pellexlem5 - Mathbox for Stefan O'Rear

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Theorem pellexlem5 42821

Description: Lemma for pellex 42823. Invoking fiphp3d 42807, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.)

**Assertion**
Ref	Expression
pellexlem5	⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ))

Distinct variable group: 𝑥,𝐷,𝑦,𝑧

Proof of Theorem pellexlem5 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pellexlem4 42820	. . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ)
2		fzfi 13937	. . . 4 ⊢ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∈ Fin
3		diffi 9139	. . . 4 ⊢ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∈ Fin → ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ∈ Fin)
4	2, 3	mp1i 13	. . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ∈ Fin)
5		elopab 5487	. . . . 5 ⊢ (𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ ∃𝑦∃𝑧(𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))))
6		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → (1^st ‘𝑎) = (1^st ‘⟨𝑦, 𝑧⟩))
7	6	oveq1d 7402	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → ((1^st ‘𝑎)↑2) = ((1^st ‘⟨𝑦, 𝑧⟩)↑2))
8		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → (2^nd ‘𝑎) = (2^nd ‘⟨𝑦, 𝑧⟩))
9	8	oveq1d 7402	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → ((2^nd ‘𝑎)↑2) = ((2^nd ‘⟨𝑦, 𝑧⟩)↑2))
10	9	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → (𝐷 · ((2^nd ‘𝑎)↑2)) = (𝐷 · ((2^nd ‘⟨𝑦, 𝑧⟩)↑2)))
11	7, 10	oveq12d 7405	. . . . . . . . . 10 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = (((1^st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2^nd ‘⟨𝑦, 𝑧⟩)↑2))))
12		vex 3451	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
13		vex 3451	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
14	12, 13	op1st 7976	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨𝑦, 𝑧⟩) = 𝑦
15	14	oveq1i 7397	. . . . . . . . . . 11 ⊢ ((1^st ‘⟨𝑦, 𝑧⟩)↑2) = (𝑦↑2)
16	12, 13	op2nd 7977	. . . . . . . . . . . . 13 ⊢ (2^nd ‘⟨𝑦, 𝑧⟩) = 𝑧
17	16	oveq1i 7397	. . . . . . . . . . . 12 ⊢ ((2^nd ‘⟨𝑦, 𝑧⟩)↑2) = (𝑧↑2)
18	17	oveq2i 7398	. . . . . . . . . . 11 ⊢ (𝐷 · ((2^nd ‘⟨𝑦, 𝑧⟩)↑2)) = (𝐷 · (𝑧↑2))
19	15, 18	oveq12i 7399	. . . . . . . . . 10 ⊢ (((1^st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2^nd ‘⟨𝑦, 𝑧⟩)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2)))
20	11, 19	eqtrdi 2780	. . . . . . . . 9 ⊢ (𝑎 = ⟨𝑦, 𝑧⟩ → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2))))
21	20	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2))))
22		simprrl 780	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℕ ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ))
23		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℕ ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → 𝐷 ∈ ℕ)
24		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))
25	24	ad2antll 729	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℕ ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))
26		nnz 12550	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
27	26	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑦 ∈ ℤ)
28		zsqcl 14094	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℤ)
29	27, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝑦↑2) ∈ ℤ)
30		nnz 12550	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℤ)
31	30	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℤ)
32		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑧 ∈ ℕ)
33	32	nnzd 12556	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑧 ∈ ℤ)
34		zsqcl 14094	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℤ)
35	33, 34	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝑧↑2) ∈ ℤ)
36	31, 35	zmulcld 12644	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝐷 · (𝑧↑2)) ∈ ℤ)
37	29, 36	zsubcld 12643	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ)
38		1re 11174	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
39		2re 12260	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
40		nnre 12193	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℝ)
41	40	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℝ)
42		nnnn0 12449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℕ₀)
43	42	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℕ₀)
44	43	nn0ge0d 12506	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 0 ≤ 𝐷)
45	41, 44	resqrtcld 15384	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (√‘𝐷) ∈ ℝ)
46		remulcl 11153	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℝ ∧ (√‘𝐷) ∈ ℝ) → (2 · (√‘𝐷)) ∈ ℝ)
47	39, 45, 46	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (2 · (√‘𝐷)) ∈ ℝ)
48		readdcl 11151	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ (2 · (√‘𝐷)) ∈ ℝ) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
49	38, 47, 48	sylancr 587	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
50	49	flcld 13760	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ)
51	50	znegcld 12640	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → -(⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ)
52	37	zred 12638	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℝ)
53	50	zred 12638	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ)
54		nn0abscl 15278	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℕ₀)
55	37, 54	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℕ₀)
56	55	nn0zd 12555	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℤ)
57	56	zred 12638	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℝ)
58		peano2re 11347	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ → ((⌊‘(1 + (2 · (√‘𝐷)))) + 1) ∈ ℝ)
59	53, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((⌊‘(1 + (2 · (√‘𝐷)))) + 1) ∈ ℝ)
60		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))
61		flltp1 13762	. . . . . . . . . . . . . . . 16 ⊢ ((1 + (2 · (√‘𝐷))) ∈ ℝ → (1 + (2 · (√‘𝐷))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1))
62	49, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (1 + (2 · (√‘𝐷))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1))
63	57, 49, 59, 60, 62	lttrd 11335	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1))
64		zleltp1 12584	. . . . . . . . . . . . . . 15 ⊢ (((abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℤ ∧ (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ) → ((abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))) ↔ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1)))
