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Theorem pellex 40195
 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 13403 . . . . . . . 8 (0...((abs‘𝑎) − 1)) ∈ Fin
2 xpfi 8836 . . . . . . . 8 (((0...((abs‘𝑎) − 1)) ∈ Fin ∧ (0...((abs‘𝑎) − 1)) ∈ Fin) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin)
31, 1, 2mp2an 691 . . . . . . 7 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4 isfinite 9162 . . . . . . 7 (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
53, 4mpbi 233 . . . . . 6 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6 nnenom 13411 . . . . . . 7 ℕ ≈ ω
76ensymi 8591 . . . . . 6 ω ≈ ℕ
8 sdomentr 8687 . . . . . 6 ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω ∧ ω ≈ ℕ) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ)
95, 7, 8mp2an 691 . . . . 5 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10 ensym 8590 . . . . . 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 728 . . . . 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 8687 . . . . 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 590 . . . 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 5618 . . . . . . . 8 {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
1514sseli 3891 . . . . . . 7 (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16 simprrl 780 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℕ)
1716nnzd 12139 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℤ)
18 simpllr 775 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19 simplr 768 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20 nnabscl 14747 . . . . . . . . . . . 12 ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
2118, 19, 20syl2anc 587 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22 zmodfz 13324 . . . . . . . . . . 11 (((1st𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2317, 21, 22syl2anc 587 . . . . . . . . . 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 781 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℕ)
2524nnzd 12139 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℤ)
26 zmodfz 13324 . . . . . . . . . . 11 (((2nd𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2725, 21, 26syl2anc 587 . . . . . . . . . 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 515 . . . . . . . . 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 416 . . . . . . . 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 7735 . . . . . . . 8 (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ)))
31 opelxp 5565 . . . . . . . 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 299 . . . . . . 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 410 . . . . 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 720 . . . 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 6664 . . . . . 6 (𝑑 = 𝑒 → (1st𝑑) = (1st𝑒))
3736oveq1d 7172 . . . . 5 (𝑑 = 𝑒 → ((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)))
38 fveq2 6664 . . . . . 6 (𝑑 = 𝑒 → (2nd𝑑) = (2nd𝑒))
3938oveq1d 7172 . . . . 5 (𝑑 = 𝑒 → ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))
4037, 39opeq12d 4775 . . . 4 (𝑑 = 𝑒 → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
4113, 35, 40fphpd 40176 . . 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 eleq1w 2835 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43 eleq1w 2835 . . . . . . . . . . . 12 (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
4442, 43bi2anan9 638 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45 oveq1 7164 . . . . . . . . . . . . 13 (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46 oveq1 7164 . . . . . . . . . . . . . 14 (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
4746oveq2d 7173 . . . . . . . . . . . . 13 (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
4845, 47oveqan12d 7176 . . . . . . . . . . . 12 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
4948eqeq1d 2761 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
5044, 49anbi12d 633 . . . . . . . . . 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 5109 . . . . . . . . 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 2844 . . . . . . . 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 219 . . . . . . 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 5389 . . . . . . . . 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 5389 . . . . . . . . . . . 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 1242 . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . 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 1243 . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . . . 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 1175 . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . 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 1239 . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . . . 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 1236 . . . . . . . . . . . . . . . 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 1199 . . . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑𝑒)
76 simp2 1135 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77 simp1 1134 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
7875, 76, 773netr3d 3028 . . . . . . . . . . . . . . . . 17 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79 vex 3414 . . . . . . . . . . . . . . . . . . 19 𝑏 ∈ V
80 vex 3414 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
