Proof of Theorem pellexlem2
| Step | Hyp | Ref
| Expression |
| 1 | | simpl3 1194 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℕ) |
| 2 | 1 | nnred 12281 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℝ) |
| 3 | 2 | resqcld 14165 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℝ) |
| 4 | 2 | sqge0d 14177 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (𝐵↑2)) |
| 5 | 3, 4 | absidd 15461 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(𝐵↑2)) = (𝐵↑2)) |
| 6 | 5 | eqcomd 2743 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) = (abs‘(𝐵↑2))) |
| 7 | 6 | oveq2d 7447 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) / (𝐵↑2)) = ((abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) / (abs‘(𝐵↑2)))) |
| 8 | | simpl2 1193 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℕ) |
| 9 | 8 | nncnd 12282 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℂ) |
| 10 | 9 | sqcld 14184 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴↑2) ∈ ℂ) |
| 11 | | simpl1 1192 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℕ) |
| 12 | 11 | nncnd 12282 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℂ) |
| 13 | 1 | nncnd 12282 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℂ) |
| 14 | 13 | sqcld 14184 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℂ) |
| 15 | 12, 14 | mulcld 11281 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐷 · (𝐵↑2)) ∈ ℂ) |
| 16 | 10, 15 | subcld 11620 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ∈ ℂ) |
| 17 | 1 | nnne0d 12316 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ≠ 0) |
| 18 | | sqne0 14163 |
. . . . . . . 8
⊢ (𝐵 ∈ ℂ → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0)) |
| 19 | 18 | biimpar 477 |
. . . . . . 7
⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵↑2) ≠
0) |
| 20 | 13, 17, 19 | syl2anc 584 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ≠ 0) |
| 21 | 16, 14, 20 | absdivd 15494 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2))) = ((abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) / (abs‘(𝐵↑2)))) |
| 22 | 7, 21 | eqtr4d 2780 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) / (𝐵↑2)) = (abs‘(((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2)))) |
| 23 | 22 | oveq2d 7447 |
. . 3
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) / (𝐵↑2))) = ((𝐵↑2) · (abs‘(((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2))))) |
| 24 | 16 | abscld 15475 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) ∈ ℝ) |
| 25 | 24 | recnd 11289 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) ∈ ℂ) |
| 26 | 25, 14, 20 | divcan2d 12045 |
. . 3
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) / (𝐵↑2))) = (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2))))) |
| 27 | 10, 15, 14, 20 | divsubdird 12082 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2)) = (((𝐴↑2) / (𝐵↑2)) − ((𝐷 · (𝐵↑2)) / (𝐵↑2)))) |
| 28 | 9, 13, 17 | sqdivd 14199 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) |
| 29 | 28 | eqcomd 2743 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) / (𝐵↑2)) = ((𝐴 / 𝐵)↑2)) |
| 30 | 11 | nnred 12281 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℝ) |
| 31 | 11 | nnnn0d 12587 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈
ℕ0) |
| 32 | 31 | nn0ge0d 12590 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ 𝐷) |
| 33 | | remsqsqrt 15295 |
. . . . . . . . 9
⊢ ((𝐷 ∈ ℝ ∧ 0 ≤
𝐷) →
((√‘𝐷) ·
(√‘𝐷)) = 𝐷) |
| 34 | 30, 32, 33 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) · (√‘𝐷)) = 𝐷) |
| 35 | 30, 32 | resqrtcld 15456 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈
ℝ) |
| 36 | 35 | recnd 11289 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈
ℂ) |
| 37 | 36 | sqvald 14183 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷)↑2) =
((√‘𝐷) ·
(√‘𝐷))) |
| 38 | 12, 14, 20 | divcan4d 12049 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = 𝐷) |
| 39 | 34, 37, 38 | 3eqtr4rd 2788 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = ((√‘𝐷)↑2)) |
| 40 | 29, 39 | oveq12d 7449 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴↑2) / (𝐵↑2)) − ((𝐷 · (𝐵↑2)) / (𝐵↑2))) = (((𝐴 / 𝐵)↑2) − ((√‘𝐷)↑2))) |
| 41 | 9, 13, 17 | divcld 12043 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℂ) |
| 42 | | subsq 14249 |
. . . . . . . 8
⊢ (((𝐴 / 𝐵) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ) →
(((𝐴 / 𝐵)↑2) − ((√‘𝐷)↑2)) = (((𝐴 / 𝐵) + (√‘𝐷)) · ((𝐴 / 𝐵) − (√‘𝐷)))) |
