Theorem eldioph4b 36393

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 36345	. 2 ⊢ (𝑆 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		eldioph4b.a	. . . . . 6 ⊢ 𝑊 ∈ V
3		ovex 6577	. . . . . 6 ⊢ (1...𝑁) ∈ V
4	2, 3	unex 6854	. . . . 5 ⊢ (𝑊 ∪ (1...𝑁)) ∈ V
5	4	jctr 563	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ0 ∧ (𝑊 ∪ (1...𝑁)) ∈ V))
6		eldioph4b.b	. . . . . . 7 ⊢ ¬ 𝑊 ∈ Fin
7	6	intnanr 952	. . . . . 6 ⊢ ¬ (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin)
8		unfir 8113	. . . . . 6 ⊢ ((𝑊 ∪ (1...𝑁)) ∈ Fin → (𝑊 ∈ Fin ∧ (1...𝑁) ∈ Fin))
9	7, 8	mto 187	. . . . 5 ⊢ ¬ (𝑊 ∪ (1...𝑁)) ∈ Fin
10		ssun2 3739	. . . . 5 ⊢ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))
11	9, 10	pm3.2i 470	. . . 4 ⊢ (¬ (𝑊 ∪ (1...𝑁)) ∈ Fin ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁)))
12		eldioph2b 36344	. . . 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 693	. . 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 7768	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ (1...𝑁) ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑𝑚 (1...𝑁)))
15	10, 14	mpan2 703	. . . . . . . . . . . . . 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 3738	. . . . . . . . . . . . . . . 16 ⊢ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))
18		elmapssres 7768	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ 𝑊 ⊆ (𝑊 ∪ (1...𝑁))) → (𝑢 ↾ 𝑊) ∈ (ℕ0 ↑𝑚 𝑊))
19	17, 18	mpan2 703	. . . . . . . . . . . . . . 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 3719	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
22		resundi 5330	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ↾ (𝑊 ∪ (1...𝑁))) = ((𝑢 ↾ 𝑊) ∪ (𝑢 ↾ (1...𝑁)))
23	21, 22	eqtr4i 2635	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = (𝑢 ↾ (𝑊 ∪ (1...𝑁)))
24		elmapi 7765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → 𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0)
25		ffn 5958	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢:(𝑊 ∪ (1...𝑁))⟶ℕ0 → 𝑢 Fn (𝑊 ∪ (1...𝑁)))
26		fnresdm 5914	. . . . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)) = 𝑢)
29	28	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = (𝑝‘𝑢))
30	29	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0 ↔ (𝑝‘𝑢) = 0))
31	30	biimpar 501	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0)
32		uneq2 3723	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑢 ↾ (1...𝑁)) ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊)))
33	32	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))))
34	33	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑢 ↾ 𝑊) → ((𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ (𝑢 ↾ 𝑊))) = 0))
35	34	rspcev 3282	. . . . . . . . . . . . . 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 691	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))) ∧ (𝑝‘𝑢) = 0) → ∃𝑤 ∈ (ℕ0 ↑𝑚 𝑊)(𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0)
37	16, 36	jca 553	. . . . . . . . . . . 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 2676	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ↔ (𝑢 ↾ (1...𝑁)) ∈ (ℕ0 ↑𝑚 (1...𝑁))))
39		uneq1 3722	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑡 ∪ 𝑤) = ((𝑢 ↾ (1...𝑁)) ∪ 𝑤))
40	39	fveq2d 6107	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → (𝑝‘(𝑡 ∪ 𝑤)) = (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)))
41	40	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑢 ↾ (1...𝑁)) → ((𝑝‘(𝑡 ∪ 𝑤)) = 0 ↔ (𝑝‘((𝑢 ↾ (1...𝑁)) ∪ 𝑤)) = 0))
42	41	rexbidv 3034	. . . . . . . . . . . . 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 743	. . . . . . . . . . . 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 236	. . . . . . . . . . 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 627	. . . . . . . . . 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 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)))
47	46	rexlimiv 3009	. . . . . . . 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 3719	. . . . . . . . . . . 12 ⊢ (𝑡 ∪ 𝑤) = (𝑤 ∪ 𝑡)
49		fz1ssnn 12243	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ⊆ ℕ
50		sslin 3801	. . . . . . . . . . . . . . . . . . . 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 3600	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑊 ∩ (1...𝑁)) ⊆ ∅
54		ss0 3926	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∩ (1...𝑁)) ⊆ ∅ → (𝑊 ∩ (1...𝑁)) = ∅)
55	53, 54	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∩ (1...𝑁)) = ∅
56	55	reseq2i 5314	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑤 ↾ ∅)
57		res0 5321	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ↾ ∅) = ∅
58	56, 57	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = ∅
59	55	reseq2i 5314	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ ∅)
60		res0 5321	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ↾ ∅) = ∅
61	59, 60	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ↾ (𝑊 ∩ (1...𝑁))) = ∅
62	58, 61	eqtr4i 2635	. . . . . . . . . . . . . 14 ⊢ (𝑤 ↾ (𝑊 ∩ (1...𝑁))) = (𝑡 ↾ (𝑊 ∩ (1...𝑁)))
63		elmapresaun 36352	. . . . . . . . . . . . . 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 1405	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁))) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
65	64	ancoms 468	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → (𝑤 ∪ 𝑡) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
66	48, 65	syl5eqel 2692	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
67	66	adantr 480	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑡 ∪ 𝑤) ∈ (ℕ0 ↑𝑚 (𝑊 ∪ (1...𝑁))))
68	48	reseq1i 5313	. . . . . . . . . . . 12 ⊢ ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) = ((𝑤 ∪ 𝑡) ↾ (1...𝑁))
69		elmapresaunres2 36353	. . . . . . . . . . . . . 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 1405	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (ℕ0 ↑𝑚 𝑊) ∧ 𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁))) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
71	70	ancoms 468	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → ((𝑤 ∪ 𝑡) ↾ (1...𝑁)) = 𝑡)
72	68, 71	syl5req 2657	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
73	72	adantr 480	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
74		simpr 476	. . . . . . . . . 10 ⊢ (((𝑡 ∈ (ℕ0 ↑𝑚 (1...𝑁)) ∧ 𝑤 ∈ (ℕ0 ↑𝑚 𝑊)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0) → (𝑝‘(𝑡 ∪ 𝑤)) = 0)
75		reseq1 5311	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑢 ↾ (1...𝑁)) = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)))
76	75	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑡 = (𝑢 ↾ (1...𝑁)) ↔ 𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁))))
77		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → (𝑝‘𝑢) = (𝑝‘(𝑡 ∪ 𝑤)))
78	77	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑝‘𝑢) = 0 ↔ (𝑝‘(𝑡 ∪ 𝑤)) = 0))
79	76, 78	anbi12d 743	. . . . . . . . . . 11 ⊢ (𝑢 = (𝑡 ∪ 𝑤) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = ((𝑡 ∪ 𝑤) ↾ (1...𝑁)) ∧ (𝑝‘(𝑡 ∪ 𝑤)) = 0)))
80	79	rspcev 3282	. . . . . . . . . 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 1316	. . . . . . . . 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 3059	. . . . . . . 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 198	. . . . . . 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 2726	. . . . . 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 2905	. . . . . 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 2635	. . . . 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 2622	. . . 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 3023	. . 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 275	. 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 672	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816  ℕcn 10897  ℕ₀cn0 11169  ...cfz 12197  mzPolycmzp 36303  Diophcdioph 36336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-mzpcl 36304  df-mzp 36305  df-dioph 36337

This theorem is referenced by:  eldioph4i  36394  diophren  36395