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Theorem pellexlem5 36211
Description: Lemma for pellex 36213. Invoking fiphp3d 36197, 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.
StepHypRef Expression
1 pellexlem4 36210 . . 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 12588 . . . 4 (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∈ Fin
3 diffi 8054 . . . 4 ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∈ Fin → ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ∈ Fin)
42, 3mp1i 13 . . 3 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ∈ Fin)
5 elopab 4898 . . . . 5 (𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ ∃𝑦𝑧(𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))))
6 fveq2 6088 . . . . . . . . . . . 12 (𝑎 = ⟨𝑦, 𝑧⟩ → (1st𝑎) = (1st ‘⟨𝑦, 𝑧⟩))
76oveq1d 6542 . . . . . . . . . . 11 (𝑎 = ⟨𝑦, 𝑧⟩ → ((1st𝑎)↑2) = ((1st ‘⟨𝑦, 𝑧⟩)↑2))
8 fveq2 6088 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑦, 𝑧⟩ → (2nd𝑎) = (2nd ‘⟨𝑦, 𝑧⟩))
98oveq1d 6542 . . . . . . . . . . . 12 (𝑎 = ⟨𝑦, 𝑧⟩ → ((2nd𝑎)↑2) = ((2nd ‘⟨𝑦, 𝑧⟩)↑2))
109oveq2d 6543 . . . . . . . . . . 11 (𝑎 = ⟨𝑦, 𝑧⟩ → (𝐷 · ((2nd𝑎)↑2)) = (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2)))
117, 10oveq12d 6545 . . . . . . . . . 10 (𝑎 = ⟨𝑦, 𝑧⟩ → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = (((1st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2))))
12 vex 3175 . . . . . . . . . . . . 13 𝑦 ∈ V
13 vex 3175 . . . . . . . . . . . . 13 𝑧 ∈ V
1412, 13op1st 7044 . . . . . . . . . . . 12 (1st ‘⟨𝑦, 𝑧⟩) = 𝑦
1514oveq1i 6537 . . . . . . . . . . 11 ((1st ‘⟨𝑦, 𝑧⟩)↑2) = (𝑦↑2)
1612, 13op2nd 7045 . . . . . . . . . . . . 13 (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
1716oveq1i 6537 . . . . . . . . . . . 12 ((2nd ‘⟨𝑦, 𝑧⟩)↑2) = (𝑧↑2)
1817oveq2i 6538 . . . . . . . . . . 11 (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2)) = (𝐷 · (𝑧↑2))
1915, 18oveq12i 6539 . . . . . . . . . 10 (((1st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2)))
2011, 19syl6eq 2659 . . . . . . . . 9 (𝑎 = ⟨𝑦, 𝑧⟩ → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2))))
2120ad2antrl 759 . . . . . . . 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))))))) → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2))))
22 simprrl 799 . . . . . . . . . . 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 471 . . . . . . . . . . 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 791 . . . . . . . . . . . 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))))
2524ad2antll 760 . . . . . . . . . . 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 11232 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
2726ad2antrr 757 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑦 ∈ ℤ)
28 zsqcl 12751 . . . . . . . . . . . . . 14 (𝑦 ∈ ℤ → (𝑦↑2) ∈ ℤ)
2927, 28syl 17 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝑦↑2) ∈ ℤ)
30 nnz 11232 . . . . . . . . . . . . . . 15 (𝐷 ∈ ℕ → 𝐷 ∈ ℤ)
3130ad2antrl 759 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℤ)
32 simplr 787 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑧 ∈ ℕ)
3332nnzd 11313 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑧 ∈ ℤ)
34 zsqcl 12751 . . . . . . . . . . . . . . 15 (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℤ)
3533, 34syl 17 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝑧↑2) ∈ ℤ)
3631, 35zmulcld 11320 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝐷 · (𝑧↑2)) ∈ ℤ)
3729, 36zsubcld 11319 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ)
38 1re 9895 . . . . . . . . . . . . . . 15 1 ∈ ℝ
39 2re 10937 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
40 nnre 10874 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ ℕ → 𝐷 ∈ ℝ)
4140ad2antrl 759 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℝ)
42 nnnn0 11146 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ ℕ → 𝐷 ∈ ℕ0)
4342ad2antrl 759 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℕ0)
4443nn0ge0d 11201 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 0 ≤ 𝐷)
4541, 44resqrtcld 13950 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (√‘𝐷) ∈ ℝ)
46 remulcl 9877 . . . . . . . . . . . . . . . 16 ((2 ∈ ℝ ∧ (√‘𝐷) ∈ ℝ) → (2 · (√‘𝐷)) ∈ ℝ)
4739, 45, 46sylancr 693 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (2 · (√‘𝐷)) ∈ ℝ)
48 readdcl 9875 . . . . . . . . . . . . . . 15 ((1 ∈ ℝ ∧ (2 · (√‘𝐷)) ∈ ℝ) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
4938, 47, 48sylancr 693 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
5049flcld 12416 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ)
5150znegcld 11316 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → -(⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ)
5237zred 11314 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℝ)
5350zred 11314 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ)
54 nn0abscl 13846 . . . . . . . . . . . . . . . . . 18 (((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℕ0)
5537, 54syl 17 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℕ0)
5655nn0zd 11312 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℤ)
5756zred 11314 . . . . . . . . . . . . . . 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 10060 . . . . . . . . . . . . . . . 16 ((⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ → ((⌊‘(1 + (2 · (√‘𝐷)))) + 1) ∈ ℝ)
5953, 58syl 17 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((⌊‘(1 + (2 · (√‘𝐷)))) + 1) ∈ ℝ)
60 simprr 791 . . . . . . . . . . . . . . 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 12418 . . . . . . . . . . . . . . . 16 ((1 + (2 · (√‘𝐷))) ∈ ℝ → (1 + (2 · (√‘𝐷))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1))
6249, 61syl 17 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (1 + (2 · (√‘𝐷))) < ((⌊‘(1 + (2 · (√‘𝐷)))) + 1))
