rmxdioph - Mathbox for Stefan O'Rear

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Theorem rmxdioph 41740

Description: X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)

**Assertion**
Ref	Expression
rmxdioph	⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2)))} ∈ (Dioph‘3)

Proof of Theorem rmxdioph Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 485	. . . . . 6 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → (𝑎‘1) ∈ (ℤ_≥‘2))
2		elmapi 8839	. . . . . . . 8 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → 𝑎:(1...3)⟶ℕ₀)
3		df-3 12272	. . . . . . . . . 10 ⊢ 3 = (2 + 1)
4		ssid 4003	. . . . . . . . . 10 ⊢ (1...3) ⊆ (1...3)
5	3, 4	jm2.27dlem5 41737	. . . . . . . . 9 ⊢ (1...2) ⊆ (1...3)
6		2nn 12281	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
7	6	jm2.27dlem3 41735	. . . . . . . . 9 ⊢ 2 ∈ (1...2)
8	5, 7	sselii 3978	. . . . . . . 8 ⊢ 2 ∈ (1...3)
9		ffvelcdm 7080	. . . . . . . 8 ⊢ ((𝑎:(1...3)⟶ℕ₀ ∧ 2 ∈ (1...3)) → (𝑎‘2) ∈ ℕ₀)
10	2, 8, 9	sylancl 586	. . . . . . 7 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘2) ∈ ℕ₀)
11	10	adantr 481	. . . . . 6 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → (𝑎‘2) ∈ ℕ₀)
12		3nn 12287	. . . . . . . . 9 ⊢ 3 ∈ ℕ
13	12	jm2.27dlem3 41735	. . . . . . . 8 ⊢ 3 ∈ (1...3)
14		ffvelcdm 7080	. . . . . . . 8 ⊢ ((𝑎:(1...3)⟶ℕ₀ ∧ 3 ∈ (1...3)) → (𝑎‘3) ∈ ℕ₀)
15	2, 13, 14	sylancl 586	. . . . . . 7 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘3) ∈ ℕ₀)
16	15	adantr 481	. . . . . 6 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → (𝑎‘3) ∈ ℕ₀)
17		rmxdiophlem 41739	. . . . . 6 ⊢ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘2) ∈ ℕ₀ ∧ (𝑎‘3) ∈ ℕ₀) → ((𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2)) ↔ ∃𝑏 ∈ ℕ₀ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)))
18	1, 11, 16, 17	syl3anc 1371	. . . . 5 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → ((𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2)) ↔ ∃𝑏 ∈ ℕ₀ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)))
19	18	pm5.32da 579	. . . 4 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2))) ↔ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ ∃𝑏 ∈ ℕ₀ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1))))
20		anass 469	. . . . . 6 ⊢ ((((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1) ↔ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)))
21	20	rexbii 3094	. . . . 5 ⊢ (∃𝑏 ∈ ℕ₀ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1) ↔ ∃𝑏 ∈ ℕ₀ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)))
22		r19.42v 3190	. . . . 5 ⊢ (∃𝑏 ∈ ℕ₀ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)) ↔ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ ∃𝑏 ∈ ℕ₀ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)))
23	21, 22	bitr2i 275	. . . 4 ⊢ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ ∃𝑏 ∈ ℕ₀ (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)) ↔ ∃𝑏 ∈ ℕ₀ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1))
24	19, 23	bitrdi 286	. . 3 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2))) ↔ ∃𝑏 ∈ ℕ₀ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)))
25	24	rabbiia 3436	. 2 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2)))} = {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ∃𝑏 ∈ ℕ₀ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)}
26		3nn0 12486	. . 3 ⊢ 3 ∈ ℕ₀
27		vex 3478	. . . . . . 7 ⊢ 𝑐 ∈ V
28	27	resex 6027	. . . . . 6 ⊢ (𝑐 ↾ (1...3)) ∈ V
29		fvex 6901	. . . . . 6 ⊢ (𝑐‘4) ∈ V
30		df-2 12271	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
31	30, 5	jm2.27dlem5 41737	. . . . . . . . . . . 12 ⊢ (1...1) ⊆ (1...3)
32		1nn 12219	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
