rmxdioph - Mathbox for Stefan O'Rear

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Theorem rmxdioph 42333

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 484	. . . . . 6 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → (𝑎‘1) ∈ (ℤ_≥‘2))
2		elmapi 8845	. . . . . . . 8 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → 𝑎:(1...3)⟶ℕ₀)
3		df-3 12280	. . . . . . . . . 10 ⊢ 3 = (2 + 1)
4		ssid 3999	. . . . . . . . . 10 ⊢ (1...3) ⊆ (1...3)
5	3, 4	jm2.27dlem5 42330	. . . . . . . . 9 ⊢ (1...2) ⊆ (1...3)
6		2nn 12289	. . . . . . . . . 10 ⊢ 2 ∈ ℕ
7	6	jm2.27dlem3 42328	. . . . . . . . 9 ⊢ 2 ∈ (1...2)
8	5, 7	sselii 3974	. . . . . . . 8 ⊢ 2 ∈ (1...3)
9		ffvelcdm 7077	. . . . . . . 8 ⊢ ((𝑎:(1...3)⟶ℕ₀ ∧ 2 ∈ (1...3)) → (𝑎‘2) ∈ ℕ₀)
10	2, 8, 9	sylancl 585	. . . . . . 7 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘2) ∈ ℕ₀)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → (𝑎‘2) ∈ ℕ₀)
12		3nn 12295	. . . . . . . . 9 ⊢ 3 ∈ ℕ
13	12	jm2.27dlem3 42328	. . . . . . . 8 ⊢ 3 ∈ (1...3)
14		ffvelcdm 7077	. . . . . . . 8 ⊢ ((𝑎:(1...3)⟶ℕ₀ ∧ 3 ∈ (1...3)) → (𝑎‘3) ∈ ℕ₀)
15	2, 13, 14	sylancl 585	. . . . . . 7 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...3)) → (𝑎‘3) ∈ ℕ₀)
16	15	adantr 480	. . . . . 6 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...3)) ∧ (𝑎‘1) ∈ (ℤ_≥‘2)) → (𝑎‘3) ∈ ℕ₀)
17		rmxdiophlem 42332	. . . . . 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 1368	. . . . 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 578	. . . 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 468	. . . . . 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 3088	. . . . 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 3184	. . . . 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 276	. . . 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 287	. . 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 3430	. 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 12494	. . 3 ⊢ 3 ∈ ℕ₀
27		vex 3472	. . . . . . 7 ⊢ 𝑐 ∈ V
28	27	resex 6023	. . . . . 6 ⊢ (𝑐 ↾ (1...3)) ∈ V
29		fvex 6898	. . . . . 6 ⊢ (𝑐‘4) ∈ V
30		df-2 12279	. . . . . . . . . . . . 13 ⊢ 2 = (1 + 1)
31	30, 5	jm2.27dlem5 42330	. . . . . . . . . . . 12 ⊢ (1...1) ⊆ (1...3)
32		1nn 12227	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ
33	32	jm2.27dlem3 42328	. . . . . . . . . . . 12 ⊢ 1 ∈ (1...1)
34	31, 33	sselii 3974	. . . . . . . . . . 11 ⊢ 1 ∈ (1...3)
35	34	jm2.27dlem1 42326	. . . . . . . . . 10 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → (𝑎‘1) = (𝑐‘1))
36	35	eleq1d 2812	. . . . . . . . 9 ⊢ (𝑎 = (𝑐 ↾ (1...3)) → ((𝑎‘1) ∈ (ℤ_≥‘2) ↔ (𝑐‘1) ∈ (ℤ_≥‘2)))
37	36	adantr 480	. . . . . . . 8 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((𝑎‘1) ∈ (ℤ_≥‘2) ↔ (𝑐‘1) ∈ (ℤ_≥‘2)))
38		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → 𝑏 = (𝑐‘4))
39	8	jm2.27dlem1 42326	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝑎 = (𝑐 ↾ (1...3)) ∧ 𝑏 = (𝑐‘4)) → ((𝑎‘1) Y_rm (𝑎‘2)) = ((𝑐‘1) Y_rm (𝑐‘2)))
42	38, 41	eqeq12d 2742	. . . . . . . 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 630	. . . . . . 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 42326	. . . . . . . . . . 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 480	. . . . . . . . 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 2728	. . . . . . 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 630	. . . . . 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 3855	. . . . 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 3432	. . . 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 12495	. . . . . 6 ⊢ 4 ∈ ℕ₀
57		rmydioph 42331	. . . . . 6 ⊢ {𝑏 ∈ (ℕ₀ ↑_m (1...3)) ∣ ((𝑏‘1) ∈ (ℤ_≥‘2) ∧ (𝑏‘3) = ((𝑏‘1) Y_rm (𝑏‘2)))} ∈ (Dioph‘3)
58		simp1 1133	. . . . . . . . 9 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → (𝑏‘1) = (𝑐‘1))
59	58	eleq1d 2812	. . . . . . . 8 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → ((𝑏‘1) ∈ (ℤ_≥‘2) ↔ (𝑐‘1) ∈ (ℤ_≥‘2)))
60		simp3 1135	. . . . . . . . 9 ⊢ (((𝑏‘1) = (𝑐‘1) ∧ (𝑏‘2) = (𝑐‘2) ∧ (𝑏‘3) = (𝑐‘4)) → (𝑏‘3) = (𝑐‘4))
61		simp2 1134	. . . . . . . . . 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 2742	. . . . . . . 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 630	. . . . . . 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 12281	. . . . . . . . . . 11 ⊢ 4 = (3 + 1)