6357, 49, 59, 60, 62lttrd 10049 . . . . . . . . . . . . . 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 11261 . . . . . . . . . . . . . . 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)))
6556, 50, 64syl2anc 690 . . . . . . . . . . . . . 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)))
6663, 65mpbird 245 . . . . . . . . . . . . 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 13849 . . . . . . . . . . . . . 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)))))))
6867biimpa 499 . . . . . . . . . . . . 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))))))
6952, 53, 66, 68syl21anc 1316 . . . . . . . . . . . 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 12158 . . . . . . . . . . . . 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)))))))
7170biimpar 500 . . . . . . . . . . . 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))))))
7237, 51, 50, 69, 71syl31anc 1320 . . . . . . . . . . 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))))))
7322, 23, 25, 72syl12anc 1315 . . . . . . . . . 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))))))
7473adantlr 746 . . . . . . . . 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 789 . . . . . . . . . 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)
7675ad2antll 760 . . . . . . . . 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 4259 . . . . . . . . 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))
7874, 76, 77sylanbrc 694 . . . . . . . 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}))
7921, 78eqeltrd 2687 . . . . . . 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))))))) → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}))
8079ex 448 . . . . . 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)))))) → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0})))
8180exlimdvv 1848 . . . . 5 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (∃𝑦𝑧(𝑎 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))) → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0})))
825, 81syl5bi 230 . . . 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)))))} → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0})))
8382imp 443 . . 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)))))}) → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}))
841, 4, 83fiphp3d 36197 . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ)
85 eldif 3549 . . . . . 6 (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ↔ (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ¬ 𝑥 ∈ {0}))
86 elfzelz 12168 . . . . . . . 8 (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) → 𝑥 ∈ ℤ)
87 simp2 1054 . . . . . . . . . 10 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → 𝑥 ∈ ℤ)
88 velsn 4140 . . . . . . . . . . . . 13 (𝑥 ∈ {0} ↔ 𝑥 = 0)
8988biimpri 216 . . . . . . . . . . . 12 (𝑥 = 0 → 𝑥 ∈ {0})
9089necon3bi 2807 . . . . . . . . . . 11 𝑥 ∈ {0} → 𝑥 ≠ 0)
91903ad2ant3 1076 . . . . . . . . . 10 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → 𝑥 ≠ 0)
9287, 91jca 552 . . . . . . . . 9 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))
93923exp 1255 . . . . . . . 8 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ℤ → (¬ 𝑥 ∈ {0} → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))))
9486, 93syl5 33 . . . . . . 7 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) → (¬ 𝑥 ∈ {0} → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))))
9594impd 445 . . . . . 6 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ¬ 𝑥 ∈ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)))
9685, 95syl5bi 230 . . . . 5 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)))
97 simp2l 1079 . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ) → 𝑥 ∈ ℤ)
98 simp2r 1080 . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ) → 𝑥 ≠ 0)
99 nnex 10873 . . . . . . . . . . 11 ℕ ∈ V
10099, 99xpex 6837 . . . . . . . . . 10 (ℕ × ℕ) ∈ V
101 opabssxp 5106 . . . . . . . . . 10 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ⊆ (ℕ × ℕ)
102 ssdomg 7864 . . . . . . . . . 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)))
103100, 101, 102mp2 9 . . . . . . . . 9 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ (ℕ × ℕ)
104 xpnnen 14724 . . . . . . . . 9 (ℕ × ℕ) ≈ ℕ
105 domentr 7878 . . . . . . . . 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)
106103, 104, 105mp2an 703 . . . . . . . 8 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ ℕ
107 ensym 7868 . . . . . . . . . 10 ({𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥})
1081073ad2ant3 1076 . . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥})
109100, 101ssexi 4726 . . . . . . . . . 10 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ∈ V
110 fveq2 6088 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑏 → (1st𝑎) = (1st𝑏))
111110oveq1d 6542 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑏 → ((1st𝑎)↑2) = ((1st𝑏)↑2))
112 fveq2 6088 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑏 → (2nd𝑎) = (2nd𝑏))
113112oveq1d 6542 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑏 → ((2nd𝑎)↑2) = ((2nd𝑏)↑2))
114113oveq2d 6543 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑏 → (𝐷 · ((2nd𝑎)↑2)) = (𝐷 · ((2nd𝑏)↑2)))
115111, 114oveq12d 6545 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))))
116115eqeq1d 2611 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → ((((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥 ↔ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥))
117116elrab 3330 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ↔ (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥))
