pellexlem2 - Mathbox for Stefan O'Rear

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Theorem pellexlem2 42922

Description: Lemma for pellex 42927. 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 1194	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℕ)
2	1	nnred 12140	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℝ)
3	2	resqcld 14032	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℝ)
4	2	sqge0d 14044	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (𝐵↑2))
5	3, 4	absidd 15330	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(𝐵↑2)) = (𝐵↑2))
6	5	eqcomd 2737	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) = (abs‘(𝐵↑2)))
7	6	oveq2d 7362	. . . . 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 12141	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℂ)
10	9	sqcld 14051	. . . . . . 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 12141	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℂ)
13	1	nncnd 12141	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ∈ ℂ)
14	13	sqcld 14051	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℂ)
15	12, 14	mulcld 11132	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐷 · (𝐵↑2)) ∈ ℂ)
16	10, 15	subcld 11472	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ∈ ℂ)
17	1	nnne0d 12175	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐵 ≠ 0)
18		sqne0 14030	. . . . . . . 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 15365	. . . . 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 2769	. . . 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 7362	. . 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 15346	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) ∈ ℝ)
25	24	recnd 11140	. . . 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 11899	. . 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 11936	. . . . . 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 14066	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
29	28	eqcomd 2737	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴↑2) / (𝐵↑2)) = ((𝐴 / 𝐵)↑2))
30	11	nnred 12140	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℝ)
31	11	nnnn0d 12442	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐷 ∈ ℕ₀)
32	31	nn0ge0d 12445	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ 𝐷)
33		remsqsqrt 15163	. . . . . . . . 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 15325	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈ ℝ)
36	35	recnd 11140	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (√‘𝐷) ∈ ℂ)
37	36	sqvald 14050	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷)↑2) = ((√‘𝐷) · (√‘𝐷)))
38	12, 14, 20	divcan4d 11903	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = 𝐷)
39	34, 37, 38	3eqtr4rd 2777	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐷 · (𝐵↑2)) / (𝐵↑2)) = ((√‘𝐷)↑2))
40	29, 39	oveq12d 7364	. . . . . 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 11897	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℂ)
42		subsq 14117	. . . . . . . 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 11131	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ∈ ℂ)
45	8	nnred 12140	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 𝐴 ∈ ℝ)
46	45, 1	nndivred 12179	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐴 / 𝐵) ∈ ℝ)
47	46, 35	resubcld 11545	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℝ)
48	47	recnd 11140	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) − (√‘𝐷)) ∈ ℂ)
49	44, 48	mulcomd 11133	. . . . . . 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 2766	. . . . . 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 2770	. . . . 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 6826	. . . 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 7362	. . 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 2774	. 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 15364	. . . 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 7362	. . 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 15346	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ∈ ℝ)
58	44	abscld 15346	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℝ)
59	57, 58	remulcld 11142	. . . . 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 11142	. . . 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 12398	. . . . . . . . 9 ⊢ 2 ∈ ℕ₀
62	61	nn0negzi 12511	. . . . . . . 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 14156	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℝ)
65	64, 58	remulcld 11142	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷)))) ∈ ℝ)
66	3, 65	remulcld 11142	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · ((𝐵↑-2) · (abs‘((𝐴 / 𝐵) + (√‘𝐷))))) ∈ ℝ)
67		1red 11113	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ∈ ℝ)
68		2re 12199	. . . . . . 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 11142	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 · (√‘𝐷)) ∈ ℝ)
71	67, 70	readdcld 11141	. . . 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 12174	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐴)
74	1	nngt0d 12174	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐵)
75	45, 2, 73, 74	divgt0d 12057	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐴 / 𝐵))
