pellexlem2 - Mathbox for Stefan O'Rear

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Theorem pellexlem2 40652

Description: Lemma for pellex 40657. 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 1192	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℕ)
2	1	nnred 11988	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℝ)
3	2	resqcld 13965	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℝ)
4	2	sqge0d 13966	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (𝐵↑2))
5	3, 4	absidd 15134	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(𝐵↑2)) = (𝐵↑2))
6	5	eqcomd 2744	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) = (abs‘(𝐵↑2)))
7	6	oveq2d 7291	. . . . 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 1191	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℕ)
9	8	nncnd 11989	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℂ)
10	9	sqcld 13862	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴↑2) ∈ ℂ)
11		simpl1 1190	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℕ)
12	11	nncnd 11989	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℂ)
13	1	nncnd 11989	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℂ)
14	13	sqcld 13862	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℂ)
15	12, 14	mulcld 10995	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐷 · (𝐵↑2)) ∈ ℂ)
16	10, 15	subcld 11332	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ∈ ℂ)
17	1	nnne0d 12023	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ≠ 0)
18		sqne0 13843	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ((𝐵↑2) ≠ 0 ↔ 𝐵 ≠ 0))
19	18	biimpar 478	. . . . . . 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 15167	. . . . 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 2781	. . . 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 7291	. . 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 15148	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) ∈ ℝ)
25	24	recnd 11003	. . . 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 11753	. . 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 11790	. . . . . 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 13877	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
29	28	eqcomd 2744	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) / (𝐵↑2)) = ((𝐴 / 𝐵)↑2))
30	11	nnred 11988	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℝ)
31	11	nnnn0d 12293	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℕ₀)
32	31	nn0ge0d 12296	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ 𝐷)
33		remsqsqrt 14968	. . . . . . . . 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 15129	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈ ℝ)
36	35	recnd 11003	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈ ℂ)
37	36	sqvald 13861	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷)↑2) = ((√‘𝐷) · (√‘𝐷)))
38	12, 14, 20	divcan4d 11757	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = 𝐷)
39	34, 37, 38	3eqtr4rd 2789	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = ((√‘𝐷)↑2))
40	29, 39	oveq12d 7293	. . . . . 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 11751	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℂ)
42		subsq 13926	. . . . . . . 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 10994	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ)
45	8	nnred 11988	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℝ)
46	45, 1	nndivred 12027	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℝ)
47	46, 35	resubcld 11403	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℝ)
48	47	recnd 11003	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℂ)
49	44, 48	mulcomd 10996	. . . . . . 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 2778	. . . . . 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 2782	. . . . 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 6778	. . . 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 7291	. . 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 2786	. 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 15166	. . . 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 7291	. . 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 15148	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ∈ ℝ)
58	44	abscld 15148	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℝ)
59	57, 58	remulcld 11005	. . . . 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 11005	. . . 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 12250	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
62	61	nn0negzi 12359	. . . . . . . 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 13964	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℝ)
65	64, 58	remulcld 11005	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ∈ ℝ)
66	3, 65	remulcld 11005	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) ∈ ℝ)
67		1red 10976	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ∈ ℝ)
68		2re 12047	. . . . . . 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 11005	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 · (√‘𝐷)) ∈ ℝ)
71	67, 70	readdcld 11004	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
72		simpr 485	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2))
73	8	nngt0d 12022	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐴)
74	1	nngt0d 12022	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐵)
75	45, 2, 73, 74	divgt0d 11910	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐴 / 𝐵))
76	11	nngt0d 12022	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐷)
77		sqrtgt0 14970	. . . . . . . . . . 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 11550	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < ((𝐴 / 𝐵) + (√‘𝐷)))
80	79	gt0ne0d 11539	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0)
81		absgt0 15036	. . . . . . . . 9 ⊢ (((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ → (((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0 ↔ 0 < (abs‘((𝐴 / 𝐵) + (√‘𝐷)))))
82	81	biimpa 477	. . . . . . . 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 11825	. . . . . . 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 1373	. . . . . 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 231	. . . . 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 13967	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐵↑2))
88		ltmul2 11826	. . . . . 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 1373	. . . . 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 231	. . . 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 13869	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℂ)
92	58	recnd 11003	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℂ)
93		mulass 10959	. . . . . . . 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 2744	. . . . . . 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 1370	. . . . . 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 13790	. . . . . . . . . 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 7291	. . . . . . . 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 11746	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (1 / (𝐵↑2))) = 1)
100	98, 99	eqtrd 2778	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (𝐵↑-2)) = 1)
101	100	oveq1d 7290	. . . . . 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 10993	. . . . . 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 2782	. . . . 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 11177	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = ((√‘𝐷) + (𝐴 / 𝐵)))
105		ppncan 11263	. . . . . . . . . 10 ⊢ (((√‘𝐷) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) → (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))) = ((√‘𝐷) + (𝐴 / 𝐵)))
106	105	eqcomd 2744	. . . . . . . . 9 ⊢ (((√‘𝐷) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) → ((√‘𝐷) + (𝐴 / 𝐵)) = (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))))
107	36, 36, 41, 106	syl3anc 1370	. . . . . . . 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 10994	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) + (√‘𝐷)) ∈ ℂ)
109	108, 48	addcomd 11177	. . . . . . . . 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 12109	. . . . . . . . . . . 12 ⊢ ((√‘𝐷) ∈ ℂ → (2 · (√‘𝐷)) = ((√‘𝐷) + (√‘𝐷)))
111	110	eqcomd 2744	. . . . . . . . . . 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 7291	. . . . . . . . 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 2778	. . . . . . . 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 2782	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))))
116	115	fveq2d 6778	. . . . . 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 11004	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈ ℝ)
118	117	recnd 11003	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈ ℂ)
119	118	abscld 15148	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) ∈ ℝ)
120	70	recnd 11003	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 · (√‘𝐷)) ∈ ℂ)
121	120	abscld 15148	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 · (√‘𝐷))) ∈ ℝ)
122	57, 121	readdcld 11004	. . . . . . 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 15168	. . . . . . 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 12075	. . . . . . . . . . . 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 15132	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (√‘𝐷))
127	69, 35, 125, 126	mulge0d 11552	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (2 · (√‘𝐷)))
128	70, 127	absidd 15134	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 · (√‘𝐷))) = (2 · (√‘𝐷)))
129	128	oveq2d 7291	. . . . . . . 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 13959	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℕ)
131	130	nnge1d 12021	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ≤ (𝐵↑2))
132		0lt1 11497	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 1)
134		lerec 11858	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ (1 / 1))
137		1div1e1 11665	. . . . . . . . . . . . 13 ⊢ (1 / 1) = 1
138	136, 137	breqtrdi 5115	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ 1)
139	97, 138	eqbrtrd 5096	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ≤ 1)
140	57, 64, 67, 72, 139	ltletrd 11135	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < 1)
141	57, 67, 140	ltled 11123	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ≤ 1)
142	57, 67, 70, 141	leadd1dd 11589	. . . . . . . 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 5096	. . . . . . 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 11132	. . . . . 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 5096	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ≤ (1 + (2 · (√‘𝐷))))
146	103, 145	eqbrtrd 5096	. . . 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 11135	. . 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 5096	. 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 5096	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ↑cexp 13782 √csqrt 14944 abscabs 14945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947

This theorem is referenced by: pellexlem3 40653

W3C validator