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Theorem pellex 37925
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 12979 . . . . . . . 8 (0...((abs‘𝑎) − 1)) ∈ Fin
2 xpfi 8387 . . . . . . . 8 (((0...((abs‘𝑎) − 1)) ∈ Fin ∧ (0...((abs‘𝑎) − 1)) ∈ Fin) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin)
31, 1, 2mp2an 672 . . . . . . 7 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4 isfinite 8713 . . . . . . 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 12987 . . . . . . 7 ℕ ≈ ω
76ensymi 8159 . . . . . 6 ω ≈ ℕ
8 sdomentr 8250 . . . . . 6 ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω ∧ ω ≈ ℕ) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ)
95, 7, 8mp2an 672 . . . . 5 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10 ensym 8158 . . . . . 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 708 . . . . 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 8250 . . . . 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 575 . . . 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 5333 . . . . . . . 8 {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
1514sseli 3748 . . . . . . 7 (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16 simprrl 766 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℕ)
1716nnzd 11683 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℤ)
18 simpllr 760 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19 simplr 752 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20 nnabscl 14273 . . . . . . . . . . . 12 ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
2118, 19, 20syl2anc 573 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22 zmodfz 12900 . . . . . . . . . . 11 (((1st𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2317, 21, 22syl2anc 573 . . . . . . . . . 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 767 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℕ)
2524nnzd 11683 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℤ)
26 zmodfz 12900 . . . . . . . . . . 11 (((2nd𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2725, 21, 26syl2anc 573 . . . . . . . . . 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 501 . . . . . . . . 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 397 . . . . . . . 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 7350 . . . . . . . 8 (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ)))
31 opelxp 5286 . . . . . . . 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 393 . . . . 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 700 . . . 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 6332 . . . . . 6 (𝑑 = 𝑒 → (1st𝑑) = (1st𝑒))
3736oveq1d 6808 . . . . 5 (𝑑 = 𝑒 → ((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)))
38 fveq2 6332 . . . . . 6 (𝑑 = 𝑒 → (2nd𝑑) = (2nd𝑒))
3938oveq1d 6808 . . . . 5 (𝑑 = 𝑒 → ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))
4037, 39opeq12d 4547 . . . 4 (𝑑 = 𝑒 → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
4113, 35, 40fphpd 37906 . . 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 2833 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43 eleq1w 2833 . . . . . . . . . . . 12 (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
4442, 43bi2anan9 620 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45 oveq1 6800 . . . . . . . . . . . . 13 (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46 oveq1 6800 . . . . . . . . . . . . . 14 (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
4746oveq2d 6809 . . . . . . . . . . . . 13 (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
4845, 47oveqan12d 6812 . . . . . . . . . . . 12 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
4948eqeq1d 2773 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
5044, 49anbi12d 616 . . . . . . . . . 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 4856 . . . . . . . . 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 2842 . . . . . . . 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 5116 . . . . . . . . 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 5116 . . . . . . . . . . . 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 1310 . . . . . . . . . . . . . . . . 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 1113 . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . 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 1311 . . . . . . . . . . . . . . . . 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 1113 . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . 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 1303 . . . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . 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 1307 . . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . . 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 1241 . . . . . . . . . . . . . . . 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 1304 . . . . . . . . . . . . . . . 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 1243 . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑𝑒)
76 simp2 1131 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77 simp1 1130 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
7875, 76, 773netr3d 3019 . . . . . . . . . . . . . . . . 17 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79 vex 3354 . . . . . . . . . . . . . . . . . . 19 𝑏 ∈ V
80 vex 3354 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
8179, 80opth 5072 . . . . . . . . . . . . . . . . . 18 (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓𝑐 = 𝑔))
8281necon3abii 2989 . . . . . . . . . . . . . . . . 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 1476 . . . . . . . . . . . . . . 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 1303 . . . . . . . . . . . . . . 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 1305 . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . 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 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))⟩)) → ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)
89 simp3r 1244 . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . 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 6823 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑑) mod (abs‘𝑎)) ∈ V
92 ovex 6823 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑑) mod (abs‘𝑎)) ∈ V
9391, 92opth 5072 . . . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . . . . . . . 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 7323 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
9997, 98syl6eq 2821 . . . . . . . . . . . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . . . . . . . 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 3354 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ V
104 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
105103, 104op1st 7323 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106102, 105syl6eq 2821 . . . . . . . . . . . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . . . . . . . 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 7324 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112110, 111syl6eq 2821 . . . . . . . . . . . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . . . . . . . . . . . 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 7324 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116114, 115syl6eq 2821 . . . . . . . . . . . . . . . . . . . . . . 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 6808 . . . . . . . . . . . . . . . . . . . . . 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 2813 . . . . . . . . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . . 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 1126 . . . . . . . . . . . . . . . . . 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 1476 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 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 37924 . . . . . . . . . . . . . 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 1112 . . . . . . . . . . . . 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 2014 . . . . . . . . . . . 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 397 . . . . . . . . . 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 2014 . . . . . . . . 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 396 . . . . . . 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 593 . . . . . 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 3185 . . . . 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 393 . . . 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 700 . . 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 667 . 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 37923 . 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 3226 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 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  wrex 3062  Vcvv 3351  cop 4322   class class class wbr 4786  {copab 4846   × cxp 5247  cfv 6031  (class class class)co 6793  ωcom 7212  1st c1st 7313  2nd c2nd 7314  cen 8106  csdm 8108  Fincfn 8109  0cc0 10138  1c1 10139   · cmul 10143  cmin 10468  cn 11222  2c2 11272  cz 11579  cq 11991  ...cfz 12533   mod cmo 12876  cexp 13067  csqrt 14181  abscabs 14182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-acn 8968  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-q 11992  df-rp 12036  df-ico 12386  df-fz 12534  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-gcd 15425  df-numer 15650  df-denom 15651
This theorem is referenced by:  pellqrex  37969
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