| 43 | 41, 36, 42 | syl2anc 584 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵)↑2) − ((√‘𝐷)↑2)) = (((𝐴 / 𝐵) + (√‘𝐷)) · ((𝐴 / 𝐵) − (√‘𝐷)))) |
| 44 | 41, 36 | addcld 11280 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ) |
| 45 | 8 | nnred 12281 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℝ) |
| 46 | 45, 1 | nndivred 12320 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℝ) |
| 47 | 46, 35 | resubcld 11691 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℝ) |
| 48 | 47 | recnd 11289 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℂ) |
| 49 | 44, 48 | mulcomd 11282 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) + (√‘𝐷)) · ((𝐴 / 𝐵) − (√‘𝐷))) = (((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷)))) |
| 50 | 43, 49 | eqtrd 2777 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵)↑2) − ((√‘𝐷)↑2)) = (((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷)))) |
| 51 | 27, 40, 50 | 3eqtrd 2781 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2)) = (((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷)))) |
| 52 | 51 | fveq2d 6910 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2))) = (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷))))) |
| 53 | 52 | oveq2d 7447 |
. . 3
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (abs‘(((𝐴↑2) − (𝐷 · (𝐵↑2))) / (𝐵↑2)))) = ((𝐵↑2) · (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 54 | 23, 26, 53 | 3eqtr3d 2785 |
. 2
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) = ((𝐵↑2) · (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 55 | 48, 44 | absmuld 15493 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷)))) = ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) |
| 56 | 55 | oveq2d 7447 |
. . 3
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷))))) = ((𝐵↑2) · ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 57 | 48 | abscld 15475 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ∈ ℝ) |
| 58 | 44 | abscld 15475 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℝ) |
| 59 | 57, 58 | remulcld 11291 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ∈ ℝ) |
| 60 | 3, 59 | remulcld 11291 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) ∈ ℝ) |
| 61 | | 2nn0 12543 |
. . . . . . . . 9
⊢ 2 ∈
ℕ0 |
| 62 | 61 | nn0negzi 12656 |
. . . . . . . 8
⊢ -2 ∈
ℤ |
| 63 | 62 | a1i 11 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → -2 ∈
ℤ) |
| 64 | 2, 17, 63 | reexpclzd 14288 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℝ) |
| 65 | 64, 58 | remulcld 11291 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ∈ ℝ) |
| 66 | 3, 65 | remulcld 11291 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) ∈ ℝ) |
| 67 | | 1red 11262 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ∈
ℝ) |
| 68 | | 2re 12340 |
. . . . . . 7
⊢ 2 ∈
ℝ |
| 69 | 68 | a1i 11 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 2 ∈
ℝ) |
| 70 | 69, 35 | remulcld 11291 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 ·
(√‘𝐷)) ∈
ℝ) |
| 71 | 67, 70 | readdcld 11290 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 + (2 ·
(√‘𝐷))) ∈
ℝ) |
| 72 | | simpr 484 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) |
| 73 | 8 | nngt0d 12315 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐴) |
| 74 | 1 | nngt0d 12315 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐵) |
| 75 | 45, 2, 73, 74 | divgt0d 12203 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐴 / 𝐵)) |
| 76 | 11 | nngt0d 12315 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐷) |
| 77 | | sqrtgt0 15297 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ ℝ ∧ 0 <
𝐷) → 0 <
(√‘𝐷)) |
| 78 | 30, 76, 77 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 <
(√‘𝐷)) |
| 79 | 46, 35, 75, 78 | addgt0d 11838 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < ((𝐴 / 𝐵) + (√‘𝐷))) |
| 80 | 79 | gt0ne0d 11827 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0) |
| 81 | | absgt0 15363 |
. . . . . . . . 9
⊢ (((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ → (((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0 ↔ 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) |
| 82 | 81 | biimpa 476 |
. . . . . . . 8
⊢ ((((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ ∧ ((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0) → 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) |
| 83 | 44, 80, 82 | syl2anc 584 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) |