33	32	jm2.27dlem3 41735	. . . . . . . . . . . 12 ⊢ 1 ∈ (1...1)
34	31, 33	sselii 3978	. . . . . . . . . . 11 ⊢ 1 ∈ (1...3)
35	34	jm2.27dlem1 41733	. . . . . . . . . 10 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → (𝑎‘1) = (𝑐‘1))
36	35	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → ((𝑎‘1) ∈ (ℤ_≥‘2) ↔ (𝑐‘1) ∈ (ℤ_≥‘2)))
37	36	adantr 481	. . . . . . . 8 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((𝑎‘1) ∈ (ℤ_≥‘2) ↔ (𝑐‘1) ∈ (ℤ_≥‘2)))
38		simpr 485	. . . . . . . . 9 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → 𝑏 = (𝑐‘4))
39	8	jm2.27dlem1 41733	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → (𝑎‘2) = (𝑐‘2))
40	35, 39	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → ((𝑎‘1) Y_rm (𝑎‘2)) = ((𝑐‘1) Y_rm (𝑐‘2)))
41	40	adantr 481	. . . . . . . . 9 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((𝑎‘1) Y_rm (𝑎‘2)) = ((𝑐‘1) Y_rm (𝑐‘2)))
42	38, 41	eqeq12d 2748	. . . . . . . 8 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → (𝑏 = ((𝑎‘1) Y_rm (𝑎‘2)) ↔ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))))
43	37, 42	anbi12d 631	. . . . . . 7 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ↔ ((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2)))))
44	13	jm2.27dlem1 41733	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → (𝑎‘3) = (𝑐‘3))
45	44	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → ((𝑎‘3)↑2) = ((𝑐‘3)↑2))
46	45	adantr 481	. . . . . . . . 9 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((𝑎‘3)↑2) = ((𝑐‘3)↑2))
47	35	oveq1d 7420	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → ((𝑎‘1)↑2) = ((𝑐‘1)↑2))
48	47	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → (((𝑎‘1)↑2) − 1) = (((𝑐‘1)↑2) − 1))
49		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑏 = (𝑐‘4) → (𝑏↑2) = ((𝑐‘4)↑2))
50	48, 49	oveqan12d 7424	. . . . . . . . 9 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((((𝑎‘1)↑2) − 1) · (𝑏↑2)) = ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2)))
51	46, 50	oveq12d 7423	. . . . . . . 8 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))))
52	51	eqeq1d 2734	. . . . . . 7 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1 ↔ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1))
53	43, 52	anbi12d 631	. . . . . 6 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1) ↔ (((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))) ∧ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1)))
54	28, 29, 53	sbc2ie 3859	. . . . 5 ⊢ ([(𝑐 ↾ (1...3)) / 𝑎][(𝑐‘4) / 𝑏](((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1) ↔ (((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))) ∧ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1))
55	54	rabbii 3438	. . . 4 ⊢ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ [(𝑐 ↾ (1...3)) / 𝑎][(𝑐‘4) / 𝑏](((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)} = {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))) ∧ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1)}
56		4nn0 12487	. . . . . 6 ⊢ 4 ∈ ℕ₀
57		rmydioph 41738	. . . . . 6 ⊢ {𝑏 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑏‘1) ∈ (ℤ_≥‘2) ∧ (𝑏‘3) = ((𝑏‘1) Y_rm (𝑏‘2)))} ∈ (Dioph‘3)
58		simp1 1136	. . . . . . . . 9 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → (𝑏‘1) = (𝑐‘1))
59	58	eleq1d 2818	. . . . . . . 8 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → ((𝑏‘1) ∈ (ℤ_≥‘2) ↔ (𝑐‘1) ∈ (ℤ_≥‘2)))
60		simp3 1138	. . . . . . . . 9 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → (𝑏‘3) = (𝑐‘4))
61		simp2 1137	. . . . . . . . . 10 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → (𝑏‘2) = (𝑐‘2))
62	58, 61	oveq12d 7423	. . . . . . . . 9 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → ((𝑏‘1) Y_rm (𝑏‘2)) = ((𝑐‘1) Y_rm (𝑐‘2)))
