Proof of Theorem flt4lem5e
| Step | Hyp | Ref
| Expression |
| 1 | | flt4lem5a.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈ ℕ) |
| 2 | 1 | nnsqcld 14283 |
. . . . . 6
⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
| 3 | | flt4lem5a.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℕ) |
| 4 | 3 | nnsqcld 14283 |
. . . . . 6
⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
| 5 | | flt4lem5a.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ ℕ) |
| 6 | | flt4lem5a.1 |
. . . . . . 7
⊢ (𝜑 → ¬ 2 ∥ 𝐴) |
| 7 | | 2prm 16729 |
. . . . . . . 8
⊢ 2 ∈
ℙ |
| 8 | 1 | nnzd 12640 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 9 | | prmdvdssq 16755 |
. . . . . . . 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 12339 |
. . . . . . . . 9
⊢ 2 ∈
ℕ |
| 14 | 13 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℕ) |
| 15 | | rplpwr 16595 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 2 ∈
ℕ) → ((𝐴 gcd
𝐶) = 1 → ((𝐴↑2) gcd 𝐶) = 1)) |
| 16 | 1, 5, 14, 15 | syl3anc 1373 |
. . . . . . 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 12282 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 19 | 18 | flt4lem 42655 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2)) |
| 20 | 3 | nncnd 12282 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 21 | 20 | flt4lem 42655 |
. . . . . . . 8
⊢ (𝜑 → (𝐵↑4) = ((𝐵↑2)↑2)) |
| 22 | 19, 21 | oveq12d 7449 |
. . . . . . 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 2779 |
. . . . . 6
⊢ (𝜑 → (((𝐴↑2)↑2) + ((𝐵↑2)↑2)) = (𝐶↑2)) |
| 25 | 2, 4, 5, 11, 17, 24 | flt4lem1 42656 |
. . . . 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 16865 |
. . . . 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 16863 |
. . . . 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 42662 |
. . . 4
⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2)) |
| 35 | 28 | nnzd 12640 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 36 | 8, 35 | gcdcomd 16551 |
. . . . 5
⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴)) |
| 37 | 31 | nnzd 12640 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 38 | 35, 37 | gcdcomd 16551 |
. . . . . . 7
⊢ (𝜑 → (𝑁 gcd 𝑀) = (𝑀 gcd 𝑁)) |
| 39 | 29, 26 | flt4lem5 42660 |
. . . . . . . 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 2777 |
. . . . . 6
⊢ (𝜑 → (𝑁 gcd 𝑀) = 1) |
| 42 | 28 | nnsqcld 14283 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁↑2) ∈ ℕ) |
| 43 | 42 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → (𝑁↑2) ∈ ℂ) |
| 44 | 2 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
| 45 | 43, 44 | addcomd 11463 |
. . . . . . 7
⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝑁↑2))) |
| 46 | 45, 34 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → ((𝑁↑2) + (𝐴↑2)) = (𝑀↑2)) |
| 47 | 28, 1, 31, 41, 46 | fltabcoprm 42652 |
. . . . 5
⊢ (𝜑 → (𝑁 gcd 𝐴) = 1) |
| 48 | 36, 47 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → (𝐴 gcd 𝑁) = 1) |
| 49 | 32, 33 | flt4lem5 42660 |
. . . 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 1393 |
. . 3
⊢ (𝜑 → (𝑅 gcd 𝑆) = 1) |
| 51 | 32 | pythagtriplem11 16863 |
. . . . 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 1393 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ ℕ) |
| 53 | 33 | pythagtriplem13 16865 |
. . . . 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 1393 |
. . . 4
⊢ (𝜑 → 𝑆 ∈ ℕ) |
| 55 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5d 42665 |
. . . 4
⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2))) |
| 56 | 31, 52, 54, 55, 50 | flt4lem5elem 42661 |
. . 3
⊢ (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1)) |
| 57 | | 3anass 1095 |
