Theorem diophrex 40597

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 2742	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
2	1	rexbidv 3226	. . . 4 ⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ 𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3		reseq1 5885	. . . . . 6 ⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
4	3	eqeq2d 2749	. . . . 5 ⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
5	4	cbvrexvw 3384	. . . 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 2811	. 2 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8		rexeq 3343	. . . . . 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 2807	. . . . 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 482	. . . 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 2742	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
12	11	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)))
13	12	rexbidv 3226	. . . . . . . . 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 3631	. . . . . . . 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 3276	. . . . . . . . . 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 1850	. . . . . . . . 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 3233	. . . . . . . . . 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 469	. . . . . . . . . . . . . 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 1850	. . . . . . . . . . . . 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 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑒 ∈ V
21	20	resex 5939	. . . . . . . . . . . . . 14 ⊢ (𝑒 ↾ (1...𝑀)) ∈ V
22		reseq1 5885	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
23	22	eqeq2d 2749	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
24	23	anbi2d 629	. . . . . . . . . . . . . 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 3479	. . . . . . . . . . . . 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 461	. . . . . . . . . . . . 13 ⊢ (((𝑐‘𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0))
28		simpl2 1191	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ≥‘𝑁))
29		fzss2 13296	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → (1...𝑁) ⊆ (1...𝑀))
30		resabs1 5921	. . . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . 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 630	. . . . . . . . . . . . 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	syl5bb 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	syl5bb 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 3226	. . . . . . . . . 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	syl5bb 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 2807	. . . . . 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 40588	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐‘𝑒) = 0)} ∈ (Dioph‘𝑁))
42	41	3ad2antl1 1184	. . . . . 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 2839	. . . . 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 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 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
45	10, 44	eqeltrd 2839	. . 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 40587	. . . . 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 497	. . . 4 ⊢ (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0 ↑m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐‘𝑒) = 0)})
48	47	3ad2ant3 1134	. . 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 3218	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ 𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
50	7, 49	eqeltrrid 2844	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ (ℤ≥‘𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢 ∈ 𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∃wrex 3065   ⊆ wss 3887   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  0cc0 10871  1c1 10872  ℕcn 11973  ℕ₀cn0 12233  ℤ_≥cuz 12582  ...cfz 13239  mzPolycmzp 40544  Diophcdioph 40577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-mzpcl 40545  df-mzp 40546  df-dioph 40578

This theorem is referenced by:  rexrabdioph  40616