Theorem pellex 38917

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 13190	. . . . . . . 8 ⊢ (0...((abs‘𝑎) − 1)) ∈ Fin
2		xpfi 8635	. . . . . . . 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 688	. . . . . . 7 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin
4		isfinite 8961	. . . . . . 7 ⊢ (((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ∈ Fin ↔ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω)
5	3, 4	mpbi 231	. . . . . 6 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ω
6		nnenom 13198	. . . . . . 7 ⊢ ℕ ≈ ω
7	6	ensymi 8407	. . . . . 6 ⊢ ω ≈ ℕ
8		sdomentr 8498	. . . . . 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 688	. . . . 5 ⊢ ((0...((abs‘𝑎) − 1)) × (0...((abs‘𝑎) − 1))) ≺ ℕ
10		ensym 8406	. . . . . 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 725	. . . . 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 8498	. . . . 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 587	. . . 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 5529	. . . . . . . 8 ⊢ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} ⊆ (ℕ × ℕ)
15	14	sseli 3885	. . . . . . 7 ⊢ (𝑑 ∈ {⟨𝑏, 𝑐⟩ ∣ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎)} → 𝑑 ∈ (ℕ × ℕ))
16		simprrl 777	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (1st ‘𝑑) ∈ ℕ)
17	16	nnzd 11935	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (1st ‘𝑑) ∈ ℤ)
18		simpllr 772	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → 𝑎 ∈ ℤ)
19		simplr 765	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → 𝑎 ≠ 0)
20		nnabscl 14519	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ ℤ ∧ 𝑎 ≠ 0) → (abs‘𝑎) ∈ ℕ)
21	18, 19, 20	syl2anc 584	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (abs‘𝑎) ∈ ℕ)
22		zmodfz 13111	. . . . . . . . . . 11 ⊢ (((1st ‘𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((1st ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
23	17, 21, 22	syl2anc 584	. . . . . . . . . 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 778	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (2nd ‘𝑑) ∈ ℕ)
25	24	nnzd 11935	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ))) → (2nd ‘𝑑) ∈ ℤ)
26		zmodfz 13111	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑑) ∈ ℤ ∧ (abs‘𝑎) ∈ ℕ) → ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ (0...((abs‘𝑎) − 1)))
27	25, 21, 26	syl2anc 584	. . . . . . . . . 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 512	. . . . . . . . 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 413	. . . . . . . 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 7580	. . . . . . . 8 ⊢ (𝑑 ∈ (ℕ × ℕ) ↔ (𝑑 ∈ (V × V) ∧ ((1st ‘𝑑) ∈ ℕ ∧ (2nd ‘𝑑) ∈ ℕ)))
31		opelxp 5479	. . . . . . . 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 297	. . . . . . 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 407	. . . . 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 717	. . . 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 6538	. . . . . 6 ⊢ (𝑑 = 𝑒 → (1st ‘𝑑) = (1st ‘𝑒))
37	36	oveq1d 7031	. . . . 5 ⊢ (𝑑 = 𝑒 → ((1st ‘𝑑) mod (abs‘𝑎)) = ((1st ‘𝑒) mod (abs‘𝑎)))
38		fveq2 6538	. . . . . 6 ⊢ (𝑑 = 𝑒 → (2nd ‘𝑑) = (2nd ‘𝑒))
39	38	oveq1d 7031	. . . . 5 ⊢ (𝑑 = 𝑒 → ((2nd ‘𝑑) mod (abs‘𝑎)) = ((2nd ‘𝑒) mod (abs‘𝑎)))
40	37, 39	opeq12d 4718	. . . 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 38898	. . 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 2865	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑓 → (𝑏 ∈ ℕ ↔ 𝑓 ∈ ℕ))
43		eleq1w 2865	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑔 → (𝑐 ∈ ℕ ↔ 𝑔 ∈ ℕ))
44	42, 43	bi2anan9 635	. . . . . . . . . . 11 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ↔ (𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ)))
45		oveq1 7023	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑓 → (𝑏↑2) = (𝑓↑2))
46		oveq1 7023	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑔 → (𝑐↑2) = (𝑔↑2))
47	46	oveq2d 7032	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑔 → (𝐷 · (𝑐↑2)) = (𝐷 · (𝑔↑2)))
48	45, 47	oveqan12d 7035	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝑓 ∧ 𝑐 = 𝑔) → ((𝑏↑2) − (𝐷 · (𝑐↑2))) = ((𝑓↑2) − (𝐷 · (𝑔↑2))))
49	48	eqeq1d 2797	. . . . . . . . . . 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 5034	. . . . . . . . 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 2874	. . . . . . . 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 217	. . . . . . 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 5304	. . . . . . . . 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 5304	. . . . . . . . . . . 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 1237	. . . . . . . . . . . . . . . . 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 1113	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝑏 ∈ ℕ)
58	57	3ad2ant1 1126	. . . . . . . . . . . . . . 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 1238	. . . . . . . . . . . . . . . . 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 1113	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → 𝑐 ∈ ℕ)
61	60	3ad2ant1 1126	. . . . . . . . . . . . . . 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 1230	. . . . . . . . . . . . . . . 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 1170	. . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . 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 1126	. . . . . . . . . . . . . . 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 1233	. . . . . . . . . . . . . . . 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 1171	. . . . . . . . . . . . . . 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 1234	. . . . . . . . . . . . . . . 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 1171	. . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑑 ≠ 𝑒)
76		simp2 1130	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑑 = ⟨𝑏, 𝑐⟩)
