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Theorem pellex 38103
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 12986 . . . . . . . 8 (0...((abs‘𝑎) − 1)) ∈ Fin
2 xpfi 8442 . . . . . . . 8 (((0...((abs‘𝑎) − 1)) ∈ Fin ∧ (0...((abs‘𝑎) − 1)) ∈ Fin) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin)
31, 1, 2mp2an 683 . . . . . . 7 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4 isfinite 8768 . . . . . . 7 (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
53, 4mpbi 221 . . . . . 6 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6 nnenom 12994 . . . . . . 7 ℕ ≈ ω
76ensymi 8214 . . . . . 6 ω ≈ ℕ
8 sdomentr 8305 . . . . . 6 ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω ∧ ω ≈ ℕ) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ)
95, 7, 8mp2an 683 . . . . 5 ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10 ensym 8213 . . . . . 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 720 . . . . 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 8305 . . . . 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 581 . . . 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 5365 . . . . . . . 8 {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
1514sseli 3759 . . . . . . 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 11734 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (1st𝑑) ∈ ℤ)
18 simpllr 793 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19 simplr 785 . . . . . . . . . . . 12 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20 nnabscl 14366 . . . . . . . . . . . 12 ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
2118, 19, 20syl2anc 579 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22 zmodfz 12907 . . . . . . . . . . 11 (((1st𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2317, 21, 22syl2anc 579 . . . . . . . . . 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 11734 . . . . . . . . . . 11 (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ))) → (2nd𝑑) ∈ ℤ)
26 zmodfz 12907 . . . . . . . . . . 11 (((2nd𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
2725, 21, 26syl2anc 579 . . . . . . . . . 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 507 . . . . . . . . 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 401 . . . . . . . 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 7405 . . . . . . . 8 (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st𝑑) ∈ ℕ ∧ (2nd𝑑) ∈ ℕ)))
31 opelxp 5315 . . . . . . . 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 287 . . . . . . 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 395 . . . . 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 712 . . . 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 6379 . . . . . 6 (𝑑 = 𝑒 → (1st𝑑) = (1st𝑒))
3736oveq1d 6861 . . . . 5 (𝑑 = 𝑒 → ((1st𝑑) mod (abs‘𝑎)) = ((1st𝑒) mod (abs‘𝑎)))
38 fveq2 6379 . . . . . 6 (𝑑 = 𝑒 → (2nd𝑑) = (2nd𝑒))
3938oveq1d 6861 . . . . 5 (𝑑 = 𝑒 → ((2nd𝑑) mod (abs‘𝑎)) = ((2nd𝑒) mod (abs‘𝑎)))
4037, 39opeq12d 4569 . . . 4 (𝑑 = 𝑒 → ⟨((1st𝑑) mod (abs‘𝑎)), ((2nd𝑑) mod (abs‘𝑎))⟩ = ⟨((1st𝑒) mod (abs‘𝑎)), ((2nd𝑒) mod (abs‘𝑎))⟩)
4113, 35, 40fphpd 38084 . . 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 2827 . . . . . . . . . . . 12 (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43 eleq1w 2827 . . . . . . . . . . . 12 (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
4442, 43bi2anan9 629 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45 oveq1 6853 . . . . . . . . . . . . 13 (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46 oveq1 6853 . . . . . . . . . . . . . 14 (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
4746oveq2d 6862 . . . . . . . . . . . . 13 (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
4845, 47oveqan12d 6865 . . . . . . . . . . . 12 ((𝑏 = 𝑓𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
4948eqeq1d 2767 . . . . . . . . . . 11 ((𝑏 = 𝑓𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
5044, 49anbi12d 624 . . . . . . . . . 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 4883 . . . . . . . . 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 2836 . . . . . . . 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 207 . . . . . . 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 5146 . . . . . . . . 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 5146 . . . . . . . . . . . 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 1325 . . . . . . . . . . . . . . . . 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 1149 . . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . 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 1326 . . . . . . . . . . . . . . . . 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 1149 . . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . 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 803 . . . . . . . . . . . . . . . 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 1163 . . . . . . . . . . . . . . 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 1321 . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . 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 1322 . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . 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 1256 . . . . . . . . . . . . . . . 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 1319 . . . . . . . . . . . . . . . 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 1258 . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑𝑒)