65	56, 50, 64	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))) ↔ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1)))
66	63, 65	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))))
67		absle 15282	. . . . . . . . . . . . . 14 ⊢ ((((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℝ ∧ (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ) → ((abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))) ↔ (-(⌊‘(1 + (2 · (√‘𝐷)))) ≤ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))))))
68	67	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℝ ∧ (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ) ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ≤ (⌊‘(1 + (2 · (√‘𝐷))))) → (-(⌊‘(1 + (2 · (√‘𝐷)))) ≤ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≤ (⌊‘(1 + (2 · (√‘𝐷))))))
69	52, 53, 66, 68	syl21anc 837	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (-(⌊‘(1 + (2 · (√‘𝐷)))) ≤ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≤ (⌊‘(1 + (2 · (√‘𝐷))))))
70		elfz 13474	. . . . . . . . . . . . 13 ⊢ ((((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ ∧ -(⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ ∧ (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ) → (((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ↔ (-(⌊‘(1 + (2 · (√‘𝐷)))) ≤ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))))))
71	70	biimpar 477	. . . . . . . . . . . 12 ⊢ (((((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ ∧ -(⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ ∧ (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ) ∧ (-(⌊‘(1 + (2 · (√‘𝐷)))) ≤ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≤ (⌊‘(1 + (2 · (√‘𝐷)))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))))
72	37, 51, 50, 69, 71	syl31anc 1375	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))))
73	22, 23, 25, 72	syl12anc 836	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℕ ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))))
74	73	adantlr 715	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))))
75		simprl 770	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0)
76	75	ad2antll 729	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0)
77		eldifsn 4750	. . . . . . . . 9 ⊢ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ↔ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0))
78	74, 76, 77	sylanbrc 583	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}))
79	21, 78	eqeltrd 2828	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}))
80	79	ex 412	. . . . . 6 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))) → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0})))
81	80	exlimdvv 1934	. . . . 5 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (∃𝑦∃𝑧(𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))) → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0})))
82	5, 81	biimtrid 242	. . . 4 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0})))
83	82	imp 406	. . 3 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}))
84	1, 4, 83	fiphp3d 42807	. 2 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}){𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ)
85		eldif 3924	. . . . . 6 ⊢ (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ↔ (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ¬ 𝑥 ∈ {0}))
86		elfzelz 13485	. . . . . . . 8 ⊢ (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) → 𝑥 ∈ ℤ)
87		simp2 1137	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → 𝑥 ∈ ℤ)
88		velsn 4605	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
89	88	biimpri 228	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → 𝑥 ∈ {0})
90	89	necon3bi 2951	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ {0} → 𝑥 ≠ 0)
91	90	3ad2ant3 1135	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → 𝑥 ≠ 0)
92	87, 91	jca 511	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))
93	92	3exp 1119	. . . . . . . 8 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ℤ → (¬ 𝑥 ∈ {0} → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))))
94	86, 93	syl5 34	. . . . . . 7 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) → (¬ 𝑥 ∈ {0} → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))))
95	94	impd 410	. . . . . 6 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ¬ 𝑥 ∈ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)))
96	85, 95	biimtrid 242	. . . . 5 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)))
97		simp2l 1200	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → 𝑥 ∈ ℤ)
98		simp2r 1201	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → 𝑥 ≠ 0)
99		nnex 12192	. . . . . . . . . . 11 ⊢ ℕ ∈ V
100	99, 99	xpex 7729	. . . . . . . . . 10 ⊢ (ℕ × ℕ) ∈ V
101		opabssxp 5731	. . . . . . . . . 10 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ⊆ (ℕ × ℕ)
102		ssdomg 8971	. . . . . . . . . 10 ⊢ ((ℕ × ℕ) ∈ V → ({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ⊆ (ℕ × ℕ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ (ℕ × ℕ)))
103	100, 101, 102	mp2 9	. . . . . . . . 9 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ (ℕ × ℕ)
104		xpnnen 16179	. . . . . . . . 9 ⊢ (ℕ × ℕ) ≈ ℕ
105		domentr 8984	. . . . . . . . 9 ⊢ (({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ (ℕ × ℕ) ∧ (ℕ × ℕ) ≈ ℕ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ ℕ)
106	103, 104, 105	mp2an 692	. . . . . . . 8 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ ℕ
107		ensym 8974	. . . . . . . . . 10 ⊢ ({𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ → ℕ ≈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥})
108	107	3ad2ant3 1135	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → ℕ ≈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥})
109	100, 101	ssexi 5277	. . . . . . . . . 10 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ∈ V
110		fveq2 6858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑏 → (1^st ‘𝑎) = (1^st ‘𝑏))
111	110	oveq1d 7402	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑏 → ((1^st ‘𝑎)↑2) = ((1^st ‘𝑏)↑2))
112		fveq2 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑏 → (2^nd ‘𝑎) = (2^nd ‘𝑏))
