Theorem eldioph4b 37375

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 37327	. 2 ⊢ (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		eldioph4b.a	. . . . . 6 ⊢ 𝑊 ∈ V
3		ovex 6678	. . . . . 6 ⊢ (1...𝑁) ∈ V
4	2, 3	unex 6956	. . . . 5 ⊢ (𝑊 ∪ (1...𝑁)) ∈ V
5	4	jctr 565	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6		eldioph4b.b	. . . . . . 7 ⊢ ¬ 𝑊 ∈ Fin
7	6	intnanr 961	. . . . . 6 ⊢ ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8		unfir 8228	. . . . . 6 ⊢ ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
9	7, 8	mto 188	. . . . 5 ⊢ ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10		ssun2 3777	. . . . 5 ⊢ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
11	9, 10	pm3.2i 471	. . . 4 ⊢ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12		eldioph2b 37326	. . . 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 694	. . 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 7882	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑𝑚 (1...𝑁)))
15	10, 14	mpan2 707	. . . . . . . . . . . . . 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 3776	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18		elmapssres 7882	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑𝑚 𝑊))
19	17, 18	mpan2 707	. . . . . . . . . . . . . . 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 3757	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22		resundi 5410	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
23	21, 22	eqtr4i 2647	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24		elmapi 7879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0 → 𝑢 Fn (𝑊 ∪ (1...𝑁)))
26		fnresdm 6000	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 Fn (𝑊 ∪ (1...𝑁)) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
27	24, 25, 26	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = 𝑢)
28	23, 27	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = 𝑢)
29	28	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = (𝑝‘𝑢))
30	29	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0 ↔ (𝑝‘𝑢) = 0))
31	30	biimpar 502	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0)
32		uneq2 3761	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)))
33	32	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))))
34	33	eqeq1d 2624	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0))
35	34	rspcev 3309	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ↾ 𝑊) ∈ (ℕ0 ↑𝑚 𝑊) ∧ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0) → ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
36	20, 31, 35	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
37	16, 36	jca 554	. . . . . . . . . . . 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))
38		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑𝑚 (1...𝑁))))
39		uneq1 3760	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
40	39	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑝‘(𝑡 ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)))
41	40	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
42	41	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
43	38, 42	anbi12d 747	. . . . . . . . . . . 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)))
44	37, 43	syl5ibrcom 237	. . . . . . . . . . 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)))
45	44	expimpd 629	. . . . . . . . . 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)))
46	45	ancomsd 470	. . . . . . . . 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)))
47	46	rexlimiv 3027	. . . . . . . 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))
48		uncom 3757	. . . . . . . . . . . 12 ⊢ (𝑡 ∪ 𝑤) = (𝑤 ∪ 𝑡)
49		fz1ssnn 12372	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ⊆ ℕ
50		sslin 3839	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...𝑁) ⊆ ℕ → (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ))
51	49, 50	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ (𝑊 ∩ ℕ)
52		eldioph4b.c	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∩ ℕ) = ∅
53	51, 52	sseqtri 3637	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ ∅
54		ss0 3974	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∩ (1...𝑁)) = ∅
56	55	reseq2i 5393	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
57		res0 5400	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ ∅) = ∅
58	56, 57	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
59	55	reseq2i 5393	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
60		res0 5400	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ ∅) = ∅
61	59, 60	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
62	58, 61	eqtr4i 2647	. . . . . . . . . . . . . 14 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
63		elmapresaun 37334	. . . . . . . . . . . . . 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))))
64	62, 63	mp3an3 1413	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁))) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
65	64	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
66	48, 65	syl5eqel 2705	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
67	66	adantr 481	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
68	48	reseq1i 5392	. . . . . . . . . . . 12 ⊢ ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) = ((𝑤 ∪ 𝑡) ↾ (1...𝑁))
69		elmapresaunres2 37335	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ (ℕ0 ↑𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
70	62, 69	mp3an3 1413	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
71	70	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
72	68, 71	syl5req 2669	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
73	72	adantr 481	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
74		simpr 477	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑝‘(𝑡 ∪ 𝑤)) = 0)
75		reseq1 5390	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
76	75	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁))))
77		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑝‘𝑢) = (𝑝‘(𝑡 ∪ 𝑤)))
78	77	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑝‘𝑢) = 0 ↔ (𝑝‘(𝑡 ∪ 𝑤)) = 0))
79	76, 78	anbi12d 747	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0)))
80	79	rspcev 3309	. . . . . . . . . 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))
81	67, 73, 74, 80	syl12anc 1324	. . . . . . . . 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))
82	81	r19.29an 3077	. . . . . . . 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))
83	47, 82	impbii 199	. . . . . . 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))
84	83	abbii 2739	. . . . . 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)}
85		df-rab 2921	. . . . . 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)}
86	84, 85	eqtr4i 2647	. . . . 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}
87	86	eqeq2i 2634	. . . 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})
88	87	rexbii 3041	. . 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})
89	13, 88	syl6bb 276	. 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}))
90	1, 89	biadan2 674	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  {crab 2916  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915   ↾ cres 5116   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  1c1 9937  ℕcn 11020  ℕ₀cn0 11292  ...cfz 12326  mzPolycmzp 37285  Diophcdioph 37318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-mzpcl 37286  df-mzp 37287  df-dioph 37319

This theorem is referenced by:  eldioph4i  37376  diophren  37377