Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pellexlem5 Structured version   Visualization version   GIF version

Theorem pellexlem5 37899
Description: Lemma for pellex 37901. Invoking fiphp3d 37885, 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 37898 . . 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 12965 . . . 4 (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∈ Fin
3 diffi 8357 . . . 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 5133 . . . . 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 6352 . . . . . . . . . . . 12 (𝑎 = ⟨𝑦, 𝑧⟩ → (1st𝑎) = (1st ‘⟨𝑦, 𝑧⟩))
76oveq1d 6828 . . . . . . . . . . 11 (𝑎 = ⟨𝑦, 𝑧⟩ → ((1st𝑎)↑2) = ((1st ‘⟨𝑦, 𝑧⟩)↑2))
8 fveq2 6352 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑦, 𝑧⟩ → (2nd𝑎) = (2nd ‘⟨𝑦, 𝑧⟩))
98oveq1d 6828 . . . . . . . . . . . 12 (𝑎 = ⟨𝑦, 𝑧⟩ → ((2nd𝑎)↑2) = ((2nd ‘⟨𝑦, 𝑧⟩)↑2))
109oveq2d 6829 . . . . . . . . . . 11 (𝑎 = ⟨𝑦, 𝑧⟩ → (𝐷 · ((2nd𝑎)↑2)) = (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2)))
117, 10oveq12d 6831 . . . . . . . . . 10 (𝑎 = ⟨𝑦, 𝑧⟩ → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = (((1st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2))))
12 vex 3343 . . . . . . . . . . . . 13 𝑦 ∈ V
13 vex 3343 . . . . . . . . . . . . 13 𝑧 ∈ V
1412, 13op1st 7341 . . . . . . . . . . . 12 (1st ‘⟨𝑦, 𝑧⟩) = 𝑦
1514oveq1i 6823 . . . . . . . . . . 11 ((1st ‘⟨𝑦, 𝑧⟩)↑2) = (𝑦↑2)
1612, 13op2nd 7342 . . . . . . . . . . . . 13 (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
1716oveq1i 6823 . . . . . . . . . . . 12 ((2nd ‘⟨𝑦, 𝑧⟩)↑2) = (𝑧↑2)
1817oveq2i 6824 . . . . . . . . . . 11 (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2)) = (𝐷 · (𝑧↑2))
1915, 18oveq12i 6825 . . . . . . . . . 10 (((1st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2)))
2011, 19syl6eq 2810 . . . . . . . . 9 (𝑎 = ⟨𝑦, 𝑧⟩ → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = ((𝑦↑2) − (𝐷 · (𝑧↑2))))
2120ad2antrl 766 . . . . . . . 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 823 . . . . . . . . . . 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 474 . . . . . . . . . . 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 813 . . . . . . . . . . . 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 767 . . . . . . . . . . 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 11591 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
2726ad2antrr 764 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑦 ∈ ℤ)
28 zsqcl 13128 . . . . . . . . . . . . . 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 11591 . . . . . . . . . . . . . . 15 (𝐷 ∈ ℕ → 𝐷 ∈ ℤ)
3130ad2antrl 766 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℤ)
32 simplr 809 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑧 ∈ ℕ)
3332nnzd 11673 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝑧 ∈ ℤ)
34 zsqcl 13128 . . . . . . . . . . . . . . 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 11680 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (𝐷 · (𝑧↑2)) ∈ ℤ)
3729, 36zsubcld 11679 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℤ)
38 1re 10231 . . . . . . . . . . . . . . 15 1 ∈ ℝ
39 2re 11282 . . . . . . . . . . . . . . . 16 2 ∈ ℝ
40 nnre 11219 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ ℕ → 𝐷 ∈ ℝ)
4140ad2antrl 766 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℝ)
42 nnnn0 11491 . . . . . . . . . . . . . . . . . . 19 (𝐷 ∈ ℕ → 𝐷 ∈ ℕ0)
4342ad2antrl 766 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 𝐷 ∈ ℕ0)
4443nn0ge0d 11546 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → 0 ≤ 𝐷)
4541, 44resqrtcld 14355 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (√‘𝐷) ∈ ℝ)
46 remulcl 10213 . . . . . . . . . . . . . . . 16 ((2 ∈ ℝ ∧ (√‘𝐷) ∈ ℝ) → (2 · (√‘𝐷)) ∈ ℝ)
4739, 45, 46sylancr 698 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (2 · (√‘𝐷)) ∈ ℝ)
48 readdcl 10211 . . . . . . . . . . . . . . 15 ((1 ∈ ℝ ∧ (2 · (√‘𝐷)) ∈ ℝ) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
4938, 47, 48sylancr 698 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (1 + (2 · (√‘𝐷))) ∈ ℝ)
5049flcld 12793 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ)
5150znegcld 11676 . . . . . . . . . . . 12 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → -(⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℤ)
5237zred 11674 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) ∈ ℝ)
5350zred 11674 . . . . . . . . . . . . 13 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (⌊‘(1 + (2 · (√‘𝐷)))) ∈ ℝ)