76	11	nngt0d 12174	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 𝐷)
77		sqrtgt0 15165	. . . . . . . . . . 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 11692	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < ((𝐴 / 𝐵) + (√‘𝐷)))
80	79	gt0ne0d 11681	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) ≠ 0)
81		absgt0 15232	. . . . . . . . 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 11971	. . . . . . 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 14157	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < (𝐵↑2))
88		ltmul2 11972	. . . . . 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 14058	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ∈ ℂ)
92	58	recnd 11140	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ∈ ℂ)
93		mulass 11094	. . . . . . . 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 2737	. . . . . . 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 13976	. . . . . . . . . 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 7362	. . . . . . . 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 11892	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (1 / (𝐵↑2))) = 1)
100	98, 99	eqtrd 2766	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐵↑2) · (𝐵↑-2)) = 1)
101	100	oveq1d 7361	. . . . . 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 11130	. . . . . 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 2770	. . . . 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 11315	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = ((√‘𝐷) + (𝐴 / 𝐵)))
105		ppncan 11403	. . . . . . . . . 10 ⊢ (((√‘𝐷) ∈ ℂ ∧ (√‘𝐷) ∈ ℂ ∧ (𝐴 / 𝐵) ∈ ℂ) → (((√‘𝐷) + (√‘𝐷)) + ((𝐴 / 𝐵) − (√‘𝐷))) = ((√‘𝐷) + (𝐴 / 𝐵)))
106	105	eqcomd 2737	. . . . . . . . 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 11131	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((√‘𝐷) + (√‘𝐷)) ∈ ℂ)
109	108, 48	addcomd 11315	. . . . . . . . 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 12256	. . . . . . . . . . . 12 ⊢ ((√‘𝐷) ∈ ℂ → (2 · (√‘𝐷)) = ((√‘𝐷) + (√‘𝐷)))
111	110	eqcomd 2737	. . . . . . . . . . 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 7362	. . . . . . . . 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 2766	. . . . . . . 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 2770	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → ((𝐴 / 𝐵) + (√‘𝐷)) = (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))))
116	115	fveq2d 6826	. . . . . 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 11141	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈ ℝ)
118	117	recnd 11140	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷))) ∈ ℂ)
119	118	abscld 15346	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(((𝐴 / 𝐵) − (√‘𝐷)) + (2 · (√‘𝐷)))) ∈ ℝ)
120	70	recnd 11140	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (2 · (√‘𝐷)) ∈ ℂ)
121	120	abscld 15346	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 · (√‘𝐷))) ∈ ℝ)
122	57, 121	readdcld 11141	. . . . . . 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 15366	. . . . . . 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 12227	. . . . . . . . . . . 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 15328	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (√‘𝐷))
127	69, 35, 125, 126	mulge0d 11694	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 ≤ (2 · (√‘𝐷)))
128	70, 127	absidd 15330	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘(2 · (√‘𝐷))) = (2 · (√‘𝐷)))
129	128	oveq2d 7362	. . . . . . . 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 14151	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑2) ∈ ℕ)
131	130	nnge1d 12173	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 1 ≤ (𝐵↑2))
132		0lt1 11639	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → 0 < 1)
134		lerec 12005	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 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 11812	. . . . . . . . . . . . 13 ⊢ (1 / 1) = 1
138	136, 137	breqtrdi 5130	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (1 / (𝐵↑2)) ≤ 1)
139	97, 138	eqbrtrd 5111	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (𝐵↑-2) ≤ 1)
140	57, 64, 67, 72, 139	ltletrd 11273	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < 1)
141	57, 67, 140	ltled 11261	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) − (√‘𝐷))) ≤ 1)
142	57, 67, 70, 141	leadd1dd 11731	. . . . . . . 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 5111	. . . . . . 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 11270	. . . . . 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 5111	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴 / 𝐵) + (√‘𝐷))) ≤ (1 + (2 · (√‘𝐷))))
146	103, 145	eqbrtrd 5111	. . . 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 11273	. . 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 5111	. 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 5111	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 < clt 11146 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ℤcz 12468 ↑cexp 13968 √csqrt 15140 abscabs 15141

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-n0 12382 df-z 12469 df-uz 12733 df-rp 12891 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143

This theorem is referenced by: pellexlem3 42923

W3C validator