63	60, 62	eqeq12d 2748	. . . . . . . 8 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → ((𝑏‘3) = ((𝑏‘1) Y_rm (𝑏‘2)) ↔ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))))
64	59, 63	anbi12d 631	. . . . . . 7 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → (((𝑏‘1) ∈ (ℤ_≥‘2) ∧ (𝑏‘3) = ((𝑏‘1) Y_rm (𝑏‘2))) ↔ ((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2)))))
65		df-4 12273	. . . . . . . . . . 11 ⊢ 4 = (3 + 1)
66		ssid 4003	. . . . . . . . . . 11 ⊢ (1...4) ⊆ (1...4)
67	65, 66	jm2.27dlem5 41737	. . . . . . . . . 10 ⊢ (1...3) ⊆ (1...4)
68	3, 67	jm2.27dlem5 41737	. . . . . . . . 9 ⊢ (1...2) ⊆ (1...4)
69	30, 68	jm2.27dlem5 41737	. . . . . . . 8 ⊢ (1...1) ⊆ (1...4)
70	69, 33	sselii 3978	. . . . . . 7 ⊢ 1 ∈ (1...4)
71	68, 7	sselii 3978	. . . . . . 7 ⊢ 2 ∈ (1...4)
72		4nn 12291	. . . . . . . 8 ⊢ 4 ∈ ℕ
73	72	jm2.27dlem3 41735	. . . . . . 7 ⊢ 4 ∈ (1...4)
74	64, 70, 71, 73	rabren3dioph 41538	. . . . . 6 ⊢ ((4 ∈ ℕ₀ ∧ {𝑏 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑏‘1) ∈ (ℤ_≥‘2) ∧ (𝑏‘3) = ((𝑏‘1) Y_rm (𝑏‘2)))} ∈ (Dioph‘3)) → {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ ((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2)))} ∈ (Dioph‘4))
75	56, 57, 74	mp2an 690	. . . . 5 ⊢ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ ((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2)))} ∈ (Dioph‘4)
76		ovex 7438	. . . . . . . . 9 ⊢ (1...4) ∈ V
77	67, 13	sselii 3978	. . . . . . . . 9 ⊢ 3 ∈ (1...4)
78		mzpproj 41460	. . . . . . . . 9 ⊢ (((1...4) ∈ V ∧ 3 ∈ (1...4)) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘3)) ∈ (mzPoly‘(1...4)))
79	76, 77, 78	mp2an 690	. . . . . . . 8 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘3)) ∈ (mzPoly‘(1...4))
80		2nn0 12485	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
81		mzpexpmpt 41468	. . . . . . . 8 ⊢ (((𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘3)) ∈ (mzPoly‘(1...4)) ∧ 2 ∈ ℕ₀) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘3)↑2)) ∈ (mzPoly‘(1...4)))
82	79, 80, 81	mp2an 690	. . . . . . 7 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘3)↑2)) ∈ (mzPoly‘(1...4))
83		mzpproj 41460	. . . . . . . . . . 11 ⊢ (((1...4) ∈ V ∧ 1 ∈ (1...4)) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘1)) ∈ (mzPoly‘(1...4)))
84	76, 70, 83	mp2an 690	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘1)) ∈ (mzPoly‘(1...4))
85		mzpexpmpt 41468	. . . . . . . . . 10 ⊢ (((𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘1)) ∈ (mzPoly‘(1...4)) ∧ 2 ∈ ℕ₀) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘1)↑2)) ∈ (mzPoly‘(1...4)))
86	84, 80, 85	mp2an 690	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘1)↑2)) ∈ (mzPoly‘(1...4))
87		1z 12588	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
88		mzpconstmpt 41463	. . . . . . . . . 10 ⊢ (((1...4) ∈ V ∧ 1 ∈ ℤ) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ 1) ∈ (mzPoly‘(1...4)))
89	76, 87, 88	mp2an 690	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ 1) ∈ (mzPoly‘(1...4))
90		mzpsubmpt 41466	. . . . . . . . 9 ⊢ (((𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘1)↑2)) ∈ (mzPoly‘(1...4)) ∧ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ 1) ∈ (mzPoly‘(1...4))) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘1)↑2) − 1)) ∈ (mzPoly‘(1...4)))
91	86, 89, 90	mp2an 690	. . . . . . . 8 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘1)↑2) − 1)) ∈ (mzPoly‘(1...4))
92		mzpproj 41460	. . . . . . . . . 10 ⊢ (((1...4) ∈ V ∧ 4 ∈ (1...4)) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘4)) ∈ (mzPoly‘(1...4)))
93	76, 73, 92	mp2an 690	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘4)) ∈ (mzPoly‘(1...4))
