Theorem eldioph4b 40633

Description: Membership in Dioph expressed using a quantified union to add witness variables instead of a restriction to remove them. (Contributed by Stefan O'Rear, 16-Oct-2014.)

Hypotheses

Ref	Expression
eldioph4b.a	⊢ 𝑊 ∈ V
eldioph4b.b	⊢ ¬ 𝑊 ∈ Fin
eldioph4b.c	⊢ (𝑊 ∩ ℕ) = ∅

Assertion

Ref	Expression
eldioph4b	⊢ (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0}))

Distinct variable groups:   𝑊,𝑝,𝑡,𝑤   𝑆,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤

Proof of Theorem eldioph4b

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldiophelnn0 40586	. 2 ⊢ (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		eldioph4b.a	. . . . . 6 ⊢ 𝑊 ∈ V
3		ovex 7308	. . . . . 6 ⊢ (1...𝑁) ∈ V
4	2, 3	unex 7596	. . . . 5 ⊢ (𝑊 ∪ (1...𝑁)) ∈ V
5	4	jctr 525	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6		eldioph4b.b	. . . . . . 7 ⊢ ¬ 𝑊 ∈ Fin
7	6	intnanr 488	. . . . . 6 ⊢ ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8		unfir 9082	. . . . . 6 ⊢ ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
9	7, 8	mto 196	. . . . 5 ⊢ ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10		ssun2 4107	. . . . 5 ⊢ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
11	9, 10	pm3.2i 471	. . . 4 ⊢ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12		eldioph2b 40585	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V) ∧ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))) → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
13	5, 11, 12	sylancl 586	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}))
14		elmapssres 8655	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
15	10, 14	mpan2 688	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
16	15	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
17		ssun1 4106	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18		elmapssres 8655	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
19	17, 18	mpan2 688	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
20	19	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
21		uncom 4087	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22		resundi 5905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
23	21, 22	eqtr4i 2769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24		elmapi 8637	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25		ffn 6600	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0 → 𝑢 Fn (𝑊 ∪ (1...𝑁)))
26		fnresdm 6551	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 Fn (𝑊 ∪ (1...𝑁)) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
27	24, 25, 26	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
28	23, 27	eqtrid 2790	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = 𝑢)
29	28	fveqeq2d 6782	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0 ↔ (𝑝‘𝑢) = 0))
30	29	biimpar 478	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0)
31		uneq2 4091	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)))
32	31	fveqeq2d 6782	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0))
33	32	rspcev 3561	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0) → ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
34	20, 30, 33	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
35	16, 34	jca 512	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
36		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
37		uneq1 4090	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
38	37	fveqeq2d 6782	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
39	38	rexbidv 3226	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
40	36, 39	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0) ↔ ((𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)))
41	35, 40	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0)))
42	41	expimpd 454	. . . . . . . . . 10 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (((𝑝‘𝑢) = 0 ∧ 𝑡 = (𝑢 ↾ (1...𝑁))) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0)))
43	42	ancomsd 466	. . . . . . . . 9 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0)))
44	43	rexlimiv 3209	. . . . . . . 8 ⊢ (∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0))
45		uncom 4087	. . . . . . . . . . . 12 ⊢ (𝑡 ∪ 𝑤) = (𝑤 ∪ 𝑡)
46		fz1ssnn 13287	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ⊆ ℕ
47		sslin 4168	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...𝑁) ⊆ ℕ → (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ))
48	46, 47	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ)
49		eldioph4b.c	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∩ ℕ) = ∅
50	48, 49	sseqtri 3957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ ∅
51		ss0 4332	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∩ (1...𝑁)) = ∅
53	52	reseq2i 5888	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
54		res0 5895	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ ∅) = ∅
55	53, 54	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
56	52	reseq2i 5888	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
57		res0 5895	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ ∅) = ∅
58	56, 57	eqtri 2766	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
59	55, 58	eqtr4i 2769	. . . . . . . . . . . . . 14 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
60		elmapresaun 8668	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
61	59, 60	mp3an3 1449	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
62	61	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
63	45, 62	eqeltrid 2843	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
64	63	adantr 481	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
65	45	reseq1i 5887	. . . . . . . . . . . 12 ⊢ ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) = ((𝑤 ∪ 𝑡) ↾ (1...𝑁))
66		elmapresaunres2 40593	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
67	59, 66	mp3an3 1449	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
68	67	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
69	65, 68	eqtr2id 2791	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
70	69	adantr 481	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
71		simpr 485	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑝‘(𝑡 ∪ 𝑤)) = 0)
72		reseq1 5885	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
73	72	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁))))
74		fveqeq2 6783	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑝‘𝑢) = 0 ↔ (𝑝‘(𝑡 ∪ 𝑤)) = 0))
75	73, 74	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0)))
76	75	rspcev 3561	. . . . . . . . . 10 ⊢ (((𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0)) → ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0))
77	64, 70, 71, 76	syl12anc 834	. . . . . . . . 9 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0))
78	77	r19.29an 3217	. . . . . . . 8 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0) → ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0))
79	44, 78	impbii 208	. . . . . . 7 ⊢ (∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0))
80	79	abbii 2808	. . . . . 6 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0)}
81		df-rab 3073	. . . . . 6 ⊢ {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0} = {𝑡 ∣ (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0)}
82	80, 81	eqtr4i 2769	. . . . 5 ⊢ {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0}
83	82	eqeq2i 2751	. . . 4 ⊢ (𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ 𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0})
84	83	rexbii 3181	. . 3 ⊢ (∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁)))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0})
85	13, 84	bitrdi 287	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑆 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0}))
86	1, 85	biadanii 819	1 ⊢ (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑝 ∈ (mzPoly‘(𝑊 ∪ (1...𝑁)))𝑆 = {𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0}))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  {crab 3068  Vcvv 3432   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256   ↾ cres 5591   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Fincfn 8733  0cc0 10871  1c1 10872  ℕcn 11973  ℕ₀cn0 12233  ...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-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:  eldioph4i  40634  diophren  40635