Theorem diophrex 42895

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 2737	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
2	1	rexbidv 3157	. . . 4 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3		reseq1 5928	. . . . . 6 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
4	3	eqeq2d 2744	. . . . 5 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
5	4	cbvrexvw 3212	. . . 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 2803	. 2 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8		rexeq 3289	. . . . . 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 2799	. . . . 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 2737	. . . . . . . . . . 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 3157	. . . . . . . . 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 3650	. . . . . . . 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 3163	. . . . . . . . . 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 3260	. . . . . . . . . 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 3441	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
21	20	resex 5984	. . . . . . . . . . . . . 14 ⊢ (𝑒 ↾ (1...𝑀)) ∈ V
22		reseq1 5928	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
23	22	eqeq2d 2744	. . . . . . . . . . . . . . 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 3487	. . . . . . . . . . . . 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 13468	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑀))
30		resabs1 5961	. . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 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 3157	. . . . . . . . . 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 2799	. . . . . 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 42886	. . . . . . 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 2833	. . . . 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 2833	. . 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 42885	. . . . 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 3141	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
50	7, 49	eqeltrrid 2838	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 2711  ∃wrex 3057   ⊆ wss 3898   ↾ cres 5623  ‘cfv 6488  (class class class)co 7354   ↑_m cmap 8758  0cc0 11015  1c1 11016  ℕcn 12134  ℕ₀cn0 12390  ℤ_≥cuz 12740  ...cfz 13411  mzPolycmzp 42842  Diophcdioph 42875

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 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-inf2 9540  ax-cnex 11071  ax-resscn 11072  ax-1cn 11073  ax-icn 11074  ax-addcl 11075  ax-addrcl 11076  ax-mulcl 11077  ax-mulrcl 11078  ax-mulcom 11079  ax-addass 11080  ax-mulass 11081  ax-distr 11082  ax-i2m1 11083  ax-1ne0 11084  ax-1rid 11085  ax-rnegex 11086  ax-rrecex 11087  ax-cnre 11088  ax-pre-lttri 11089  ax-pre-lttrn 11090  ax-pre-ltadd 11091  ax-pre-mulgt0 11092

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-of 7618  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-oadd 8397  df-er 8630  df-map 8760  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-dju 9803  df-card 9841  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161  df-sub 11355  df-neg 11356  df-nn 12135  df-n0 12391  df-z 12478  df-uz 12741  df-fz 13412  df-hash 14242  df-mzpcl 42843  df-mzp 42844  df-dioph 42876

This theorem is referenced by:  rexrabdioph  42914