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Theorem pellex 37716
Description: Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)
Assertion
Ref Expression
pellex ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
Distinct variable group:   𝑥,𝐷,𝑦

Proof of Theorem pellex
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfi 12811 . . . . . . . 8 (0...((abs‘𝑎) − 1)) ∈ Fin
2 xpfi 8272 . . . . . . . 8 (((0...((abs‘𝑎) − 1)) ∈ Fin ∧ (0...((abs‘𝑎) − 1)) ∈ Fin) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin)
31, 1, 2mp2an 708 . . . . . . 7 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4 isfinite 8587 . . . . . . 7 (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
53, 4mpbi 220 . . . . . 6 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6 nnenom 12819 . . . . . . 7 ℕ ≈ ω
76ensymi 8047 . . . . . 6 ω ≈ ℕ
8 sdomentr 8135 . . . . . 6 ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω ∧ ω ≈ ℕ) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ)
95, 7, 8mp2an 708 . . . . 5 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10 ensym 8046 . . . . . 6 ({⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ → ℕ ≈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)})
1110ad2antll 765 . . . . 5 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) → ℕ ≈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)})
12 sdomentr 8135 . . . . 5 ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ ∧ ℕ ≈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)})
139, 11, 12sylancr 696 . . . 4 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)})
14 opabssxp 5227 . . . . . . . 8 {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
1514sseli 3632 . . . . . . 7 (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16 simprrl 821 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℕ)
1716nnzd 11519 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℤ)
18 simpllr 815 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19 simplr 807 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20 nnabscl 14109 . . . . . . . . . . . 12 ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
2118, 19, 20syl2anc 694 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22 zmodfz 12732 . . . . . . . . . . 11 (((1st𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2317, 21, 22syl2anc 694 . . . . . . . . . 10 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → ((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
24 simprrr 822 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℕ)
2524nnzd 11519 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℤ)
26 zmodfz 12732 . . . . . . . . . . 11 (((2nd𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2725, 21, 26syl2anc 694 . . . . . . . . . 10 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2823, 27jca 553 . . . . . . . . 9 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)) ∧ ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1))))
2928ex 449 . . . . . . . 8 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ)) → (((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)) ∧ ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))))
30 elxp7 7245 . . . . . . . 8 (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ)))
31 opelxp 5180 . . . . . . . 8 (⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ↔ (((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)) ∧ ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1))))
3229, 30, 313imtr4g 285 . . . . . . 7 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (𝑑 ∈ (ℕ × ℕ) → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1)))))
3315, 32syl5 34 . . . . . 6 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1)))))
3433imp 444 . . . . 5 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ 𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}) → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))))
3534adantlrr 757 . . . 4 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) ∧ 𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}) → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))))
36 fveq2 6229 . . . . . 6 (𝑑 = 𝑒 → (1st𝑑) = (1st𝑒))
3736oveq1d 6705 . . . . 5 (𝑑 = 𝑒 → ((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)))
38 fveq2 6229 . . . . . 6 (𝑑 = 𝑒 → (2nd𝑑) = (2nd𝑒))
3938oveq1d 6705 . . . . 5 (𝑑 = 𝑒 → ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))
4037, 39opeq12d 4441 . . . 4 (𝑑 = 𝑒 → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
4113, 35, 40fphpd 37697 . . 3 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) → ∃𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}∃𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩))
42 eleq1 2718 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43 eleq1 2718 . . . . . . . . . . . 12 (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
4442, 43bi2anan9 935 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45 oveq1 6697 . . . . . . . . . . . . 13 (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46 oveq1 6697 . . . . . . . . . . . . . 14 (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
4746oveq2d 6706 . . . . . . . . . . . . 13 (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
4845, 47oveqan12d 6709 . . . . . . . . . . . 12 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
4948eqeq1d 2653 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
5044, 49anbi12d 747 . . . . . . . . . 10 ((𝑏 = 𝑓𝑐 = 𝑔) → (((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎) ↔ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)))
5150cbvopabv 4755 . . . . . . . . 9 {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)}
5251eleq2i 2722 . . . . . . . 8 (𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ↔ 𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)})
5352biimpi 206 . . . . . . 7 (𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)})
54 elopab 5012 . . . . . . . . 9 (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ↔ ∃𝑏𝑐(𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)))
55 elopab 5012 . . . . . . . . . . . 12 (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} ↔ ∃𝑓𝑔(𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)))
56 simp3ll 1152 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → 𝑏 ∈ ℕ)
57563expb 1285 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝑏 ∈ ℕ)
58573ad2ant1 1102 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑏 ∈ ℕ)
59 simp3lr 1153 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → 𝑐 ∈ ℕ)
60593expb 1285 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝑐 ∈ ℕ)
61603ad2ant1 1102 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑐 ∈ ℕ)
62 simp1lr 1145 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑎 ∈ ℤ)
63623adant1r 1359 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑎 ∈ ℤ)
64 simp-4l 823 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝐷 ∈ ℕ)
65643ad2ant1 1102 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝐷 ∈ ℕ)
66 simp-4r 824 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → ¬ (√‘𝐷) ∈ ℚ)
67663ad2ant1 1102 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ¬ (√‘𝐷) ∈ ℚ)
68 simp2ll 1148 . . . . . . . . . . . . . . . 16 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑓 ∈ ℕ)
69683adant2l 1360 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑓 ∈ ℕ)
70 simp2lr 1149 . . . . . . . . . . . . . . . 16 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑔 ∈ ℕ)
71703adant2l 1360 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑔 ∈ ℕ)
72 simp2l 1107 . . . . . . . . . . . . . . . 16 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑒 = ⟨𝑓, 𝑔⟩)
73 simp1rl 1146 . . . . . . . . . . . . . . . 16 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑑 = ⟨𝑏, 𝑐⟩)
