Theorem pellex 43417

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 13987	. . . . . . . 8 ⊢ (0...((abs‘𝑎) − 1)) ∈ Fin
2		xpfi 9266	. . . . . . . 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 702	. . . . . . 7 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4		isfinite 9609	. . . . . . 7 ⊢ (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
5	3, 4	mpbi 232	. . . . . 6 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6		nnenom 13995	. . . . . . 7 ⊢ ℕ ≈ ω
7	6	ensymi 8987	. . . . . 6 ⊢ ω ≈ ℕ
8		sdomentr 9085	. . . . . 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 702	. . . . 5 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10		ensym 8986	. . . . . 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 739	. . . . 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 9085	. . . . 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 596	. . . 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 5741	. . . . . . . 8 ⊢ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
15	14	sseli 3934	. . . . . . 7 ⊢ (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16		simprrl 790	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (1st ‘𝑑) ∈ ℕ)
17	16	nnzd 12596	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (1st ‘𝑑) ∈ ℤ)
18		simpllr 785	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19		simplr 778	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20		nnabscl 15355	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
21	18, 19, 20	syl2anc 593	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22		zmodfz 13905	. . . . . . . . . . 11 ⊢ (((1st ‘𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
23	17, 21, 22	syl2anc 593	. . . . . . . . . 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 791	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (2nd ‘𝑑) ∈ ℕ)
25	24	nnzd 12596	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (2nd ‘𝑑) ∈ ℤ)
26		zmodfz 13905	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
27	25, 21, 26	syl2anc 593	. . . . . . . . . 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 519	. . . . . . . . 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 416	. . . . . . . 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 8007	. . . . . . . 8 ⊢ (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ)))
31		opelxp 5685	. . . . . . . 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 298	. . . . . . 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 410	. . . . 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 731	. . . 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 6869	. . . . . 6 ⊢ (𝑑 = 𝑒 → (1st ‘𝑑) = (1st ‘𝑒))
37	36	oveq1d 7413	. . . . 5 ⊢ (𝑑 = 𝑒 → ((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)))
38		fveq2 6869	. . . . . 6 ⊢ (𝑑 = 𝑒 → (2nd ‘𝑑) = (2nd ‘𝑒))
39	38	oveq1d 7413	. . . . 5 ⊢ (𝑑 = 𝑒 → ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))
40	37, 39	opeq12d 4841	. . . 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 43398	. . 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 2847	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43		eleq1w 2847	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
44	42, 43	bi2anan9 647	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45		oveq1 7405	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46		oveq1 7405	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
47	46	oveq2d 7414	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
48	45, 47	oveqan12d 7417	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
49	48	eqeq1d 2766	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → (((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎 ↔ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎))
50	44, 49	anbi12d 641	. . . . . . . . . 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 5175	. . . . . . . . 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 2856	. . . . . . . 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 218	. . . . . . 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 5499	. . . . . . . . 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 5499	. . . . . . . . . . . 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 1259	. . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . 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 1260	. . . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . 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 1147	. . . . . . . . . . . . . . 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 1255	. . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . 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 1256	. . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . 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 1214	. . . . . . . . . . . . . . . 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 1253	. . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑑 ≠ 𝑒)
76		simp2 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77		simp1 1150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
78	75, 76, 77	3netr3d 3035	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79		vex 3460	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑏 ∈ V
80		vex 3460	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑐 ∈ V
81	79, 80	opth 5446	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
82	81	necon3abii 3005	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
83	78, 82	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
84	72, 73, 74, 83	syl3anc 1392	. . . . . . . . . . . . . . 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 1252	. . . . . . . . . . . . . . 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 1254	. . . . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . 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 1258	. . . . . . . . . . . . . . 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 1217	. . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑑) mod (abs‘𝑎)) ∈ V
92		ovex 7431	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ V
93	91, 92	opth 5446	. . . . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . . . . 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 7980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
99	97, 98	eqtrdi 2815	. . . . . . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . . . . 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 3460	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
104		vex 3460	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
105	103, 104	op1st 7980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106	102, 105	eqtrdi 2815	. . . . . . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . . . . 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 7981	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112	110, 111	eqtrdi 2815	. . . . . . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . . . . 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 7981	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116	114, 115	eqtrdi 2815	. . . . . . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . . . . . . 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 519	. . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (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 499	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) ∧ (𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) ∧ (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 43416	. . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 1956	. . . . . . . . . . . 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 244	. . . . . . . . . . 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 416	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)) → (𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)} → ((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 1956	. . . . . . . . 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 244	. . . . . . . 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 414	. . . . . . 7 ⊢ ((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) → ((𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ∧ 𝑒 ∈ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)}) → ((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 615	. . . . . 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 3220	. . . . 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 410	. . . 4 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ ∃𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)}∃𝑒 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} (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 731	. . 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 697	. 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 43415	. 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 3172	1 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088  Vcvv 3456  ⟨cop 4590   class class class wbr 5102  {copab 5164   × cxp 5647  ‘cfv 6523  (class class class)co 7398  ωcom 7848  1^st c1st 7970  2^nd c2nd 7971   ≈ cen 8926   ≺ csdm 8928  Fincfn 8929  0cc0 11075  1c1 11076   · cmul 11080   − cmin 11416  ℕcn 12212  2c2 12274  ℤcz 12570  ℚcq 12951  ...cfz 13514   mod cmo 13881  ↑cexp 14076  √csqrt 15262  abscabs 15263

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-inf2 9598  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-oadd 8443  df-omul 8444  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9899  df-acn 9902  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-n0 12484  df-xnn0 12557  df-z 12571  df-uz 12842  df-q 12952  df-rp 12996  df-ico 13357  df-fz 13515  df-fl 13804  df-mod 13882  df-seq 14017  df-exp 14077  df-hash 14346  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-dvds 16289  df-gcd 16531  df-numer 16772  df-denom 16773

This theorem is referenced by:  pellqrex  43461