8179, 80opth 5341 . . . . . . . . . . . . . . . . . 18 (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓𝑐 = 𝑔))
8281necon3abii 2998 . . . . . . . . . . . . . . . . 17 (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8378, 82sylib 221 . . . . . . . . . . . . . . . 16 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8472, 73, 74, 83syl3anc 1369 . . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . . . 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 1241 . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . . . . . . 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 7190 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑑) mod (abs‘𝑎)) ∈ V
92 ovex 7190 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑑) mod (abs‘𝑎)) ∈ V
9391, 92opth 5341 . . . . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . . . . . . . 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 7708 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
9997, 98eqtrdi 2810 . . . . . . . . . . . . . . . . . . . . . . 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 7172 . . . . . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . . . . . . . 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 3414 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ V
104 vex 3414 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
105103, 104op1st 7708 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106102, 105eqtrdi 2810 . . . . . . . . . . . . . . . . . . . . . . 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 7172 . . . . . . . . . . . . . . . . . . . . . 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 2802 . . . . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . . . . . . . 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 7709 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112110, 111eqtrdi 2810 . . . . . . . . . . . . . . . . . . . . . . 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 7172 . . . . . . . . . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . . . . . . . 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 7709 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116114, 115eqtrdi 2810 . . . . . . . . . . . . . . . . . . . . . . 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 7172 . . . . . . . . . . . . . . . . . . . . . 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 2802 . . . . . . . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . 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 40194 . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . 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 1936 . . . . . . . . . . . 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 245 . . . . . . . . . . 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 416 . . . . . . . . . 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 1936 . . . . . . . . 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 245 . . . . . . . 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 414 . . . . . . 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 608 . . . . . 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 3218 . . . . 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 410 . . . 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 720 . . 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 686 . 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 40193 . 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 3214 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 399   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112   ≠ wne 2952  ∃wrex 3072  Vcvv 3410  ⟨cop 4532   class class class wbr 5037  {copab 5099   × cxp 5527  ‘cfv 6341  (class class class)co 7157  ωcom 7586  1st c1st 7698  2nd c2nd 7699   ≈ cen 8538   ≺ csdm 8540  Fincfn 8541  0cc0 10589  1c1 10590   · cmul 10594   − cmin 10922  ℕcn 11688  2c2 11743  ℤcz 12034  ℚcq 12402  ...cfz 12953   mod cmo 13300  ↑cexp 13493  √csqrt 14654  abscabs 14655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466  ax-inf2 9151  ax-cnex 10645  ax-resscn 10646  ax-1cn 10647  ax-icn 10648  ax-addcl 10649  ax-addrcl 10650  ax-mulcl 10651  ax-mulrcl 10652  ax-mulcom 10653  ax-addass 10654  ax-mulass 10655  ax-distr 10656  ax-i2m1 10657  ax-1ne0 10658  ax-1rid 10659  ax-rnegex 10660  ax-rrecex 10661  ax-cnre 10662  ax-pre-lttri 10663  ax-pre-lttrn 10664  ax-pre-ltadd 10665  ax-pre-mulgt0 10666  ax-pre-sup 10667 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-se 5489  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-lim 6180  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-isom 6350  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-om 7587  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-rdg 8063  df-1o 8119  df-oadd 8123  df-omul 8124  df-er 8306  df-map 8425  df-en 8542  df-dom 8543  df-sdom 8544  df-fin 8545  df-sup 8953  df-inf 8954  df-oi 9021  df-card 9415  df-acn 9418  df-pnf 10729  df-mnf 10730  df-xr 10731  df-ltxr 10732  df-le 10733  df-sub 10924  df-neg 10925  df-div 11350  df-nn 11689  df-2 11751  df-3 11752  df-n0 11949  df-xnn0 12021  df-z 12035  df-uz 12297  df-q 12403  df-rp 12445  df-ico 12799  df-fz 12954  df-fl 13225  df-mod 13301  df-seq 13433  df-exp 13494  df-hash 13755  df-cj 14520  df-re 14521  df-im 14522  df-sqrt 14656  df-abs 14657  df-dvds 15670  df-gcd 15908  df-numer 16145  df-denom 16146 This theorem is referenced by:  pellqrex  40239
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