. . 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 1129 |
. 2
⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ)) |
| 60 | | sq2 14236 |
. . . . . . 7
⊢
(2↑2) = 4 |
| 61 | | 4cn 12351 |
. . . . . . 7
⊢ 4 ∈
ℂ |
| 62 | 60, 61 | eqeltri 2837 |
. . . . . 6
⊢
(2↑2) ∈ ℂ |
| 63 | 62 | a1i 11 |
. . . . 5
⊢ (𝜑 → (2↑2) ∈
ℂ) |
| 64 | 52, 54 | nnmulcld 12319 |
. . . . . . 7
⊢ (𝜑 → (𝑅 · 𝑆) ∈ ℕ) |
| 65 | 31, 64 | nnmulcld 12319 |
. . . . . 6
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℕ) |
| 66 | 65 | nncnd 12282 |
. . . . 5
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℂ) |
| 67 | | 4ne0 12374 |
. . . . . . 7
⊢ 4 ≠
0 |
| 68 | 60, 67 | eqnetri 3011 |
. . . . . 6
⊢
(2↑2) ≠ 0 |
| 69 | 68 | a1i 11 |
. . . . 5
⊢ (𝜑 → (2↑2) ≠
0) |
| 70 | | 2cn 12341 |
. . . . . . . 8
⊢ 2 ∈
ℂ |
| 71 | 70 | sqvali 14219 |
. . . . . . 7
⊢
(2↑2) = (2 · 2) |
| 72 | 71 | oveq1i 7441 |
. . . . . 6
⊢
((2↑2) · (𝑀 · (𝑅 · 𝑆))) = ((2 · 2) · (𝑀 · (𝑅 · 𝑆))) |
| 73 | | 2cnd 12344 |
. . . . . . . 8
⊢ (𝜑 → 2 ∈
ℂ) |
| 74 | 31 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 75 | 64 | nncnd 12282 |
. . . . . . . 8
⊢ (𝜑 → (𝑅 · 𝑆) ∈ ℂ) |
| 76 | 73, 73, 74, 75 | mul4d 11473 |
. . . . . . 7
⊢ (𝜑 → ((2 · 2) ·
(𝑀 · (𝑅 · 𝑆))) = ((2 · 𝑀) · (2 · (𝑅 · 𝑆)))) |
| 77 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5c 42664 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆))) |
| 78 | 77, 28 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) ∈ ℕ) |
| 79 | 78 | nncnd 12282 |
. . . . . . . . 9
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) ∈ ℂ) |
| 80 | 73, 74, 79 | mulassd 11284 |
. . . . . . . 8
⊢ (𝜑 → ((2 · 𝑀) · (2 · (𝑅 · 𝑆))) = (2 · (𝑀 · (2 · (𝑅 · 𝑆))))) |
| 81 | 77 | eqcomd 2743 |
. . . . . . . . . 10
⊢ (𝜑 → (2 · (𝑅 · 𝑆)) = 𝑁) |
| 82 | 81 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 · (2 · (𝑅 · 𝑆))) = (𝑀 · 𝑁)) |
| 83 | 82 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → (2 · (𝑀 · (2 · (𝑅 · 𝑆)))) = (2 · (𝑀 · 𝑁))) |
| 84 | 80, 83 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((2 · 𝑀) · (2 · (𝑅 · 𝑆))) = (2 · (𝑀 · 𝑁))) |
| 85 | 29, 26, 32, 33, 1, 3, 5, 6, 12,
23 | flt4lem5b 42663 |
. . . . . . 7
⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2)) |
| 86 | 76, 84, 85 | 3eqtrd 2781 |
. . . . . 6
⊢ (𝜑 → ((2 · 2) ·
(𝑀 · (𝑅 · 𝑆))) = (𝐵↑2)) |
| 87 | 72, 86 | eqtrid 2789 |
. . . . 5
⊢ (𝜑 → ((2↑2) ·
(𝑀 · (𝑅 · 𝑆))) = (𝐵↑2)) |
| 88 | 63, 66, 69, 87 | mvllmuld 12099 |
. . . 4
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) = ((𝐵↑2) / (2↑2))) |
| 89 | | 2ne0 12370 |
. . . . . 6
⊢ 2 ≠
0 |
| 90 | 89 | a1i 11 |
. . . . 5
⊢ (𝜑 → 2 ≠ 0) |
| 91 | 20, 73, 90 | sqdivd 14199 |
. . . 4
⊢ (𝜑 → ((𝐵 / 2)↑2) = ((𝐵↑2) / (2↑2))) |
| 92 | 88, 91 | eqtr4d 2780 |
. . 3
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2)) |
| 93 | 65 | nnzd 12640 |
. . . . 5
⊢ (𝜑 → (𝑀 · (𝑅 · 𝑆)) ∈ ℤ) |
| 94 | 92, 93 | eqeltrrd 2842 |
. . . 4
⊢ (𝜑 → ((𝐵 / 2)↑2) ∈
ℤ) |
| 95 | 3 | nnzd 12640 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℤ) |
| 96 | | znq 12994 |
. . . . 5
⊢ ((𝐵 ∈ ℤ ∧ 2 ∈
ℕ) → (𝐵 / 2)
∈ ℚ) |
| 97 | 95, 13, 96 | sylancl 586 |
. . . 4
⊢ (𝜑 → (𝐵 / 2) ∈ ℚ) |
| 98 | 3 | nngt0d 12315 |
. . . . 5
⊢ (𝜑 → 0 < 𝐵) |
| 99 | 3 | nnred 12281 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 100 | | halfpos2 12495 |
. . . . . 6
⊢ (𝐵 ∈ ℝ → (0 <
𝐵 ↔ 0 < (𝐵 / 2))) |
| 101 | 99, 100 | syl 17 |
. . . . 5
⊢ (𝜑 → (0 < 𝐵 ↔ 0 < (𝐵 / 2))) |
| 102 | 98, 101 | mpbid 232 |
. . . 4
⊢ (𝜑 → 0 < (𝐵 / 2)) |
| 103 | 94, 97, 102 | posqsqznn 42371 |
. . 3
⊢ (𝜑 → (𝐵 / 2) ∈ ℕ) |
| 104 | 92, 103 | jca 511 |
. 2
⊢ (𝜑 → ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ)) |
| 105 | 58, 59, 104 | 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))) |