Theorem diophrex 42337

Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)

Assertion

Ref	Expression
diophrex	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Distinct variable groups:   𝑡,𝑁,𝑢   𝑡,𝑆,𝑢

Allowed substitution hints:   𝑀(𝑢,𝑡)

Proof of Theorem diophrex

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2729	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
2	1	rexbidv 3168	. . . 4 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3		reseq1 5979	. . . . . 6 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
4	3	eqeq2d 2736	. . . . 5 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
5	4	cbvrexvw 3225	. . . 4 ⊢ (∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
6	2, 5	bitrdi 286	. . 3 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
7	6	cbvabv 2798	. 2 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8		rexeq 3310	. . . . . 6 ⊢ (𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)} → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))))
9	8	abbidv 2794	. . . . 5 ⊢ (𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)} → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))})
10	9	adantl 480	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))})
11		eqeq1 2729	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
12	11	anbi1d 629	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)))
13	12	rexbidv 3168	. . . . . . . . 9 ⊢ (𝑑 = 𝑏 → (∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ↔ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)))
14	13	rexab 3686	. . . . . . . 8 ⊢ (∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏(∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
15		r19.41v 3178	. . . . . . . . . 10 ⊢ (∃𝑒 ∈ (ℕ0 ↑m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
16	15	exbii 1842	. . . . . . . . 9 ⊢ (∃𝑏∃𝑒 ∈ (ℕ0 ↑m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏(∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
17		rexcom4 3275	. . . . . . . . . 10 ⊢ (∃𝑒 ∈ (ℕ0 ↑m ℕ)∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏∃𝑒 ∈ (ℕ0 ↑m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
18		anass 467	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐‘𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
19	18	exbii 1842	. . . . . . . . . . . . 13 ⊢ (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐‘𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
20		vex 3465	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
21	20	resex 6034	. . . . . . . . . . . . . 14 ⊢ (𝑒 ↾ (1...𝑀)) ∈ V
22		reseq1 5979	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
23	22	eqeq2d 2736	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
24	23	anbi2d 628	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (((𝑐‘𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))))
25	21, 24	ceqsexv 3514	. . . . . . . . . . . . 13 ⊢ (∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐‘𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) ↔ ((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
26	19, 25	bitri 274	. . . . . . . . . . . 12 ⊢ (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27		ancom 459	. . . . . . . . . . . . 13 ⊢ (((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0))
28		simpl2 1189	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ≥‘𝑁))
29		fzss2 13576	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑀))
30		resabs1 6012	. . . . . . . . . . . . . . . 16 ⊢ ((1...𝑁) ⊆ (1...𝑀) → ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑒 ↾ (1...𝑁)))
31	28, 29, 30	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑒 ↾ (1...𝑁)))
32	31	eqeq2d 2736	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ↔ 𝑎 = (𝑒 ↾ (1...𝑁))))
33	32	anbi1d 629	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → ((𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
34	27, 33	bitrid 282	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
35	26, 34	bitrid 282	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
36	35	rexbidv 3168	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑒 ∈ (ℕ0 ↑m ℕ)∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
37	17, 36	bitr3id 284	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏∃𝑒 ∈ (ℕ0 ↑m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
38	16, 37	bitr3id 284	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏(∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
39	14, 38	bitrid 282	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)))
40	39	abbidv 2794	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)})
41		eldioph3 42328	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)} ∈ (Dioph‘𝑁))
42	41	3ad2antl1 1182	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)} ∈ (Dioph‘𝑁))
43	40, 42	eqeltrd 2825	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
44	43	adantr 479	. . . 4 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
45	10, 44	eqeltrd 2825	. . 3 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
46		eldioph3b 42327	. . . . 5 ⊢ (𝑆 ∈ (Dioph‘𝑀) ↔ (𝑀 ∈ ℕ0 ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}))
47	46	simprbi 495	. . . 4 ⊢ (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)})
48	47	3ad2ant3 1132	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)})
49	45, 48	r19.29a 3151	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
50	7, 49	eqeltrrid 2830	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∃wrex 3059   ⊆ wss 3944   ↾ cres 5680  ‘cfv 6549  (class class class)co 7419   ↑_m cmap 8845  0cc0 11140  1c1 11141  ℕcn 12245  ℕ₀cn0 12505  ℤ_≥cuz 12855  ...cfz 13519  mzPolycmzp 42284  Diophcdioph 42317

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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-hash 14326  df-mzpcl 42285  df-mzp 42286  df-dioph 42318

This theorem is referenced by:  rexrabdioph  42356