| 84 | | ltmul1 12117 |
. . . . . . 7
⊢
(((abs‘((𝐴 /
𝐵) −
(√‘𝐷))) ∈
ℝ ∧ (𝐵↑-2)
∈ ℝ ∧ ((abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℝ ∧ 0 <
(abs‘((𝐴 / 𝐵) + (√‘𝐷))))) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2) ↔ ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) < ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 85 | 57, 64, 58, 83, 84 | syl112anc 1376 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2) ↔ ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) < ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 86 | 72, 85 | mpbid 232 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) < ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) |
| 87 | 2, 17 | sqgt0d 14289 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐵↑2)) |
| 88 | | ltmul2 12118 |
. . . . . 6
⊢
((((abs‘((𝐴 /
𝐵) −
(√‘𝐷)))
· (abs‘((𝐴 /
𝐵) + (√‘𝐷)))) ∈ ℝ ∧
((𝐵↑-2) ·
(abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ∈ ℝ ∧
((𝐵↑2) ∈ ℝ
∧ 0 < (𝐵↑2)))
→ (((abs‘((𝐴 /
𝐵) −
(√‘𝐷)))
· (abs‘((𝐴 /
𝐵) + (√‘𝐷)))) < ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ↔ ((𝐵↑2) · ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) < ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))))) |
| 89 | 59, 65, 3, 87, 88 | syl112anc 1376 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) < ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ↔ ((𝐵↑2) · ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) < ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))))) |
| 90 | 86, 89 | mpbid 232 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) < ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 91 | 13, 17, 63 | expclzd 14191 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℂ) |
| 92 | 58 | recnd 11289 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℂ) |
| 93 | | mulass 11243 |
. . . . . . . 8
⊢ (((𝐵↑2) ∈ ℂ ∧
(𝐵↑-2) ∈ ℂ
∧ (abs‘((𝐴 /
𝐵) + (√‘𝐷))) ∈ ℂ) →
(((𝐵↑2) ·
(𝐵↑-2)) ·
(abs‘((𝐴 / 𝐵) + (√‘𝐷)))) = ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))) |
| 94 | 93 | eqcomd 2743 |
. . . . . . 7
⊢ (((𝐵↑2) ∈ ℂ ∧
(𝐵↑-2) ∈ ℂ
∧ (abs‘((𝐴 /
𝐵) + (√‘𝐷))) ∈ ℂ) →
((𝐵↑2) ·
((𝐵↑-2) ·
(abs‘((𝐴 / 𝐵) + (√‘𝐷))))) = (((𝐵↑2) · (𝐵↑-2)) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) |
| 95 | 14, 91, 92, 94 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) = (((𝐵↑2) · (𝐵↑-2)) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) |
| 96 | | expneg 14110 |
. . . . . . . . . 10
⊢ ((𝐵 ∈ ℂ ∧ 2 ∈
ℕ0) → (𝐵↑-2) = (1 / (𝐵↑2))) |
| 97 | 13, 61, 96 | sylancl 586 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) = (1 / (𝐵↑2))) |
| 98 | 97 | oveq2d 7447 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (𝐵↑-2)) = ((𝐵↑2) · (1 / (𝐵↑2)))) |
| 99 | 14, 20 | recidd 12038 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (1 / (𝐵↑2))) = 1) |
| 100 | 98, 99 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (𝐵↑-2)) = 1) |
| 101 | 100 | oveq1d 7446 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐵↑2) · (𝐵↑-2)) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) = (1 · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) |
| 102 | 92 | mullidd 11279 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 ·
(abs‘((𝐴 / 𝐵) + (√‘𝐷)))) = (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) |
| 103 | 95, 101, 102 | 3eqtrd 2781 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) = (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) |
| 104 | 41, 36 | addcomd 11463 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = ((√‘𝐷) + (𝐴 / 𝐵))) |
| 105 | | ppncan 11551 |
. . . . . . . . . 10
⊢
(((√‘𝐷)
∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) →
(((√‘𝐷) +
(√‘𝐷)) +
((𝐴 / 𝐵) − (√‘𝐷))) = ((√‘𝐷) + (𝐴 / 𝐵))) |
| 106 | 105 | eqcomd 2743 |
. . . . . . . . 9
⊢
(((√‘𝐷)
∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) →
((√‘𝐷) + (𝐴 / 𝐵)) = (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷)))) |
| 107 | 36, 36, 41, 106 | syl3anc 1373 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) + (𝐴 / 𝐵)) = (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷)))) |
| 108 | 36, 36 | addcld 11280 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) + (√‘𝐷)) ∈
ℂ) |
| 109 | 108, 48 | addcomd 11463 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))) = (((𝐴 / 𝐵) − (√‘𝐷)) + ((√‘𝐷) + (√‘𝐷)))) |
| 110 | | 2times 12402 |
. . . . . . . . . . . 12
⊢
((√‘𝐷)
∈ ℂ → (2 · (√‘𝐷)) = ((√‘𝐷) + (√‘𝐷))) |
| 111 | 110 | eqcomd 2743 |
. . . . . . . . . . 11