77		simp1 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → 𝑒 = ⟨𝑓, 𝑔⟩)
78	75, 76, 77	3netr3d 3060	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → ⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩)
79		vex 3440	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑏 ∈ V
80		vex 3440	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑐 ∈ V
81	79, 80	opth 5260	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑏, 𝑐⟩ = ⟨𝑓, 𝑔⟩ ↔ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
82	81	necon3abii 3030	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑏, 𝑐⟩ ≠ ⟨𝑓, 𝑔⟩ ↔ ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
83	78, 82	sylib 219	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ 𝑑 = ⟨𝑏, 𝑐⟩ ∧ 𝑑 ≠ 𝑒) → ¬ (𝑏 = 𝑓 ∧ 𝑐 = 𝑔))
84	72, 73, 74, 83	syl3anc 1364	. . . . . . . . . . . . . . 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 1230	. . . . . . . . . . . . . . 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 1232	. . . . . . . . . . . . . . . 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 1169	. . . . . . . . . . . . . . 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 1236	. . . . . . . . . . . . . . 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 1195	. . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . . 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 7048	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑑) mod (abs‘𝑎)) ∈ V
92		ovex 7048	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑑) mod (abs‘𝑎)) ∈ V
93	91, 92	opth 5260	. . . . . . . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . . . . . . . . . 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 6542	. . . . . . . . . . . . . . . . . . . . . . . 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 7553	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑏, 𝑐⟩) = 𝑏
99	97, 98	syl6eq 2847	. . . . . . . . . . . . . . . . . . . . . . 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 7031	. . . . . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . . . . 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 6542	. . . . . . . . . . . . . . . . . . . . . . . 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 3440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
104		vex 3440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑔 ∈ V
105	103, 104	op1st 7553	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1st ‘⟨𝑓, 𝑔⟩) = 𝑓
106	102, 105	syl6eq 2847	. . . . . . . . . . . . . . . . . . . . . . 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 7031	. . . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . 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 6542	. . . . . . . . . . . . . . . . . . . . . . . 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 7554	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑏, 𝑐⟩) = 𝑐
112	110, 111	syl6eq 2847	. . . . . . . . . . . . . . . . . . . . . . 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 7031	. . . . . . . . . . . . . . . . . . . . . 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 6542	. . . . . . . . . . . . . . . . . . . . . . . 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 7554	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2nd ‘⟨𝑓, 𝑔⟩) = 𝑔
116	114, 115	syl6eq 2847	. . . . . . . . . . . . . . . . . . . . . . 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 7031	. . . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . . . 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 1364	. . . . . . . . . . . . . . . 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 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))⟩)) → (𝑏 mod (abs‘𝑎)) = (𝑓 mod (abs‘𝑎)))
125	123	simprd 496	. . . . . . . . . . . . . . 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 38916	. . . . . . . . . . . . . 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 1112	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ ℤ) ∧ 𝑎 ≠ 0) ∧ (𝑑 = ⟨𝑏, 𝑐⟩ ∧ ((𝑏 ∈ ℕ ∧ 𝑐 ∈ ℕ) ∧ ((𝑏↑2) − (𝐷 · (𝑐↑2))) = 𝑎))) → ((𝑒 = ⟨𝑓, 𝑔⟩ ∧ ((𝑓 ∈ ℕ ∧ 𝑔 ∈ ℕ) ∧ ((𝑓↑2) − (𝐷 · (𝑔↑2))) = 𝑎)) → ((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 1912	. . . . . . . . . . . 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	syl5bi 243	. . . . . . . . . . 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 413	. . . . . . . . . 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 1912	. . . . . . . . 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	syl5bi 243	. . . . . . . 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 411	. . . . . . 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 3256	. . . . 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 407	. . . 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 717	. . 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 683	. 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 38915	. 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 3252	1 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ≠ wne 2984  ∃wrex 3106  Vcvv 3437  ⟨cop 4478   class class class wbr 4962  {copab 5024   × cxp 5441  ‘cfv 6225  (class class class)co 7016  ωcom 7436  1^st c1st 7543  2^nd c2nd 7544   ≈ cen 8354   ≺ csdm 8356  Fincfn 8357  0cc0 10383  1c1 10384   · cmul 10388   − cmin 10717  ℕcn 11486  2c2 11540  ℤcz 11829  ℚcq 12197  ...cfz 12742   mod cmo 13087  ↑cexp 13279  √csqrt 14426  abscabs 14427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-omul 7958  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-acn 9217  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-xnn0 11816  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-ico 12594  df-fz 12743  df-fl 13012  df-mod 13088  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-dvds 15441  df-gcd 15677  df-numer 15904  df-denom 15905

This theorem is referenced by:  pellqrex  38961