Theorem pellex 42227

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.

Step	Hyp	Ref	Expression
1		fzfi 13963	. . . . . . . 8 ⊢ (0...((abs‘𝑎) − 1)) ∈ Fin
2		xpfi 9335	. . . . . . . 8 ⊢ (((0...((abs‘𝑎) − 1)) ∈ Fin ∧ (0...((abs‘𝑎) − 1)) ∈ Fin) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin)
3	1, 1, 2	mp2an 691	. . . . . . 7 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4		isfinite 9669	. . . . . . 7 ⊢ (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
5	3, 4	mpbi 229	. . . . . 6 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6		nnenom 13971	. . . . . . 7 ⊢ ℕ ≈ ω
7	6	ensymi 9018	. . . . . 6 ⊢ ω ≈ ℕ
8		sdomentr 9129	. . . . . 6 ⊢ ((((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω ∧ ω ≈ ℕ) → ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ)
9	5, 7, 8	mp2an 691	. . . . 5 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10		ensym 9017	. . . . . 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)})
11	10	ad2antll 728	. . . . 5 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) → ℕ ≈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)})
12		sdomentr 9129	. . . . 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)})
13	9, 11, 12	sylancr 586	. . . 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 5764	. . . . . . . 8 ⊢ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
15	14	sseli 3974	. . . . . . 7 ⊢ (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16		simprrl 780	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (1st ‘𝑑) ∈ ℕ)
17	16	nnzd 12609	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (1st ‘𝑑) ∈ ℤ)
18		simpllr 775	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20		nnabscl 15298	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
21	18, 19, 20	syl2anc 583	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22		zmodfz 13884	. . . . . . . . . . 11 ⊢ (((1st ‘𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
23	17, 21, 22	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → ((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
24		simprrr 781	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (2nd ‘𝑑) ∈ ℕ)
25	24	nnzd 12609	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (2nd ‘𝑑) ∈ ℤ)
26		zmodfz 13884	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
27	25, 21, 26	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
28	23, 27	jca 511	. . . . . . . . 9 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1))))
29	28	ex 412	. . . . . . . 8 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ)) → (((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))))
30		elxp7 8022	. . . . . . . 8 ⊢ (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ)))
31		opelxp 5708	. . . . . . . 8 ⊢ (⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ↔ (((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1))))
32	29, 30, 31	3imtr4g 296	. . . . . . 7 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (𝑑 ∈ (ℕ × ℕ) → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1)))))
33	15, 32	syl5 34	. . . . . 6 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1)))))
34	33	imp 406	. . . . 5 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ 𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}) → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))))
35	34	adantlrr 720	. . . 4 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) ∧ 𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}) → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ ∈ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))))
36		fveq2 6891	. . . . . 6 ⊢ (𝑑 = 𝑒 → (1st ‘𝑑) = (1st ‘𝑒))
37	36	oveq1d 7429	. . . . 5 ⊢ (𝑑 = 𝑒 → ((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)))
38		fveq2 6891	. . . . . 6 ⊢ (𝑑 = 𝑒 → (2nd ‘𝑑) = (2nd ‘𝑒))
39	38	oveq1d 7429	. . . . 5 ⊢ (𝑑 = 𝑒 → ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))
40	37, 39	opeq12d 4877	. . . 4 ⊢ (𝑑 = 𝑒 → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)
41	13, 35, 40	fphpd 42208	. . 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)} (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩))
42		eleq1w 2811	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43		eleq1w 2811	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
44	42, 43	bi2anan9 637	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45		oveq1 7421	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46		oveq1 7421	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
47	46	oveq2d 7430	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
48	45, 47	oveqan12d 7433	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
49	48	eqeq1d 2729	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
50	44, 49	anbi12d 630	. . . . . . . . . 10 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → (((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎) ↔ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)))
51	50	cbvopabv 5215	. . . . . . . . 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)}
52	51	eleq2i 2820	. . . . . . . 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)})
53	52	biimpi 215	. . . . . . 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 5523	. . . . . . . . 9 ⊢ (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ↔ ∃𝑏∃𝑐(𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)))
55		elopab 5523	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} ↔ ∃𝑓∃𝑔(𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)))
56		simp3ll 1242	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → 𝑏 ∈ ℕ)
57	56	3expb 1118	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝑏 ∈ ℕ)
58	57	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑏 ∈ ℕ)
59		simp3lr 1243	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → 𝑐 ∈ ℕ)
60	59	3expb 1118	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝑐 ∈ ℕ)
61	60	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑐 ∈ ℕ)
62		simp1lr 1235	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑎 ∈ ℤ)
63	62	3adant1r 1175	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑎 ∈ ℤ)
64		simp-4l 782	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝐷 ∈ ℕ)
65	64	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝐷 ∈ ℕ)
66		simp-4r 783	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → ¬ (√‘𝐷) ∈ ℚ)
67	66	3ad2ant1 1131	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ¬ (√‘𝐷) ∈ ℚ)
68		simp2ll 1238	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑓 ∈ ℕ)
69	68	3adant2l 1176	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑓 ∈ ℕ)
70		simp2lr 1239	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑔 ∈ ℕ)
71	70	3adant2l 1176	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑔 ∈ ℕ)
72		simp2l 1197	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑒 = ⟨𝑓, 𝑔⟩)
73		simp1rl 1236	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑑 = ⟨𝑏, 𝑐⟩)
74		simp3l 1199	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑑 ≠ 𝑒)
75		simp3 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑑 ≠ 𝑒)
76		simp2 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77		simp1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
78	75, 76, 77	3netr3d 3012	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79		vex 3473	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑏 ∈ V
80		vex 3473	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑐 ∈ V
81	79, 80	opth 5472	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
82	81	necon3abii 2982	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
83	78, 82	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