54 nn0abscl 14251 . . . . . . . . . . . . . . . . . 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 11672 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝐷 ∈ ℕ ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) ∈ ℤ)
5756zred 11674 . . . . . . . . . . . . . . 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 10401 . . . . . . . . . . . . . . . 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 813 . . . . . . . . . . . . . . 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 12795 . . . . . . . . . . . . . . . 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 10390 . . . . . . . . . . . . . 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 11620 . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . 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 14254 . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . 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 1476 . . . . . . . . . . . 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 12525 . . . . . . . . . . . . 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 503 . . . . . . . . . . . 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 1480 . . . . . . . . . . 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 1475 . . . . . . . . . 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 753 . . . . . . . . 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 811 . . . . . . . . . 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 767 . . . . . . . . 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 4462 . . . . . . . . 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 701 . . . . . . . 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 2839 . . . . . . 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 449 . . . . . 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 2011 . . . . 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 232 . . . 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 444 . . 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 37885 . 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 3725 . . . . . 6 (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) ↔ (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ¬ 𝑥 ∈ {0}))
86 elfzelz 12535 . . . . . . . 8 (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) → 𝑥 ∈ ℤ)
87 simp2 1132 . . . . . . . . . 10 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → 𝑥 ∈ ℤ)
88 velsn 4337 . . . . . . . . . . . . 13 (𝑥 ∈ {0} ↔ 𝑥 = 0)
8988biimpri 218 . . . . . . . . . . . 12 (𝑥 = 0 → 𝑥 ∈ {0})
9089necon3bi 2958 . . . . . . . . . . 11 𝑥 ∈ {0} → 𝑥 ≠ 0)
91903ad2ant3 1130 . . . . . . . . . 10 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → 𝑥 ≠ 0)
9287, 91jca 555 . . . . . . . . 9 (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑥 ∈ ℤ ∧ ¬ 𝑥 ∈ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))
93923exp 1113 . . . . . . . 8 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ℤ → (¬ 𝑥 ∈ {0} → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))))
9486, 93syl5 34 . . . . . . 7 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) → (¬ 𝑥 ∈ {0} → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0))))
9594impd 446 . . . . . 6 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑥 ∈ (-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∧ ¬ 𝑥 ∈ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)))
9685, 95syl5bi 232 . . . . 5 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑥 ∈ ((-(⌊‘(1 + (2 · (√‘𝐷))))...(⌊‘(1 + (2 · (√‘𝐷))))) ∖ {0}) → (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)))
97 simp2l 1242 . . . . . . 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 1243 . . . . . . 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 11218 . . . . . . . . . . 11 ℕ ∈ V
10099, 99xpex 7127 . . . . . . . . . 10 (ℕ × ℕ) ∈ V
101 opabssxp 5350 . . . . . . . . . 10 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ⊆ (ℕ × ℕ)
102 ssdomg 8167 . . . . . . . . . 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 15138 . . . . . . . . 9 (ℕ × ℕ) ≈ ℕ
105 domentr 8180 . . . . . . . . 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 710 . . . . . . . 8 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≼ ℕ
107 ensym 8170 . . . . . . . . . 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 1130 . . . . . . . . 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 4955 . . . . . . . . . 10 {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ∈ V