113	112	oveq1d 7402	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑏 → ((2^nd ‘𝑎)↑2) = ((2^nd ‘𝑏)↑2))
114	113	oveq2d 7403	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑏 → (𝐷 · ((2^nd ‘𝑎)↑2)) = (𝐷 · ((2^nd ‘𝑏)↑2)))
115	111, 114	oveq12d 7405	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑏 → (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))))
116	115	eqeq1d 2731	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → ((((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥 ↔ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥))
117	116	elrab 3659	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ↔ (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥))
118		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → 𝑏 = ⟨𝑦, 𝑧⟩)
119		simprrl 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ))
120		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → (1^st ‘𝑏) = (1^st ‘⟨𝑦, 𝑧⟩))
121	120	oveq1d 7402	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → ((1^st ‘𝑏)↑2) = ((1^st ‘⟨𝑦, 𝑧⟩)↑2))
122		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → (2^nd ‘𝑏) = (2^nd ‘⟨𝑦, 𝑧⟩))
123	122	oveq1d 7402	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → ((2^nd ‘𝑏)↑2) = ((2^nd ‘⟨𝑦, 𝑧⟩)↑2))
124	123	oveq2d 7403	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → (𝐷 · ((2^nd ‘𝑏)↑2)) = (𝐷 · ((2^nd ‘⟨𝑦, 𝑧⟩)↑2)))
125	121, 124	oveq12d 7405	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = (((1^st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2^nd ‘⟨𝑦, 𝑧⟩)↑2))))
126	125, 19	eqtr2di 2781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = ⟨𝑦, 𝑧⟩ → ((𝑦↑2) − (𝐷 · (𝑧↑2))) = (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))))
127	126	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) = (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))))
128		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥)
129	127, 128	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)
130	118, 119, 129	jca32 515	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)))
131	130	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) → ((𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))) → (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥))))
132	131	2eximdv 1919	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) → (∃𝑦∃𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))) → ∃𝑦∃𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥))))
133		elopab 5487	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ ∃𝑦∃𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))))
134		elopab 5487	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ↔ ∃𝑦∃𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)))
135	132, 133, 134	3imtr4g 296	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) → (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} → 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
136	135	expimpd 453	. . . . . . . . . . . . . 14 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (((((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥 ∧ 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}) → 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
137	136	ancomsd 465	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → ((𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∧ (((1^st ‘𝑏)↑2) − (𝐷 · ((2^nd ‘𝑏)↑2))) = 𝑥) → 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
138	117, 137	biimtrid 242	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑏 ∈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} → 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
139	138	ssrdv 3952	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ⊆ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
140	139	3adant3 1132	. . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ⊆ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
141		ssdomg 8971	. . . . . . . . . 10 ⊢ ({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ∈ V → ({𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ⊆ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} → {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
142	109, 140, 141	mpsyl 68	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
143		endomtr 8983	. . . . . . . . 9 ⊢ ((ℕ ≈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}) → ℕ ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
144	108, 142, 143	syl2anc 584	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → ℕ ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
145		sbth 9061	. . . . . . . 8 ⊢ (({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ ℕ ∧ ℕ ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)
146	106, 144, 145	sylancr 587	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)
147	97, 98, 146	jca32 515	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))
148	147	3exp 1119	. . . . 5 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) → ({𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))))
149	96, 148	syld 47	. . . 4 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) → ({𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))))
150	149	impd 410	. . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ) → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ))))
151	150	reximdv2 3143	. 2 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (∃𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}){𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1^st ‘𝑎)↑2) − (𝐷 · ((2^nd ‘𝑎)↑2))) = 𝑥} ≈ ℕ → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))
152	84, 151	mpd 15	1 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 {crab 3405 Vcvv 3447 ∖ cdif 3911 ⊆ wss 3914 {csn 4589 ⟨cop 4595 class class class wbr 5107 {copab 5169 × cxp 5636 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 ≈ cen 8915 ≼ cdom 8916 Fincfn 8918 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 -cneg 11406 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℚcq 12907 ...cfz 13468 ⌊cfl 13752 ↑cexp 14026 √csqrt 15199 abscabs 15200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-omul 8439 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-ico 13312 df-fz 13469 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-numer 16705 df-denom 16706

This theorem is referenced by: pellex 42823

W3C validator