84	72, 73, 74, 83	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
85		simp1lr 1235	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → 𝑎 ≠ 0)
86		simp1rr 1237	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)
87	86	3adant1l 1174	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)
88		simp2rr 1241	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)
89		simp3r 1200	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)
90		simp3 1136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)
91		ovex 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑑) mod (abs‘𝑎)) ∈ V
92		ovex 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ V
93	91, 92	opth 5472	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩ ↔ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎))))
94	90, 93	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎))))
95		simprl 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)))
96		simpll 766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → 𝑑 = ⟨𝑏, 𝑐⟩)
97	96	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (1st ‘𝑑) = (1st ‘⟨𝑏, 𝑐⟩))
98	79, 80	op1st 7995	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
99	97, 98	eqtrdi 2783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (1st ‘𝑑) = 𝑏)
100	99	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((1st ‘𝑑) mod (abs‘𝑎)) = (𝑏 mod (abs‘𝑎)))
101		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → 𝑒 = ⟨𝑓, 𝑔⟩)
102	101	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (1st ‘𝑒) = (1st ‘⟨𝑓, 𝑔⟩))
103		vex 3473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
104		vex 3473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
105	103, 104	op1st 7995	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106	102, 105	eqtrdi 2783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (1st ‘𝑒) = 𝑓)
107	106	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((1st ‘𝑒) mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
108	95, 100, 107	3eqtr3d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
109		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))
110	96	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (2nd ‘𝑑) = (2nd ‘⟨𝑏, 𝑐⟩))
111	79, 80	op2nd 7996	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112	110, 111	eqtrdi 2783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (2nd ‘𝑑) = 𝑐)
113	112	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((2nd ‘𝑑) mod (abs‘𝑎)) = (𝑐 mod (abs‘𝑎)))
114	101	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (2nd ‘𝑒) = (2nd ‘⟨𝑓, 𝑔⟩))
115	103, 104	op2nd 7996	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116	114, 115	eqtrdi 2783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (2nd ‘𝑒) = 𝑔)
117	116	oveq1d 7429	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((2nd ‘𝑒) mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))
118	109, 113, 117	3eqtr3d 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))
119	108, 118	jca 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) ∧ (((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎))))
120	119	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩) → ((((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎))) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))))
121	120	3adant3 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ((((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)) ∧ ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎))) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))))
122	94, 121	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑒 = ⟨𝑓, 𝑔⟩ ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎))))
123	73, 72, 89, 122	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ((𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)) ∧ (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎))))
124	123	simpld 494	. . . . . . . . . . . . . . 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)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → (𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
125	123	simprd 495	. . . . . . . . . . . . . . 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)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → (𝑐 mod (abs‘𝑎)) = (𝑔 mod (abs‘𝑎)))
126	58, 61, 63, 65, 67, 69, 71, 84, 85, 87, 88, 124, 125	pellexlem6 42226	. . . . . . . . . . . . . 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)) ∧ (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
127	126	3exp 1117	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
128	127	exlimdvv 1930	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → (∃𝑓∃𝑔(𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
129	55, 128	biimtrid 241	. . . . . . . . . . 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)} → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
130	129	ex 412	. . . . . . . . . 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)} → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))))
131	130	exlimdvv 1930	. . . . . . . . 9 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → (∃𝑏∃𝑐(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)} → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))))
132	54, 131	biimtrid 241	. . . . . . . 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)} → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))))
133	132	impd 410	. . . . . . 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)}) → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
134	53, 133	sylan2i 605	. . . . . 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)}) → ((𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)))
135	134	rexlimdvv 3205	. . . . 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)} (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1))
136	135	imp 406	. . . 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)} (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
137	136	adantlrr 720	. . 3 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) ∧ ∃𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}∃𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} (𝑑 ≠ 𝑒 ∧ ⟨((1st ‘𝑑) mod (abs‘𝑎)), ((2nd ‘𝑑) mod (abs‘𝑎))⟩ = ⟨((1st ‘𝑒) mod (abs‘𝑎)), ((2nd ‘𝑒) mod (abs‘𝑎))⟩)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
138	41, 137	mpdan 686	. 2 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ)) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)
139		pellexlem5 42225	. 2 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑎 ∈ ℤ (𝑎 ≠ 0 ∧ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ≈ ℕ))
140	138, 139	r19.29a 3157	1 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2935  ∃wrex 3065  Vcvv 3469  ⟨cop 4630   class class class wbr 5142  {copab 5204   × cxp 5670  ‘cfv 6542  (class class class)co 7414  ωcom 7864  1^st c1st 7985  2^nd c2nd 7986   ≈ cen 8954   ≺ csdm 8956  Fincfn 8957  0cc0 11132  1c1 11133   · cmul 11137   − cmin 11468  ℕcn 12236  2c2 12291  ℤcz 12582  ℚcq 12956  ...cfz 13510   mod cmo 13860  ↑cexp 14052  √csqrt 15206  abscabs 15207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-inf 9460  df-oi 9527  df-card 9956  df-acn 9959  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-2 12299  df-3 12300  df-n0 12497  df-xnn0 12569  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-ico 13356  df-fz 13511  df-fl 13783  df-mod 13861  df-seq 13993  df-exp 14053  df-hash 14316  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-dvds 16225  df-gcd 16463  df-numer 16700  df-denom 16701

This theorem is referenced by:  pellqrex  42271