76 simp2 1167 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77 simp1 1166 . . . . . . . . . . . . . . . . . 18 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
7875, 76, 773netr3d 3013 . . . . . . . . . . . . . . . . 17 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79 vex 3353 . . . . . . . . . . . . . . . . . . 19 𝑏 ∈ V
80 vex 3353 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
8179, 80opth 5102 . . . . . . . . . . . . . . . . . 18 (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓𝑐 = 𝑔))
8281necon3abii 2983 . . . . . . . . . . . . . . . . 17 (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8378, 82sylib 209 . . . . . . . . . . . . . . . 16 ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑𝑒) → ¬ (𝑏 = 𝑓𝑐 = 𝑔))
8472, 73, 74, 83syl3anc 1490 . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . 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 1320 . . . . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . . . 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 1324 . . . . . . . . . . . . . . 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 1259 . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . . . . 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 6878 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑑) mod (abs‘𝑎)) ∈ V
92 ovex 6878 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑑) mod (abs‘𝑎)) ∈ V
9391, 92opth 5102 . . . . . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . . . . . . 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 783 . . . . . . . . . . . . . . . . . . . . . . . . 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 6383 . . . . . . . . . . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
9997, 98syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . . . . . . . . . . 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 6383 . . . . . . . . . . . . . . . . . . . . . . . 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 3353 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑓 ∈ V
104 vex 3353 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑔 ∈ V
105103, 104op1st 7378 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106102, 105syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . . . . . . 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 2807 . . . . . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . . . . 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 6383 . . . . . . . . . . . . . . . . . . . . . . . 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 7379 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112110, 111syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . . . . . . 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 6383 . . . . . . . . . . . . . . . . . . . . . . . 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 7379 . . . . . . . . . . . . . . . . . . . . . . . 24 (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116114, 115syl6eq 2815 . . . . . . . . . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . . . . . . 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 2807 . . . . . . . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . . 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 1162 . . . . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . 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 38102 . . . . . . . . . . . . . 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 1148 . . . . . . . . . . . . 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 2029 . . . . . . . . . . . 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 233 . . . . . . . . . . 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 401 . . . . . . . . . 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 2029 . . . . . . . . 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 233 . . . . . . . 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 398 . . . . . . 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 599 . . . . . 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 3184 . . . . 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 395 . . . 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 712 . . 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 678 . 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 38101 . 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 3225 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 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wrex 3056  Vcvv 3350  cop 4342   class class class wbr 4811  {copab 4873   × cxp 5277  cfv 6070  (class class class)co 6846  ωcom 7267  1st c1st 7368  2nd c2nd 7369  cen 8161  csdm 8163  Fincfn 8164  0cc0 10193  1c1 10194   · cmul 10198  cmin 10525  cn 11279  2c2 11332  cz 11629  cq 11996  ...cfz 12540   mod cmo 12883  cexp 13074  csqrt 14274  abscabs 14275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-omul 7773  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-acn 9023  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10527  df-neg 10528  df-div 10944  df-nn 11280  df-2 11340  df-3 11341  df-n0 11544  df-xnn0 11616  df-z 11630  df-uz 11894  df-q 11997  df-rp 12036  df-ico 12390  df-fz 12541  df-fl 12808  df-mod 12884  df-seq 13016  df-exp 13075  df-hash 13329  df-cj 14140  df-re 14141  df-im 14142  df-sqrt 14276  df-abs 14277  df-dvds 15282  df-gcd 15514  df-numer 15738  df-denom 15739
This theorem is referenced by:  pellqrex  38147
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