118 simprl 789 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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 799 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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 6088 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ⟨𝑦, 𝑧⟩ → (1st𝑏) = (1st ‘⟨𝑦, 𝑧⟩))
121120oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ⟨𝑦, 𝑧⟩ → ((1st𝑏)↑2) = ((1st ‘⟨𝑦, 𝑧⟩)↑2))
122 fveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = ⟨𝑦, 𝑧⟩ → (2nd𝑏) = (2nd ‘⟨𝑦, 𝑧⟩))
123122oveq1d 6542 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ⟨𝑦, 𝑧⟩ → ((2nd𝑏)↑2) = ((2nd ‘⟨𝑦, 𝑧⟩)↑2))
124123oveq2d 6543 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ⟨𝑦, 𝑧⟩ → (𝐷 · ((2nd𝑏)↑2)) = (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2)))
125121, 124oveq12d 6545 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ⟨𝑦, 𝑧⟩ → (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = (((1st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2))))
126125, 19syl6req 2660 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = ⟨𝑦, 𝑧⟩ → ((𝑦↑2) − (𝐷 · (𝑧↑2))) = (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))))
127126ad2antrl 759 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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))) = (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))))
128 simplr 787 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥) ∧ (𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))) → (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥)
129127, 128eqtrd 2643 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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)
130118, 119, 129jca32 555 . . . . . . . . . . . . . . . . . 18 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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)))
131130ex 448 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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))))
1321312eximdv 1834 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥) → (∃𝑦𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))) → ∃𝑦𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥))))
133 elopab 4898 . . . . . . . . . . . . . . . 16 (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ ∃𝑦𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))))
134 elopab 4898 . . . . . . . . . . . . . . . 16 (𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ↔ ∃𝑦𝑧(𝑏 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)))
135132, 133, 1343imtr4g 283 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑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)}))
136135expimpd 626 . . . . . . . . . . . . . 14 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (((((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥𝑏 ∈ {⟨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)}))
137136ancomsd 468 . . . . . . . . . . . . 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)))))} ∧ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥) → 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
138117, 137syl5bi 230 . . . . . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} → 𝑏 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
139138ssrdv 3573 . . . . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ⊆ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
1401393adant3 1073 . . . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ⊆ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
141 ssdomg 7864 . . . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)}))
142109, 140, 141mpsyl 65 . . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
143 endomtr 7877 . . . . . . . . 9 ((ℕ ≈ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ∧ {𝑎 ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑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)})
144108, 142, 143syl2anc 690 . . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ) → ℕ ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)})
145 sbth 7942 . . . . . . . 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)
146106, 144, 145sylancr 693 . . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)
14797, 98, 146jca32 555 . . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ) → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))
1481473exp 1255 . . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))))
14996, 148syld 45 . . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))))
150149impd 445 . . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ) → (𝑥 ∈ ℤ ∧ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ))))
151150reximdv2 2996 . 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)))))} ∣ (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥} ≈ ℕ → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ)))
15284, 151mpd 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 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  wrex 2896  {crab 2899  Vcvv 3172  cdif 3536  wss 3539  {csn 4124  cop 4130   class class class wbr 4577  {copab 4636   × cxp 5026  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  cen 7815  cdom 7816  Fincfn 7818  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797   < clt 9930  cle 9931  cmin 10117  -cneg 10118  cn 10867  2c2 10917  0cn0 11139  cz 11210  cq 11620  ...cfz 12152  cfl 12408  cexp 12677  csqrt 13767  abscabs 13768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-acn 8628  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-ico 12008  df-fz 12153  df-fl 12410  df-mod 12486  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-dvds 14768  df-gcd 15001  df-numer 15227  df-denom 15228
This theorem is referenced by:  pellex  36213
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