Theorem diophin 38296

Description: If two sets are Diophantine, so is their intersection. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Assertion

Ref	Expression
diophin	⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁))

Proof of Theorem diophin

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldiophelnn0 38287	. . 3 ⊢ (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2		id 22	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
3		zex 11737	. . . . . . 7 ⊢ ℤ ∈ V
4		difexg 5045	. . . . . . 7 ⊢ (ℤ ∈ V → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ V)
5	3, 4	mp1i 13	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ V)
6		ominf 8460	. . . . . . 7 ⊢ ¬ ω ∈ Fin
7		nn0z 11752	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
8		lzenom 38293	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ≈ ω)
9		enfi 8464	. . . . . . . 8 ⊢ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ≈ ω → ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ Fin ↔ ω ∈ Fin))
10	7, 8, 9	3syl 18	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ Fin ↔ ω ∈ Fin))
11	6, 10	mtbiri 319	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ¬ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ Fin)
12		fz1eqin 38292	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ))
13		inss1 4053	. . . . . . 7 ⊢ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ) ⊆ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))
14	12, 13	syl6eqss 3874	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))
15		eldioph2b 38286	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ V) ∧ (¬ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∈ Fin ∧ (1...𝑁) ⊆ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)}))
16	2, 5, 11, 14, 15	syl22anc 829	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)}))
17		nnex 11381	. . . . . . 7 ⊢ ℕ ∈ V
18	17	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ℕ ∈ V)
19		1z 11759	. . . . . . 7 ⊢ 1 ∈ ℤ
20		nnuz 12029	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
21	20	uzinf 13083	. . . . . . 7 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
22	19, 21	mp1i 13	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → ¬ ℕ ∈ Fin)
23		elfznn 12687	. . . . . . . 8 ⊢ (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
24	23	ssriv 3825	. . . . . . 7 ⊢ (1...𝑁) ⊆ ℕ
25	24	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ)
26		eldioph2b 38286	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}))
27	2, 18, 22, 25, 26	syl22anc 829	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}))
28	16, 27	anbi12d 624	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) ↔ (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)})))
29		reeanv 3293	. . . . 5 ⊢ (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑏 ∈ (mzPoly‘ℕ)(𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) ↔ (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}))
30		inab 4121	. . . . . . . . 9 ⊢ ({𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) = {𝑐 ∣ (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))}
31		reeanv 3293	. . . . . . . . . . 11 ⊢ (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) ↔ (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)))
32		simplrl 767	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → 𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
33		simplrr 768	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))
34	12	eqcomd 2784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ) = (1...𝑁))
35	34	reseq2d 5642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → (𝑑 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
36	35	ad3antrrr 720	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑑 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
37	34	reseq2d 5642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → (𝑒 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ (1...𝑁)))
38	37	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑒 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ (1...𝑁)))
39		simprrl 771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → 𝑐 = (𝑒 ↾ (1...𝑁)))
40		simprll 769	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → 𝑐 = (𝑑 ↾ (1...𝑁)))
41	38, 39, 40	3eqtr2d 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑒 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
42	36, 41	eqtr4d 2817	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑑 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)))
43		elmapresaun 38294	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (𝑑 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ))) → (𝑑 ∪ 𝑒) ∈ (ℕ0 ↑𝑚 ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ ℕ)))
44	32, 33, 42, 43	syl3anc 1439	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑑 ∪ 𝑒) ∈ (ℕ0 ↑𝑚 ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ ℕ)))
45	20	uneq2i 3987	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ ℕ) = ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ (ℤ≥‘1))
46	19	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℤ)
47		nn0p1nn 11683	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
48	47	nnge1d 11423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ (𝑁 + 1))
49		lzunuz 38291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 1 ≤ (𝑁 + 1)) → ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ (ℤ≥‘1)) = ℤ)
50	7, 46, 48, 49	syl3anc 1439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ (ℤ≥‘1)) = ℤ)
51	45, 50	syl5eq 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ ℕ) = ℤ)
52	51	oveq2d 6938	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ0 → (ℕ0 ↑𝑚 ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ ℕ)) = (ℕ0 ↑𝑚 ℤ))