⊢
((√‘𝐷)
∈ ℂ → ((√‘𝐷) + (√‘𝐷)) = (2 · (√‘𝐷))) |
| 112 | 36, 111 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) + (√‘𝐷)) = (2 ·
(√‘𝐷))) |
| 113 | 112 | oveq2d 7447 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + ((√‘𝐷) + (√‘𝐷))) = (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) |
| 114 | 109, 113 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))) = (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) |
| 115 | 104, 107,
114 | 3eqtrd 2781 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) |
| 116 | 115 | fveq2d 6910 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) = (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))))) |
| 117 | 47, 70 | readdcld 11290 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈
ℝ) |
| 118 | 117 | recnd 11289 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈
ℂ) |
| 119 | 118 | abscld 15475 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) ∈
ℝ) |
| 120 | 70 | recnd 11289 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 ·
(√‘𝐷)) ∈
ℂ) |
| 121 | 120 | abscld 15475 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 ·
(√‘𝐷))) ∈
ℝ) |
| 122 | 57, 121 | readdcld 11290 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) + (abs‘(2 ·
(√‘𝐷)))) ∈
ℝ) |
| 123 | 48, 120 | abstrid 15495 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) ≤ ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) + (abs‘(2 ·
(√‘𝐷))))) |
| 124 | | 0le2 12368 |
. . . . . . . . . . . 12
⊢ 0 ≤
2 |
| 125 | 124 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ 2) |
| 126 | 30, 32 | sqrtge0d 15459 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤
(√‘𝐷)) |
| 127 | 69, 35, 125, 126 | mulge0d 11840 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (2 ·
(√‘𝐷))) |
| 128 | 70, 127 | absidd 15461 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 ·
(√‘𝐷))) = (2
· (√‘𝐷))) |
| 129 | 128 | oveq2d 7447 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) + (abs‘(2 ·
(√‘𝐷)))) =
((abs‘((𝐴 / 𝐵) − (√‘𝐷))) + (2 ·
(√‘𝐷)))) |
| 130 | 1 | nnsqcld 14283 |
. . . . . . . . . . . . . . 15
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℕ) |
| 131 | 130 | nnge1d 12314 |
. . . . . . . . . . . . . 14
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ≤ (𝐵↑2)) |
| 132 | | 0lt1 11785 |
. . . . . . . . . . . . . . . 16
⊢ 0 <
1 |
| 133 | 132 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 1) |
| 134 | | lerec 12151 |
. . . . . . . . . . . . . . 15
⊢ (((1
∈ ℝ ∧ 0 < 1) ∧ ((𝐵↑2) ∈ ℝ ∧ 0 < (𝐵↑2))) → (1 ≤ (𝐵↑2) ↔ (1 / (𝐵↑2)) ≤ (1 /
1))) |
| 135 | 67, 133, 3, 87, 134 | syl22anc 839 |
. . . . . . . . . . . . . 14
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 ≤ (𝐵↑2) ↔ (1 / (𝐵↑2)) ≤ (1 / 1))) |
| 136 | 131, 135 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ (1 / 1)) |
| 137 | | 1div1e1 11958 |
. . . . . . . . . . . . 13
⊢ (1 / 1) =
1 |
| 138 | 136, 137 | breqtrdi 5184 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ 1) |
| 139 | 97, 138 | eqbrtrd 5165 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ≤ 1) |
| 140 | 57, 64, 67, 72, 139 | ltletrd 11421 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < 1) |
| 141 | 57, 67, 140 | ltled 11409 |
. . . . . . . . 9
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ≤ 1) |
| 142 | 57, 67, 70, 141 | leadd1dd 11877 |
. . . . . . . 8
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) + (2 · (√‘𝐷))) ≤ (1 + (2 ·
(√‘𝐷)))) |
| 143 | 129, 142 | eqbrtrd 5165 |
. . . . . . 7
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) + (abs‘(2 ·
(√‘𝐷)))) ≤
(1 + (2 · (√‘𝐷)))) |
| 144 | 119, 122,
71, 123, 143 | letrd 11418 |
. . . . . 6
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) ≤ (1 + (2 ·
(√‘𝐷)))) |
| 145 | 116, 144 | eqbrtrd 5165 |
. . . . 5
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ≤ (1 + (2 ·
(√‘𝐷)))) |
| 146 | 103, 145 | eqbrtrd 5165 |
. . . 4
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) ≤ (1 + (2 ·
(√‘𝐷)))) |
| 147 | 60, 66, 71, 90, 146 | ltletrd 11421 |
. . 3
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((abs‘((𝐴 / 𝐵) − (√‘𝐷))) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) < (1 + (2 ·
(√‘𝐷)))) |
| 148 | 56, 147 | eqbrtrd 5165 |
. 2
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) · ((𝐴 / 𝐵) + (√‘𝐷))))) < (1 + (2 ·
(√‘𝐷)))) |
| 149 | 54, 148 | eqbrtrd 5165 |
1
⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧
(abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 ·
(√‘𝐷)))) |