pellexlem2 - Mathbox for Stefan O'Rear

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Theorem pellexlem2 39447

Description: Lemma for pellex 39452. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)

**Assertion**
Ref	Expression
pellexlem2	⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 · (√‘𝐷))))

**Proof of Theorem pellexlem2**
Step	Hyp	Ref	Expression
1		simpl3 1189	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℕ)
2	1	nnred 11653	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℝ)
3	2	resqcld 13612	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℝ)
4	2	sqge0d 13613	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (𝐵↑2))
5	3, 4	absidd 14782	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(𝐵↑2)) = (𝐵↑2))
6	5	eqcomd 2827	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) = (abs‘(𝐵↑2)))
7	6	oveq2d 7172	. . . . 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 1188	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℕ)
9	8	nncnd 11654	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℂ)
10	9	sqcld 13509	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴↑2) ∈ ℂ)
11		simpl1 1187	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℕ)
12	11	nncnd 11654	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℂ)
13	1	nncnd 11654	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℂ)
14	13	sqcld 13509	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℂ)
15	12, 14	mulcld 10661	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐷 · (𝐵↑2)) ∈ ℂ)
16	10, 15	subcld 10997	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ∈ ℂ)
17	1	nnne0d 11688	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ≠ 0)
18		sqne0 13490	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0))
19	18	biimpar 480	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐵↑2) ≠ 0)
20	13, 17, 19	syl2anc 586	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ≠ 0)
21	16, 14, 20	absdivd 14815	. . . . 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 2859	. . . 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 7172	. . 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 14796	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) ∈ ℝ)
25	24	recnd 10669	. . . 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 11418	. . 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 11455	. . . . . 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 13524	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
29	28	eqcomd 2827	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) / (𝐵↑2)) = ((𝐴 / 𝐵)↑2))
30	11	nnred 11653	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℝ)
31	11	nnnn0d 11956	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℕ₀)
32	31	nn0ge0d 11959	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ 𝐷)
33		remsqsqrt 14616	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℝ ∧ 0 ≤ 𝐷) → ((√‘𝐷) · (√‘𝐷)) = 𝐷)
34	30, 32, 33	syl2anc 586	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) · (√‘𝐷)) = 𝐷)
35	30, 32	resqrtcld 14777	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈ ℝ)
36	35	recnd 10669	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈ ℂ)
37	36	sqvald 13508	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷)↑2) = ((√‘𝐷) · (√‘𝐷)))
38	12, 14, 20	divcan4d 11422	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = 𝐷)
39	34, 37, 38	3eqtr4rd 2867	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = ((√‘𝐷)↑2))
40	29, 39	oveq12d 7174	. . . . . 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 11416	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℂ)
42		subsq 13573	. . . . . . . 8 ⊢ (((𝐴 / 𝐵) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ) → (((𝐴 / 𝐵)↑2) − ((√‘𝐷)↑2)) = (((𝐴 / 𝐵) + (√‘𝐷)) · ((𝐴 / 𝐵) − (√‘𝐷))))
43	41, 36, 42	syl2anc 586	. . . . . . 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 10660	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ)
45	8	nnred 11653	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℝ)
46	45, 1	nndivred 11692	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℝ)
47	46, 35	resubcld 11068	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℝ)
48	47	recnd 10669	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℂ)
49	44, 48	mulcomd 10662	. . . . . . 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 2856	. . . . . 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 2860	. . . . 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 6674	. . . 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 7172	. . 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 2864	. 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 14814	. . . 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 7172	. . 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 14796	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ∈ ℝ)
58	44	abscld 14796	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℝ)
59	57, 58	remulcld 10671	. . . . 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 10671	. . . 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 11915	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
62	61	nn0negzi 12022	. . . . . . . 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 13611	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℝ)
65	64, 58	remulcld 10671	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ∈ ℝ)
66	3, 65	remulcld 10671	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) ∈ ℝ)
67		1red 10642	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ∈ ℝ)
68		2re 11712	. . . . . . 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 10671	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 · (√‘𝐷)) ∈ ℝ)
71	67, 70	readdcld 10670	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
72		simpr 487	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2))
73	8	nngt0d 11687	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐴)
74	1	nngt0d 11687	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐵)
75	45, 2, 73, 74	divgt0d 11575	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐴 / 𝐵))
76	11	nngt0d 11687	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐷)