53	52	ad3antrrr 720	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (ℕ0 ↑𝑚 ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∪ ℕ)) = (ℕ0 ↑𝑚 ℤ))
54	44, 53	eleqtrd 2861	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑑 ∪ 𝑒) ∈ (ℕ0 ↑𝑚 ℤ))
55		unidm 3979	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 ∪ 𝑐) = 𝑐
56	40, 39	uneq12d 3991	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑐 ∪ 𝑐) = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁))))
57	55, 56	syl5eqr 2828	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → 𝑐 = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁))))
58		resundir 5661	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∪ 𝑒) ↾ (1...𝑁)) = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁)))
59	57, 58	syl6eqr 2832	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → 𝑐 = ((𝑑 ∪ 𝑒) ↾ (1...𝑁)))
60		uncom 3980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∪ 𝑒) = (𝑒 ∪ 𝑑)
61	60	reseq1i 5638	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = ((𝑒 ∪ 𝑑) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))
62		incom 4028	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)
63	62, 34	syl5eq 2826	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = (1...𝑁))
64	63	reseq2d 5642	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ0 → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
65	64	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
66	63	reseq2d 5642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ0 → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑑 ↾ (1...𝑁)))
67	66	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑑 ↾ (1...𝑁)))
68	67, 40, 39	3eqtr2d 2820	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
69	65, 68	eqtr4d 2817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))))
70		elmapresaunres2 38295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑒 ∈ (ℕ0 ↑𝑚 ℕ) ∧ 𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))) → ((𝑒 ∪ 𝑑) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = 𝑑)
71	33, 32, 69, 70	syl3anc 1439	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → ((𝑒 ∪ 𝑑) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = 𝑑)
72	61, 71	syl5eq 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → ((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = 𝑑)
73	72	fveq2d 6450	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑎‘𝑑))
74		simprlr 770	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑎‘𝑑) = 0)
75	73, 74	eqtrd 2814	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0)
76		elmapresaunres2 38295	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ) ∧ (𝑑 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ ((ℤ ∖ (ℤ≥‘(𝑁 + 1))) ∩ ℕ))) → ((𝑑 ∪ 𝑒) ↾ ℕ) = 𝑒)
77	32, 33, 42, 76	syl3anc 1439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → ((𝑑 ∪ 𝑒) ↾ ℕ) = 𝑒)
78	77	fveq2d 6450	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = (𝑏‘𝑒))
79		simprrr 772	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑏‘𝑒) = 0)
80	78, 79	eqtrd 2814	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = 0)
81	59, 75, 80	jca32 511	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → (𝑐 = ((𝑑 ∪ 𝑒) ↾ (1...𝑁)) ∧ ((𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = 0)))
82		reseq1 5636	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → (𝑓 ↾ (1...𝑁)) = ((𝑑 ∪ 𝑒) ↾ (1...𝑁)))
83	82	eqeq2d 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → (𝑐 = (𝑓 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑑 ∪ 𝑒) ↾ (1...𝑁))))
84		reseq1 5636	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = ((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
85	84	fveqeq2d 6454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ↔ (𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0))
86		reseq1 5636	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → (𝑓 ↾ ℕ) = ((𝑑 ∪ 𝑒) ↾ ℕ))
87	86	fveqeq2d 6454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → ((𝑏‘(𝑓 ↾ ℕ)) = 0 ↔ (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = 0))
88	85, 87	anbi12d 624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0) ↔ ((𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = 0)))
89	83, 88	anbi12d 624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑑 ∪ 𝑒) → ((𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) ↔ (𝑐 = ((𝑑 ∪ 𝑒) ↾ (1...𝑁)) ∧ ((𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = 0))))
90	89	rspcev 3511	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 ∪ 𝑒) ∈ (ℕ0 ↑𝑚 ℤ) ∧ (𝑐 = ((𝑑 ∪ 𝑒) ↾ (1...𝑁)) ∧ ((𝑎‘((𝑑 ∪ 𝑒) ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑 ∪ 𝑒) ↾ ℕ)) = 0))) → ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
91	54, 81, 90	syl2anc 579	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))) → ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
92	91	ex 403	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0 ↑𝑚 ℕ))) → (((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) → ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
93	92	rexlimdvva 3221	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) → ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
94		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → 𝑓 ∈ (ℕ0 ↑𝑚 ℤ))
95		difss 3960	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ⊆ ℤ
96		elmapssres 8165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (ℕ0 ↑𝑚 ℤ) ∧ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
97	94, 95, 96	sylancl 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
98	97	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
99		nnssz 11748	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ⊆ ℤ
100		elmapssres 8165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (ℕ0 ↑𝑚 ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℕ0 ↑𝑚 ℕ))