74 simp3l 1109 . . . . . . . . . . . . . . . 16 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑑𝑒)
75 simp3 1083 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑𝑒)
76 simp2 1082 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77 simp1 1081 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
7875, 76, 773netr3d 2899 . . . . . . . . . . . . . . . . 17 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑏 ∈ V
80 vex 3234 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
8179, 80opth 4974 . . . . . . . . . . . . . . . . . 18 (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓𝑐 = 𝑔))
8281necon3abii 2869 . . . . . . . . . . . . . . . . 17 (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8378, 82sylib 208 . . . . . . . . . . . . . . . 16 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8472, 73, 74, 83syl3anc 1366 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ¬ (𝑏 = 𝑓𝑐 = 𝑔))
85 simp1lr 1145 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → 𝑎 ≠ 0)
86 simp1rr 1147 . . . . . . . . . . . . . . . 16 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)
87863adant1l 1358 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)
88 simp2rr 1151 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)
89 simp3r 1110 . . . . . . . . . . . . . . . . 17 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
90 simp3 1083 . . . . . . . . . . . . . . . . . . 19 ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
91 ovex 6718 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑑) mod (abs‘𝑎)) ∈ V
92 ovex 6718 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑑) mod (abs‘𝑎)) ∈ V
9391, 92opth 4974 . . . . . . . . . . . . . . . . . . 19 (⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩ ↔ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎))))
9490, 93sylib 208 . . . . . . . . . . . . . . . . . 18 ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎))))
95 simprl 809 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)))
96 simpll 805 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → 𝑑 = ⟨𝑏, 𝑐⟩)
9796fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (1st𝑑) = (1st ‘⟨𝑏, 𝑐⟩))
9879, 80op1st 7218 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
9997, 98syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (1st𝑑) = 𝑏)
10099oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((1st𝑑) mod (abs‘𝑎)) = (𝑏 mod (abs‘𝑎)))
101 simplr 807 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → 𝑒 = ⟨𝑓, 𝑔⟩)
102101fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (1st𝑒) = (1st ‘⟨𝑓, 𝑔⟩))
103 vex 3234 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ V
104 vex 3234 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
105103, 104op1st 7218 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106102, 105syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (1st𝑒) = 𝑓)
107106oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((1st𝑒) mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
10895, 100, 1073eqtr3d 2693 . . . . . . . . . . . . . . . . . . . . 21 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
109 simprr 811 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))
11096fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (2nd𝑑) = (2nd ‘⟨𝑏, 𝑐⟩))
11179, 80op2nd 7219 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112110, 111syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (2nd𝑑) = 𝑐)
113112oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((2nd𝑑) mod (abs‘𝑎)) = (𝑐 mod (abs‘𝑎)))
114101fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (2nd𝑒) = (2nd ‘⟨𝑓, 𝑔⟩))
115103, 104op2nd 7219 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116114, 115syl6eq 2701 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (2nd𝑒) = 𝑔)
117116oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((2nd𝑒) mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))
118109, 113, 1173eqtr3d 2693 . . . . . . . . . . . . . . . . . . . . 21 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))
119108, 118jca 553 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎))))
120119ex 449 . . . . . . . . . . . . . . . . . . 19 ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) → ((((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎))) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))))
1211203adant3 1101 . . . . . . . . . . . . . . . . . 18 ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ((((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)) ∧ ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎))) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))))
12294, 121mpd 15 . . . . . . . . . . . . . . . . 17 ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎))))
12373, 72, 89, 122syl3anc 1366 . . . . . . . . . . . . . . . 16 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎))))
124123simpld 474 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → (𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
125123simprd 478 . . . . . . . . . . . . . . 15 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))
12658, 61, 63, 65, 67, 69, 71, 84, 85, 87, 88, 124, 125pellexlem6 37715 . . . . . . . . . . . . . 14 ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
1271263exp 1283 . . . . . . . . . . . . 13 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
128127exlimdvv 1902 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → (∃𝑓𝑔(𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
12955, 128syl5bi 232 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
130129ex 449 . . . . . . . . . 10 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))))
131130exlimdvv 1902 . . . . . . . . 9 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (∃𝑏𝑐(𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))))
13254, 131syl5bi 232 . . . . . . . 8 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))))
133132impd 446 . . . . . . 7 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ∧ 𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)}) → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
13453, 133sylan2i 688 . . . . . 6 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ∧ 𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}) → ((𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
135134rexlimdvv 3066 . . . . 5 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (∃𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}∃𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))
136135imp 444 . . . 4 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ ∃𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}∃𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
137136adantlrr 757 . . 3 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) ∧ ∃𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}∃𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} (𝑑𝑒 ∧ ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
13841, 137mpdan 703 . 2 ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
139 pellexlem5 37714 . 2 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑎 ∈ ℤ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ))
140138, 139r19.29a 3107 1 ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  cop 4216   class class class wbr 4685  {copab 4745   × cxp 5141  cfv 5926  (class class class)co 6690  ωcom 7107  1st c1st 7208  2nd c2nd 7209  cen 7994  csdm 7996  Fincfn 7997  0cc0 9974  1c1 9975   · cmul 9979  cmin 10304  cn 11058  2c2 11108  cz 11415  cq 11826  ...cfz 12364   mod cmo 12708  cexp 12900  csqrt 14017  abscabs 14018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ico 12219  df-fz 12365  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-gcd 15264  df-numer 15490  df-denom 15491
This theorem is referenced by:  pellqrex  37760
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