110 fveq2 6352 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑏 → (1st𝑎) = (1st𝑏))
111110oveq1d 6828 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑏 → ((1st𝑎)↑2) = ((1st𝑏)↑2))
112 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑏 → (2nd𝑎) = (2nd𝑏))
113112oveq1d 6828 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑏 → ((2nd𝑎)↑2) = ((2nd𝑏)↑2))
114113oveq2d 6829 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑏 → (𝐷 · ((2nd𝑎)↑2)) = (𝐷 · ((2nd𝑏)↑2)))
115111, 114oveq12d 6831 . . . . . . . . . . . . . . 15 (𝑎 = 𝑏 → (((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))))
116115eqeq1d 2762 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → ((((1st𝑎)↑2) − (𝐷 · ((2nd𝑎)↑2))) = 𝑥 ↔ (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = 𝑥))
117116elrab 3504 . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . . . 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 823 . . . . . . . . . . . . . . . . . . 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 6352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ⟨𝑦, 𝑧⟩ → (1st𝑏) = (1st ‘⟨𝑦, 𝑧⟩))
121120oveq1d 6828 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ⟨𝑦, 𝑧⟩ → ((1st𝑏)↑2) = ((1st ‘⟨𝑦, 𝑧⟩)↑2))
122 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = ⟨𝑦, 𝑧⟩ → (2nd𝑏) = (2nd ‘⟨𝑦, 𝑧⟩))
123122oveq1d 6828 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = ⟨𝑦, 𝑧⟩ → ((2nd𝑏)↑2) = ((2nd ‘⟨𝑦, 𝑧⟩)↑2))
124123oveq2d 6829 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = ⟨𝑦, 𝑧⟩ → (𝐷 · ((2nd𝑏)↑2)) = (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2)))
125121, 124oveq12d 6831 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = ⟨𝑦, 𝑧⟩ → (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))) = (((1st ‘⟨𝑦, 𝑧⟩)↑2) − (𝐷 · ((2nd ‘⟨𝑦, 𝑧⟩)↑2))))
126125, 19syl6req 2811 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = ⟨𝑦, 𝑧⟩ → ((𝑦↑2) − (𝐷 · (𝑧↑2))) = (((1st𝑏)↑2) − (𝐷 · ((2nd𝑏)↑2))))
127126ad2antrl 766 . . . . . . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . . . . 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 559 . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . 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 1997 . . . . . . . . . . . . . . . 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 5133 . . . . . . . . . . . . . . . 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 5133 . . . . . . . . . . . . . . . 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 285 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 3750 . . . . . . . . . . 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 1127 . . . . . . . . . 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 8167 . . . . . . . . . 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 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)))))} ∣ (((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 8179 . . . . . . . . 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 696 . . . . . . . 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 8245 . . . . . . . 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 698 . . . . . . 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 559 . . . . . 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 1113 . . . . 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 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)))))} ∣ (((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 446 . . 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 3152 . 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 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wne 2932  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  {csn 4321  cop 4327   class class class wbr 4804  {copab 4864   × cxp 5264  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  cen 8118  cdom 8119  Fincfn 8121  cr 10127  0cc0 10128  1c1 10129   + caddc 10131   · cmul 10133   < clt 10266  cle 10267  cmin 10458  -cneg 10459  cn 11212  2c2 11262  0cn0 11484  cz 11569  cq 11981  ...cfz 12519  cfl 12785  cexp 13054  csqrt 14172  abscabs 14173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-acn 8958  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-ico 12374  df-fz 12520  df-fl 12787  df-mod 12863  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-dvds 15183  df-gcd 15419  df-numer 15645  df-denom 15646
This theorem is referenced by:  pellex  37901
  Copyright terms: Public domain W3C validator