Theorem eldioph4b 42804

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 42757	. 2 ⊢ (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		eldioph4b.a	. . . . . 6 ⊢ 𝑊 ∈ V
3		ovex 7382	. . . . . 6 ⊢ (1...𝑁) ∈ V
4	2, 3	unex 7680	. . . . 5 ⊢ (𝑊 ∪ (1...𝑁)) ∈ V
5	4	jctr 524	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6		eldioph4b.b	. . . . . . 7 ⊢ ¬ 𝑊 ∈ Fin
7	6	intnanr 487	. . . . . 6 ⊢ ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8		unfir 9197	. . . . . 6 ⊢ ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
9	7, 8	mto 197	. . . . 5 ⊢ ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10		ssun2 4130	. . . . 5 ⊢ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
11	9, 10	pm3.2i 470	. . . 4 ⊢ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12		eldioph2b 42756	. . . 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 8794	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
15	10, 14	mpan2 691	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
16	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
17		ssun1 4129	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18		elmapssres 8794	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
19	17, 18	mpan2 691	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑m 𝑊))
21		uncom 4109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22		resundi 5944	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
23	21, 22	eqtr4i 2755	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24		elmapi 8776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25		ffn 6652	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0 → 𝑢 Fn (𝑊 ∪ (1...𝑁)))
26		fnresdm 6601	. . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = 𝑢)
29	28	fveqeq2d 6830	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0 ↔ (𝑝‘𝑢) = 0))
30	29	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0)
31		uneq2 4113	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)))
32	31	fveqeq2d 6830	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0))
33	32	rspcev 3577	. . . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 2816	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁))))
37		uneq1 4112	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
38	37	fveqeq2d 6830	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
39	38	rexbidv 3153	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 247	. . . . . . . . . . 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 453	. . . . . . . . . 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 465	. . . . . . . . 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 3123	. . . . . . . 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 4109	. . . . . . . . . . . 12 ⊢ (𝑡 ∪ 𝑤) = (𝑤 ∪ 𝑡)
46		fz1ssnn 13458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ⊆ ℕ
47		sslin 4194	. . . . . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ ∅
51		ss0 4353	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∩ (1...𝑁)) = ∅
53	52	reseq2i 5927	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
54		res0 5934	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ ∅) = ∅
55	53, 54	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
56	52	reseq2i 5927	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
57		res0 5934	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ ∅) = ∅
58	56, 57	eqtri 2752	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
59	55, 58	eqtr4i 2755	. . . . . . . . . . . . . 14 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
60		elmapresaun 8807	. . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
62	61	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
63	45, 62	eqeltrid 2832	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
64	63	adantr 480	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑m (𝑊 ∪ (1...𝑁))))
65	45	reseq1i 5926	. . . . . . . . . . . 12 ⊢ ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) = ((𝑤 ∪ 𝑡) ↾ (1...𝑁))
66		elmapresaunres2 42764	. . . . . . . . . . . . . 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 1452	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑m 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑m (1...𝑁))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
68	67	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
69	65, 68	eqtr2id 2777	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
70	69	adantr 480	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
71		simpr 484	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑m 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑝‘(𝑡 ∪ 𝑤)) = 0)
72		reseq1 5924	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
73	72	eqeq2d 2740	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁))))
74		fveqeq2 6831	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑝‘𝑢) = 0 ↔ (𝑝‘(𝑡 ∪ 𝑤)) = 0))
75	73, 74	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0)))
76	75	rspcev 3577	. . . . . . . . . 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 836	. . . . . . . . 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 3133	. . . . . . . 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 209	. . . . . . 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 2796	. . . . . 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 3395	. . . . . 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 2755	. . . . 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 2742	. . . 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 3076	. . 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 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  {crab 3394  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284   ↾ cres 5621   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  Fincfn 8872  0cc0 11009  1c1 11010  ℕcn 12128  ℕ₀cn0 12384  ...cfz 13410  mzPolycmzp 42715  Diophcdioph 42748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-hash 14238  df-mzpcl 42716  df-mzp 42717  df-dioph 42749

This theorem is referenced by:  eldioph4i  42805  diophren  42806