101	94, 99, 100	sylancl 580	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑓 ↾ ℕ) ∈ (ℕ0 ↑𝑚 ℕ))
102	101	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑓 ↾ ℕ) ∈ (ℕ0 ↑𝑚 ℕ))
103		simprl 761	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → 𝑐 = (𝑓 ↾ (1...𝑁)))
104	14	ad3antrrr 720	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (1...𝑁) ⊆ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))
105	104	resabs1d 5677	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
106	103, 105	eqtr4d 2817	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → 𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)))
107		simprrl 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0)
108	106, 107	jca 507	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0))
109		resabs1 5676	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...𝑁) ⊆ ℕ → ((𝑓 ↾ ℕ) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
110	24, 109	mp1i 13	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ((𝑓 ↾ ℕ) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
111	103, 110	eqtr4d 2817	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → 𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)))
112		simprrr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑏‘(𝑓 ↾ ℕ)) = 0)
113	108, 111, 112	jca32 511	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0) ∧ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
114		reseq1 5636	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) → (𝑑 ↾ (1...𝑁)) = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)))
115	114	eqeq2d 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) → (𝑐 = (𝑑 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁))))
116		fveqeq2 6455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) → ((𝑎‘𝑑) = 0 ↔ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0))
117	115, 116	anbi12d 624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) → ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ↔ (𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0)))
118	117	anbi1d 623	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) → (((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) ↔ ((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))))
119		reseq1 5636	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = (𝑓 ↾ ℕ) → (𝑒 ↾ (1...𝑁)) = ((𝑓 ↾ ℕ) ↾ (1...𝑁)))
120	119	eqeq2d 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = (𝑓 ↾ ℕ) → (𝑐 = (𝑒 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁))))
121		fveqeq2 6455	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = (𝑓 ↾ ℕ) → ((𝑏‘𝑒) = 0 ↔ (𝑏‘(𝑓 ↾ ℕ)) = 0))
122	120, 121	anbi12d 624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = (𝑓 ↾ ℕ) → ((𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0) ↔ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
123	122	anbi2d 622	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝑓 ↾ ℕ) → (((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) ↔ ((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0) ∧ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
124	118, 123	rspc2ev 3526	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ (𝑓 ↾ ℕ) ∈ (ℕ0 ↑𝑚 ℕ) ∧ ((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0) ∧ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)))
125	98, 102, 113, 124	syl3anc 1439	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)))
126	125	rexlimdva2 3216	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) → ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))))
127	93, 126	impbid 204	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) ↔ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
128		simplrl 767	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
129		mzpf 38259	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) → 𝑎:(ℤ ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))⟶ℤ)
130	128, 129	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → 𝑎:(ℤ ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))⟶ℤ)
131		nn0ssz 11750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ0 ⊆ ℤ
132		mapss 8186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0 ↑𝑚 ℤ) ⊆ (ℤ ↑𝑚 ℤ))
133	3, 131, 132	mp2an 682	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℕ0 ↑𝑚 ℤ) ⊆ (ℤ ↑𝑚 ℤ)
134	133	sseli 3817	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 ℤ) → 𝑓 ∈ (ℤ ↑𝑚 ℤ))
135		elmapssres 8165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℤ ↑𝑚 ℤ) ∧ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
136	134, 95, 135	sylancl 580	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
137	136	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
138	130, 137	ffvelrnd 6624	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) ∈ ℤ)
139	138	zred 11834	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) ∈ ℝ)
140		simplrr 768	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → 𝑏 ∈ (mzPoly‘ℕ))
141		mzpf 38259	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ (mzPoly‘ℕ) → 𝑏:(ℤ ↑𝑚 ℕ)⟶ℤ)
142	140, 141	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → 𝑏:(ℤ ↑𝑚 ℕ)⟶ℤ)
143		elmapssres 8165	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∈ (ℤ ↑𝑚 ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
144	134, 99, 143	sylancl 580	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (ℕ0 ↑𝑚 ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
145	144	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
146	142, 145	ffvelrnd 6624	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℤ)
147	146	zred 11834	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℝ)