77		sqrtgt0 14618	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℝ ∧ 0 < 𝐷) → 0 < (√‘𝐷))
78	30, 76, 77	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (√‘𝐷))
79	46, 35, 75, 78	addgt0d 11215	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < ((𝐴 / 𝐵) + (√‘𝐷)))
80	79	gt0ne0d 11204	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0)
81		absgt0 14684	. . . . . . . . 9 ⊢ (((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ → (((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0 ↔ 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))
82	81	biimpa 479	. . . . . . . 8 ⊢ ((((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ ∧ ((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0) → 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷))))
83	44, 80, 82	syl2anc 586	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷))))
84		ltmul1 11490	. . . . . . 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 1370	. . . . . 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 234	. . . . 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 13614	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐵↑2))
88		ltmul2 11491	. . . . . 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 1370	. . . . 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 234	. . . 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 13516	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℂ)
92	58	recnd 10669	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℂ)
93		mulass 10625	. . . . . . . 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 2827	. . . . . . 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 1367	. . . . . 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 13438	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℂ ∧ 2 ∈ ℕ₀) → (𝐵↑-2) = (1 / (𝐵↑2)))
97	13, 61, 96	sylancl 588	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) = (1 / (𝐵↑2)))
98	97	oveq2d 7172	. . . . . . . 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 11411	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (1 / (𝐵↑2))) = 1)
100	98, 99	eqtrd 2856	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (𝐵↑-2)) = 1)
101	100	oveq1d 7171	. . . . . 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	mulid2d 10659	. . . . . 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 2860	. . . . 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 10842	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = ((√‘𝐷) + (𝐴 / 𝐵)))
105		ppncan 10928	. . . . . . . . . 10 ⊢ (((√‘𝐷) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) → (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))) = ((√‘𝐷) + (𝐴 / 𝐵)))
106	105	eqcomd 2827	. . . . . . . . 9 ⊢ (((√‘𝐷) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) → ((√‘𝐷) + (𝐴 / 𝐵)) = (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))))
107	36, 36, 41, 106	syl3anc 1367	. . . . . . . 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 10660	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) + (√‘𝐷)) ∈ ℂ)
109	108, 48	addcomd 10842	. . . . . . . . 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 11774	. . . . . . . . . . . 12 ⊢ ((√‘𝐷) ∈ ℂ → (2 · (√‘𝐷)) = ((√‘𝐷) + (√‘𝐷)))
111	110	eqcomd 2827	. . . . . . . . . . 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 7172	. . . . . . . . 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 2856	. . . . . . . 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 2860	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))))
116	115	fveq2d 6674	. . . . . 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 10670	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈ ℝ)
118	117	recnd 10669	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈ ℂ)
119	118	abscld 14796	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) ∈ ℝ)
120	70	recnd 10669	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 · (√‘𝐷)) ∈ ℂ)
121	120	abscld 14796	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 · (√‘𝐷))) ∈ ℝ)
122	57, 121	readdcld 10670	. . . . . . 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 14816	. . . . . . 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 11740	. . . . . . . . . . . 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 14780	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (√‘𝐷))
127	69, 35, 125, 126	mulge0d 11217	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (2 · (√‘𝐷)))
128	70, 127	absidd 14782	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 · (√‘𝐷))) = (2 · (√‘𝐷)))
129	128	oveq2d 7172	. . . . . . . 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 13606	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℕ)
131	130	nnge1d 11686	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ≤ (𝐵↑2))
132		0lt1 11162	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 1)
134		lerec 11523	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ (1 / 1))
137		1div1e1 11330	. . . . . . . . . . . . 13 ⊢ (1 / 1) = 1
138	136, 137	breqtrdi 5107	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ 1)
139	97, 138	eqbrtrd 5088	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ≤ 1)
140	57, 64, 67, 72, 139	ltletrd 10800	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < 1)
141	57, 67, 140	ltled 10788	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ≤ 1)
142	57, 67, 70, 141	leadd1dd 11254	. . . . . . . 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 5088	. . . . . . 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 10797	. . . . . 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 5088	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ≤ (1 + (2 · (√‘𝐷))))
146	103, 145	eqbrtrd 5088	. . . 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 10800	. . 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 5088	. 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 5088	1 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 · (√‘𝐷))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 / cdiv 11297 ℕcn 11638 2c2 11693 ℕ₀cn0 11898 ℤcz 11982 ↑cexp 13430 √csqrt 14592 abscabs 14593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595

This theorem is referenced by: pellexlem3 39448

W3C validator