. . . . . . . 8
⊢ (𝜑 → (𝑅 · 𝑆) ∈ ℂ) |
77 | 74, 74, 75, 76 | mul4d 10932 |
. . . . . . 7
⊢ (𝜑 → ((2 · 2) ·
(𝑀 · (𝑅 · 𝑆))) = ((2 · 𝑀) · (2 · (𝑅 · 𝑆)))) |
78 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5c 40085 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆))) |
79 | 78 | eqcomd 2744 |
. . . . . . . . . . 11
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) = 𝑁) |
80 | 79, 28 | eqeltrd 2833 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) ∈ ℕ) |
81 | 80 | nncnd 11734 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) ∈ ℂ) |
82 | 74, 75, 81 | mulassd 10744 |
. . . . . . . 8
⊢ (𝜑 → ((2 · 𝑀) · (2 · (𝑅 · 𝑆))) = (2 · (𝑀 · (2 · (𝑅 · 𝑆))))) |
83 | 79 | oveq2d 7188 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 · (2 · (𝑅 · 𝑆))) = (𝑀 · 𝑁)) |
84 | 83 | oveq2d 7188 |
. . . . . . . 8
⊢ (𝜑 → (2 · (𝑀 · (2 · (𝑅 · 𝑆)))) = (2 · (𝑀 · 𝑁))) |
85 | 82, 84 | eqtrd 2773 |
. . . . . . 7
⊢ (𝜑 → ((2 · 𝑀) · (2 · (𝑅 · 𝑆))) = (2 · (𝑀 · 𝑁))) |
86 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5b 40084 |
. . . . . . 7
⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2)) |
87 | 77, 85, 86 | 3eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
(𝑀 · (𝑅 · 𝑆))) = (𝐵↑2)) |
88 | 73, 87 | syl5eq 2785 |
. . . . 5
⊢ (𝜑 → ((2↑2) ·
(𝑀 · (𝑅 · 𝑆))) = (𝐵↑2)) |
89 | 64, 67, 70, 88 | mvllmuld 11552 |
. . . 4
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) = ((𝐵↑2) / (2↑2))) |
90 | | 2ne0 11822 |
. . . . . 6
⊢ 2 ≠
0 |
91 | 90 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ≠ 0) |
92 | 20, 74, 91 | sqdivd 13617 |
. . . 4
⊢ (𝜑 → ((𝐵 / 2)↑2) = ((𝐵↑2) / (2↑2))) |
93 | 89, 92 | eqtr4d 2776 |
. . 3
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2)) |
94 | 66 | nnzd 12169 |
. . . . 5
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℤ) |
95 | 93, 94 | eqeltrrd 2834 |
. . . 4
⊢ (𝜑 → ((𝐵 / 2)↑2) ∈
ℤ) |
96 | 3 | nnzd 12169 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℤ) |
97 | | znq 12436 |
. . . . 5
⊢ ((𝐵 ∈ ℤ ∧ 2 ∈
ℕ) → (𝐵 / 2)
∈ ℚ) |
98 | 96, 13, 97 | sylancl 589 |
. . . 4
⊢ (𝜑 → (𝐵 / 2) ∈ ℚ) |
99 | 3 | nngt0d 11767 |
. . . . 5
⊢ (𝜑 → 0 < 𝐵) |
100 | 3 | nnred 11733 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
101 | | halfpos2 11947 |
. . . . . 6
⊢ (𝐵 ∈ ℝ → (0 <
𝐵 ↔ 0 < (𝐵 / 2))) |
102 | 100, 101 | syl 17 |
. . . . 5
⊢ (𝜑 → (0 < 𝐵 ↔ 0 < (𝐵 / 2))) |
103 | 99, 102 | mpbid 235 |
. . . 4
⊢ (𝜑 → 0 < (𝐵 / 2)) |
104 | 95, 98, 103 | posqsqznn 39942 |
. . 3
⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
105 | 93, 104 | jca 515 |
. 2
⊢ (𝜑 → ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ)) |
106 | 59, 60, 105 | 3jca 1129 |
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))) |