94		mzpexpmpt 41468	. . . . . . . . 9 ⊢ (((𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘4)) ∈ (mzPoly‘(1...4)) ∧ 2 ∈ ℕ₀) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘4)↑2)) ∈ (mzPoly‘(1...4)))
95	93, 80, 94	mp2an 690	. . . . . . . 8 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘4)↑2)) ∈ (mzPoly‘(1...4))
96		mzpmulmpt 41465	. . . . . . . 8 ⊢ (((𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘1)↑2) − 1)) ∈ (mzPoly‘(1...4)) ∧ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘4)↑2)) ∈ (mzPoly‘(1...4))) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) ∈ (mzPoly‘(1...4)))
97	91, 95, 96	mp2an 690	. . . . . . 7 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) ∈ (mzPoly‘(1...4))
98		mzpsubmpt 41466	. . . . . . 7 ⊢ (((𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘3)↑2)) ∈ (mzPoly‘(1...4)) ∧ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) ∈ (mzPoly‘(1...4))) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2)))) ∈ (mzPoly‘(1...4)))
99	82, 97, 98	mp2an 690	. . . . . 6 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2)))) ∈ (mzPoly‘(1...4))
100		eqrabdioph 41500	. . . . . 6 ⊢ ((4 ∈ ℕ₀ ∧ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2)))) ∈ (mzPoly‘(1...4)) ∧ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ 1) ∈ (mzPoly‘(1...4))) → {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1} ∈ (Dioph‘4))
101	56, 99, 89, 100	mp3an 1461	. . . . 5 ⊢ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1} ∈ (Dioph‘4)
102		anrabdioph 41503	. . . . 5 ⊢ (({𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ ((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2)))} ∈ (Dioph‘4) ∧ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1} ∈ (Dioph‘4)) → {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))) ∧ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1)} ∈ (Dioph‘4))
103	75, 101, 102	mp2an 690	. . . 4 ⊢ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘1) ∈ (ℤ_≥‘2) ∧ (𝑐‘4) = ((𝑐‘1) Y_rm (𝑐‘2))) ∧ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1)} ∈ (Dioph‘4)
104	55, 103	eqeltri 2829	. . 3 ⊢ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ [(𝑐 ↾ (1...3)) / 𝑎][(𝑐‘4) / 𝑏](((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)} ∈ (Dioph‘4)
105	65	rexfrabdioph 41518	. . 3 ⊢ ((3 ∈ ℕ₀ ∧ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ [(𝑐 ↾ (1...3)) / 𝑎][(𝑐‘4) / 𝑏](((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)} ∈ (Dioph‘4)) → {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ∃𝑏 ∈ ℕ₀ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)} ∈ (Dioph‘3))
106	26, 104, 105	mp2an 690	. 2 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ∃𝑏 ∈ ℕ₀ (((𝑎‘1) ∈ (ℤ_≥‘2) ∧ 𝑏 = ((𝑎‘1) Y_rm (𝑎‘2))) ∧ (((𝑎‘3)↑2) − ((((𝑎‘1)↑2) − 1) · (𝑏↑2))) = 1)} ∈ (Dioph‘3)
107	25, 106	eqeltri 2829	1 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ_≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) X_rm (𝑎‘2)))} ∈ (Dioph‘3)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 {crab 3432 Vcvv 3474 [wsbc 3776 ↦ cmpt 5230 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 1c1 11107 · cmul 11111 − cmin 11440 2c2 12263 3c3 12264 4c4 12265 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 ↑cexp 14023 mzPolycmzp 41445 Diophcdioph 41478 X_rm crmx 41623 Y_rm crmy 41624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-numer 16667 df-denom 16668 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-mzpcl 41446 df-mzp 41447 df-dioph 41479 df-squarenn 41564 df-pell1qr 41565 df-pell14qr 41566 df-pell1234qr 41567 df-pellfund 41568 df-rmx 41625 df-rmy 41626

This theorem is referenced by: expdiophlem2 41746

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