148		sumsqeq0 13261	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) ∈ ℝ ∧ (𝑏‘(𝑓 ↾ ℕ)) ∈ ℝ) → (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0) ↔ (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) = 0))
149	139, 147, 148	syl2anc 579	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0) ↔ (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) = 0))
150	134	adantl 475	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → 𝑓 ∈ (ℤ ↑𝑚 ℤ))
151		reseq1 5636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑓 → (𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))) = (𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
152	151	fveq2d 6450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑓 → (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))))
153	152	oveq1d 6937	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) = ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2))
154		reseq1 5636	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = 𝑓 → (𝑔 ↾ ℕ) = (𝑓 ↾ ℕ))
155	154	fveq2d 6450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = 𝑓 → (𝑏‘(𝑔 ↾ ℕ)) = (𝑏‘(𝑓 ↾ ℕ)))
156	155	oveq1d 6937	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → ((𝑏‘(𝑔 ↾ ℕ))↑2) = ((𝑏‘(𝑓 ↾ ℕ))↑2))
157	153, 156	oveq12d 6940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑓 → (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)) = (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)))
158		eqid 2778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2))) = (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))
159		ovex 6954	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) ∈ V
160	157, 158, 159	fvmpt 6542	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ (ℤ ↑𝑚 ℤ) → ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)))
161	150, 160	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)))
162	161	eqeq1d 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0 ↔ (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) = 0))
163	149, 162	bitr4d 274	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0) ↔ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0))
164	163	anbi2d 622	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0 ↑𝑚 ℤ)) → ((𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) ↔ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
165	164	rexbidva 3234	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) ↔ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
166	127, 165	bitrd 271	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) ↔ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
167	31, 166	syl5bbr 277	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ((∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)) ↔ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
168	167	abbidv 2906	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → {𝑐 ∣ (∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0))} = {𝑐 ∣ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)})
169	30, 168	syl5eq 2826	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ({𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) = {𝑐 ∣ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)})
170		simpl 476	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
171		fzssuz 12699	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ (ℤ≥‘1)
172		uzssz 12012	. . . . . . . . . . . 12 ⊢ (ℤ≥‘1) ⊆ ℤ
173	171, 172	sstri 3830	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℤ
174	3, 173	pm3.2i 464	. . . . . . . . . 10 ⊢ (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ)
175	174	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ))
176	3	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ℤ ∈ V)
177	95	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ⊆ ℤ)
178		simprl 761	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))))
179		mzpresrename 38273	. . . . . . . . . . . 12 ⊢ ((ℤ ∈ V ∧ (ℤ ∖ (ℤ≥‘(𝑁 + 1))) ⊆ ℤ ∧ 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))) ∈ (mzPoly‘ℤ))
180	176, 177, 178, 179	syl3anc 1439	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))) ∈ (mzPoly‘ℤ))
181		2nn0 11661	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ0
182		mzpexpmpt 38268	. . . . . . . . . . 11 ⊢ (((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))) ∈ (mzPoly‘ℤ) ∧ 2 ∈ ℕ0) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2)) ∈ (mzPoly‘ℤ))
183	180, 181, 182	sylancl 580	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2)) ∈ (mzPoly‘ℤ))
184	99	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ℕ ⊆ ℤ)
185		simprr 763	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑏 ∈ (mzPoly‘ℕ))
186		mzpresrename 38273	. . . . . . . . . . . 12 ⊢ ((ℤ ∈ V ∧ ℕ ⊆ ℤ ∧ 𝑏 ∈ (mzPoly‘ℕ)) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
187	176, 184, 185, 186	syl3anc 1439	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
188		mzpexpmpt 38268	. . . . . . . . . . 11 ⊢ (((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ) ∧ 2 ∈ ℕ0) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
189	187, 181, 188	sylancl 580	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
190		mzpaddmpt 38264	. . . . . . . . . 10 ⊢ (((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2)) ∈ (mzPoly‘ℤ) ∧ (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ)) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2))) ∈ (mzPoly‘ℤ))
191	183, 189, 190	syl2anc 579	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2))) ∈ (mzPoly‘ℤ))
192		eldioph2 38285	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ) ∧ (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2))) ∈ (mzPoly‘ℤ)) → {𝑐 ∣ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)} ∈ (Dioph‘𝑁))
193	170, 175, 191, 192	syl3anc 1439	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → {