Theorem diophrex 43017

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 2740	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
2	1	rexbidv 3160	. . . 4 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3		reseq1 5932	. . . . . 6 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
4	3	eqeq2d 2747	. . . . 5 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
5	4	cbvrexvw 3215	. . . 4 ⊢ (∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
6	2, 5	bitrdi 287	. . 3 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
7	6	cbvabv 2806	. 2 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8		rexeq 3292	. . . . . 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 2802	. . . . 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 481	. . . 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 2740	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
12	11	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)))
13	12	rexbidv 3160	. . . . . . . . 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 3653	. . . . . . . 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 3166	. . . . . . . . . 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 1849	. . . . . . . . 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 3263	. . . . . . . . . 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 468	. . . . . . . . . . . . . 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 1849	. . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
21	20	resex 5988	. . . . . . . . . . . . . 14 ⊢ (𝑒 ↾ (1...𝑀)) ∈ V
22		reseq1 5932	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
23	22	eqeq2d 2747	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
24	23	anbi2d 630	. . . . . . . . . . . . . 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 3490	. . . . . . . . . . . . 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 275	. . . . . . . . . . . 12 ⊢ (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27		ancom 460	. . . . . . . . . . . . 13 ⊢ (((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0))
28		simpl2 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ≥‘𝑁))
29		fzss2 13480	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑀))
30		resabs1 5965	. . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . 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 631	. . . . . . . . . . . . 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 283	. . . . . . . . . . . 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 283	. . . . . . . . . . 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 3160	. . . . . . . . . 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 285	. . . . . . . . 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 285	. . . . . . . 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 283	. . . . . . 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 2802	. . . . . 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 43008	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)} ∈ (Dioph‘𝑁))
42	41	3ad2antl1 1186	. . . . . 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 2836	. . . . 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 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 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
45	10, 44	eqeltrd 2836	. . 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 43007	. . . . 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 496	. . . 4 ⊢ (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)})
48	47	3ad2ant3 1135	. . 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 3144	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
50	7, 49	eqeltrrid 2841	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∃wrex 3060   ⊆ wss 3901   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  0cc0 11026  1c1 11027  ℕcn 12145  ℕ₀cn0 12401  ℤ_≥cuz 12751  ...cfz 13423  mzPolycmzp 42964  Diophcdioph 42997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254  df-mzpcl 42965  df-mzp 42966  df-dioph 42998

This theorem is referenced by:  rexrabdioph  43036