66		ssid 3999	. . . . . . . . . . 11 ⊢ (1...4) ⊆ (1...4)
67	65, 66	jm2.27dlem5 42330	. . . . . . . . . 10 ⊢ (1...3) ⊆ (1...4)
68	3, 67	jm2.27dlem5 42330	. . . . . . . . 9 ⊢ (1...2) ⊆ (1...4)
69	30, 68	jm2.27dlem5 42330	. . . . . . . 8 ⊢ (1...1) ⊆ (1...4)
70	69, 33	sselii 3974	. . . . . . 7 ⊢ 1 ∈ (1...4)
71	68, 7	sselii 3974	. . . . . . 7 ⊢ 2 ∈ (1...4)
72		4nn 12299	. . . . . . . 8 ⊢ 4 ∈ ℕ
73	72	jm2.27dlem3 42328	. . . . . . 7 ⊢ 4 ∈ (1...4)
74	64, 70, 71, 73	rabren3dioph 42131	. . . . . 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 689	. . . . 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 3974	. . . . . . . . 9 ⊢ 3 ∈ (1...4)
78		mzpproj 42053	. . . . . . . . 9 ⊢ (((1...4) ∈ V ∧ 3 ∈ (1...4)) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘3)) ∈ (mzPoly‘(1...4)))
79	76, 77, 78	mp2an 689	. . . . . . . 8 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘3)) ∈ (mzPoly‘(1...4))
80		2nn0 12493	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
81		mzpexpmpt 42061	. . . . . . . 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 689	. . . . . . 7 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘3)↑2)) ∈ (mzPoly‘(1...4))
83		mzpproj 42053	. . . . . . . . . . 11 ⊢ (((1...4) ∈ V ∧ 1 ∈ (1...4)) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘1)) ∈ (mzPoly‘(1...4)))
84	76, 70, 83	mp2an 689	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘1)) ∈ (mzPoly‘(1...4))
85		mzpexpmpt 42061	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘1)↑2)) ∈ (mzPoly‘(1...4))
87		1z 12596	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
88		mzpconstmpt 42056	. . . . . . . . . 10 ⊢ (((1...4) ∈ V ∧ 1 ∈ ℤ) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ 1) ∈ (mzPoly‘(1...4)))
89	76, 87, 88	mp2an 689	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ 1) ∈ (mzPoly‘(1...4))
90		mzpsubmpt 42059	. . . . . . . . 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 689	. . . . . . . 8 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘1)↑2) − 1)) ∈ (mzPoly‘(1...4))
92		mzpproj 42053	. . . . . . . . . 10 ⊢ (((1...4) ∈ V ∧ 4 ∈ (1...4)) → (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘4)) ∈ (mzPoly‘(1...4)))
93	76, 73, 92	mp2an 689	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (𝑐‘4)) ∈ (mzPoly‘(1...4))
94		mzpexpmpt 42061	. . . . . . . . 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 689	. . . . . . . 8 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((𝑐‘4)↑2)) ∈ (mzPoly‘(1...4))
96		mzpmulmpt 42058	. . . . . . . 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 689	. . . . . . 7 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) ∈ (mzPoly‘(1...4))
98		mzpsubmpt 42059	. . . . . . 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 689	. . . . . 6 ⊢ (𝑐 ∈ (ℤ ↑_m (1...4)) ↦ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2)))) ∈ (mzPoly‘(1...4))
100		eqrabdioph 42093	. . . . . 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 1457	. . . . 5 ⊢ {𝑐 ∈ (ℕ₀ ↑_m (1...4)) ∣ (((𝑐‘3)↑2) − ((((𝑐‘1)↑2) − 1) · ((𝑐‘4)↑2))) = 1} ∈ (Dioph‘4)
102		anrabdioph 42096	. . . . 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 689	. . . 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 2823	. . 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 42111	. . 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 689	. 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 2823	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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 {crab 3426 Vcvv 3468 [wsbc 3772 ↦ cmpt 5224 ↾ cres 5671 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ↑_m cmap 8822 1c1 11113 · cmul 11117 − cmin 11448 2c2 12271 3c3 12272 4c4 12273 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13490 ↑cexp 14032 mzPolycmzp 42038 Diophcdioph 42071 X_rm crmx 42216 Y_rm crmy 42217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-dvds 16205 df-gcd 16443 df-prm 16616 df-numer 16680 df-denom 16681 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445 df-mzpcl 42039 df-mzp 42040 df-dioph 42072 df-squarenn 42157 df-pell1qr 42158 df-pell14qr 42159 df-pell1234qr 42160 df-pellfund 42161 df-rmx 42218 df-rmy 42219

This theorem is referenced by: expdiophlem2 42339

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