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Theorem pellex 36216
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 12584 . . . . . . . 8 (0...((abs‘𝑎) − 1)) ∈ Fin
2 xpfi 8089 . . . . . . . 8 (((0...((abs‘𝑎) − 1)) ∈ Fin ∧ (0...((abs‘𝑎) − 1)) ∈ Fin) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin)
31, 1, 2mp2an 703 . . . . . . 7 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4 isfinite 8405 . . . . . . 7 (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
53, 4mpbi 218 . . . . . 6 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6 nnenom 12592 . . . . . . 7 ℕ ≈ ω
76ensymi 7865 . . . . . 6 ω ≈ ℕ
8 sdomentr 7952 . . . . . 6 ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω ∧ ω ≈ ℕ) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ)
95, 7, 8mp2an 703 . . . . 5 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10 ensym 7864 . . . . . 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 760 . . . . 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 7952 . . . . 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 693 . . . 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 5102 . . . . . . . 8 {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
1514sseli 3559 . . . . . . 7 (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16 simprrl 799 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℕ)
1716nnzd 11309 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℤ)
18 simpllr 794 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19 simplr 787 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20 nnabscl 13855 . . . . . . . . . . . 12 ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
2118, 19, 20syl2anc 690 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22 zmodfz 12505 . . . . . . . . . . 11 (((1st𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2317, 21, 22syl2anc 690 . . . . . . . . . 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 800 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℕ)
2524nnzd 11309 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℤ)
26 zmodfz 12505 . . . . . . . . . . 11 (((2nd𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2725, 21, 26syl2anc 690 . . . . . . . . . 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 552 . . . . . . . . 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 448 . . . . . . . 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 7065 . . . . . . . 8 (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ)))
31 opelxp 5056 . . . . . . . 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 283 . . . . . . 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 33 . . . . . 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 443 . . . . 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 752 . . . 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 6084 . . . . . 6 (𝑑 = 𝑒 → (1st𝑑) = (1st𝑒))
3736oveq1d 6538 . . . . 5 (𝑑 = 𝑒 → ((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)))
38 fveq2 6084 . . . . . 6 (𝑑 = 𝑒 → (2nd𝑑) = (2nd𝑒))
3938oveq1d 6538 . . . . 5 (𝑑 = 𝑒 → ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))
4037, 39opeq12d 4338 . . . 4 (𝑑 = 𝑒 → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
4113, 35, 40fphpd 36197 . . 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 2671 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43 eleq1 2671 . . . . . . . . . . . 12 (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
4442, 43bi2anan9 912 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45 oveq1 6530 . . . . . . . . . . . . 13 (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46 oveq1 6530 . . . . . . . . . . . . . 14 (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
4746oveq2d 6539 . . . . . . . . . . . . 13 (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
4845, 47oveqan12d 6542 . . . . . . . . . . . 12 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
4948eqeq1d 2607 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
5044, 49anbi12d 742 . . . . . . . . . 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 4644 . . . . . . . . 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 2675 . . . . . . . 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 204 . . . . . . 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 4894 . . . . . . . . 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 4894 . . . . . . . . . . . 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 1124 . . . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . . . . 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 1310 . . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . . . . 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 1311 . . . . . . . . . . . . . . 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 1121 . . . . . . . . . . . . . . . 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 1311 . . . . . . . . . . . . . . 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 1079 . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . 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 1081 . . . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑𝑒)
76 simp2 1054 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77 simp1 1053 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
7875, 76, 773netr3d 2853 . . . . . . . . . . . . . . . . 17 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79 vex 3171 . . . . . . . . . . . . . . . . . . 19 𝑏 ∈ V
80 vex 3171 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
8179, 80opth 4861 . . . . . . . . . . . . . . . . . 18 (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓𝑐 = 𝑔))
8281necon3abii 2823 . . . . . . . . . . . . . . . . 17 (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8378, 82sylib 206 . . . . . . . . . . . . . . . 16 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8472, 73, 74, 83syl3anc 1317 . . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . . 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 1309 . . . . . . . . . . . . . . 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 1123 . . . . . . . . . . . . . . 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 1082 . . . . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . . . . . 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 6551 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑑) mod (abs‘𝑎)) ∈ V
92 ovex 6551 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑑) mod (abs‘𝑎)) ∈ V
9391, 92opth 4861 . . . . . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . . . . . . . . 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 7040 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
9997, 98syl6eq 2655 . . . . . . . . . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ V
104 vex 3171 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
105103, 104op1st 7040 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106102, 105syl6eq 2655 . . . . . . . . . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . . . . . . 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 2647 . . . . . . . . . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . . . . . . . . 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 7041 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112110, 111syl6eq 2655 . . . . . . . . . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . . . . . . . . . . . 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 7041 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116114, 115syl6eq 2655 . . . . . . . . . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . . . . . . . . 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 2647 . . . . . . . . . . . . . . . . . . . . 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 552 . . . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 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 36215 . . . . . . . . . . . . . 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 1255 . . . . . . . . . . . . 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 1847 . . . . . . . . . . . 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 230 . . . . . . . . . . 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 448 . . . . . . . . . 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 1847 . . . . . . . . 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 230 . . . . . . . 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 445 . . . . . . 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 684 . . . . . 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 3014 . . . . 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 443 . . . 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 752 . . 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 698 . 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 36214 . 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 3055 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 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wne 2775  wrex 2892  Vcvv 3168  cop 4126   class class class wbr 4573  {copab 4632   × cxp 5022  cfv 5786  (class class class)co 6523  ωcom 6930  1st c1st 7030  2nd c2nd 7031  cen 7811  csdm 7813  Fincfn 7814  0cc0 9788  1c1 9789   · cmul 9793  cmin 10113  cn 10863  2c2 10913  cz 11206  cq 11616  ...cfz 12148   mod cmo 12481  cexp 12673  csqrt 13763  abscabs 13764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-omul 7425  df-er 7602  df-map 7719  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-oi 8271  df-card 8621  df-acn 8624  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-q 11617  df-rp 11661  df-ico 12004  df-fz 12149  df-fl 12406  df-mod 12482  df-seq 12615  df-exp 12674  df-hash 12931  df-cj 13629  df-re 13630  df-im 13631  df-sqrt 13765  df-abs 13766  df-dvds 14764  df-gcd 14997  df-numer 15223  df-denom 15224
This theorem is referenced by:  pellqrex  36260
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