Theorem diophrex 43320

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 2765	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
2	1	rexbidv 3185	. . . 4 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3		reseq1 5957	. . . . . 6 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
4	3	eqeq2d 2772	. . . . 5 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
5	4	cbvrexvw 3240	. . . 4 ⊢ (∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
6	2, 5	bitrdi 289	. . 3 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
7	6	cbvabv 2831	. 2 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8		rexeq 3315	. . . . . 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 2827	. . . . 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 485	. . . 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 2765	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
12	11	anbi1d 640	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)))
13	12	rexbidv 3185	. . . . . . . . 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 3657	. . . . . . . 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 3191	. . . . . . . . . 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 1867	. . . . . . . . 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 3288	. . . . . . . . . 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 472	. . . . . . . . . . . . . 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 1867	. . . . . . . . . . . . 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 3457	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
21	20	resex 6013	. . . . . . . . . . . . . 14 ⊢ (𝑒 ↾ (1...𝑀)) ∈ V
22		reseq1 5957	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
23	22	eqeq2d 2772	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
24	23	anbi2d 639	. . . . . . . . . . . . . 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 3501	. . . . . . . . . . . . 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 277	. . . . . . . . . . . 12 ⊢ (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27		ancom 464	. . . . . . . . . . . . 13 ⊢ (((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0))
28		simpl2 1205	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ≥‘𝑁))
29		fzss2 13566	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑀))
30		resabs1 5990	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . 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 640	. . . . . . . . . . . . 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 285	. . . . . . . . . . . 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 285	. . . . . . . . . . 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 3185	. . . . . . . . . 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 287	. . . . . . . . 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 287	. . . . . . . 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 285	. . . . . . 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 2827	. . . . . 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 43311	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)} ∈ (Dioph‘𝑁))
42	41	3ad2antl1 1198	. . . . . 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 2861	. . . . 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 484	. . . 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 2861	. . 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 43310	. . . . 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 501	. . . 4 ⊢ (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)})
48	47	3ad2ant3 1147	. . 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 3169	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
50	7, 49	eqeltrrid 2866	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739  ∃wrex 3085   ⊆ wss 3904   ↾ cres 5647  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803  0cc0 11070  1c1 11071  ℕcn 12207  ℕ₀cn0 12478  ℤ_≥cuz 12836  ...cfz 13509  mzPolycmzp 43267  Diophcdioph 43300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-oadd 8436  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-hash 14341  df-mzpcl 43268  df-mzp 43269  df-dioph 43301

This theorem is referenced by:  rexrabdioph  43335