Theorem eldioph4b 39752

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 39705	. 2 ⊢ (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		eldioph4b.a	. . . . . 6 ⊢ 𝑊 ∈ V
3		ovex 7168	. . . . . 6 ⊢ (1...𝑁) ∈ V
4	2, 3	unex 7449	. . . . 5 ⊢ (𝑊 ∪ (1...𝑁)) ∈ V
5	4	jctr 528	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6		eldioph4b.b	. . . . . . 7 ⊢ ¬ 𝑊 ∈ Fin
7	6	intnanr 491	. . . . . 6 ⊢ ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8		unfir 8770	. . . . . 6 ⊢ ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
9	7, 8	mto 200	. . . . 5 ⊢ ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10		ssun2 4100	. . . . 5 ⊢ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
11	9, 10	pm3.2i 474	. . . 4 ⊢ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12		eldioph2b 39704	. . . 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 589	. . 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 8414	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
15	10, 14	mpan2 690	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
16	15	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
17		ssun1 4099	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18		elmapssres 8414	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
19	17, 18	mpan2 690	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
20	19	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
21		uncom 4080	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22		resundi 5832	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
23	21, 22	eqtr4i 2824	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24		elmapi 8411	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25		ffn 6487	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0 → 𝑢 Fn (𝑊 ∪ (1...𝑁)))
26		fnresdm 6438	. . . . . . . . . . . . . . . . . 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	syl5eq 2845	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = 𝑢)
29	28	fveqeq2d 6653	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0 ↔ (𝑝‘𝑢) = 0))
30	29	biimpar 481	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0)
31		uneq2 4084	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)))
32	31	fveqeq2d 6653	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0))
33	32	rspcev 3571	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → ∃𝑤 ∈ (ℕ0 ↑m 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
35	16, 34	jca 515	. . . . . . . . . . . 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 2877	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
37		uneq1 4083	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
38	37	fveqeq2d 6653	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
39	38	rexbidv 3256	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 250	. . . . . . . . . . 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 457	. . . . . . . . . 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 469	. . . . . . . . 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 3239	. . . . . . . 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 4080	. . . . . . . . . . . 12 ⊢ (𝑡 ∪ 𝑤) = (𝑤 ∪ 𝑡)
46		fz1ssnn 12933	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ⊆ ℕ
47		sslin 4161	. . . . . . . . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ ∅
51		ss0 4306	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∩ (1...𝑁)) = ∅
53	52	reseq2i 5815	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
54		res0 5822	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ ∅) = ∅
55	53, 54	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
56	52	reseq2i 5815	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
57		res0 5822	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ ∅) = ∅
58	56, 57	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
59	55, 58	eqtr4i 2824	. . . . . . . . . . . . . 14 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
60		elmapresaun 8427	. . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
62	61	ancoms 462	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
63	45, 62	eqeltrid 2894	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
64	63	adantr 484	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
65	45	reseq1i 5814	. . . . . . . . . . . 12 ⊢ ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) = ((𝑤 ∪ 𝑡) ↾ (1...𝑁))
66		elmapresaunres2 39712	. . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
68	67	ancoms 462	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
69	65, 68	syl5req 2846	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
70	69	adantr 484	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
71		simpr 488	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑝‘(𝑡 ∪ 𝑤)) = 0)
72		reseq1 5812	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
73	72	eqeq2d 2809	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁))))
74		fveqeq2 6654	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑝‘𝑢) = 0 ↔ (𝑝‘(𝑡 ∪ 𝑤)) = 0))
75	73, 74	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0)))
76	75	rspcev 3571	. . . . . . . . . 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 835	. . . . . . . . 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 3247	. . . . . . . 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 212	. . . . . . 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 2863	. . . . . 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 3115	. . . . . 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 2824	. . . . 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 2811	. . . 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 3210	. . 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	syl6bb 290	. 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 821	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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  {crab 3110  Vcvv 3441   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243   ↾ cres 5521   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  Fincfn 8492  0cc0 10526  1c1 10527  ℕcn 11625  ℕ₀cn0 11885  ...cfz 12885  mzPolycmzp 39663  Diophcdioph 39696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-mzpcl 39664  df-mzp 39665  df-dioph 39697

This theorem is referenced by:  eldioph4i  39753  diophren  39754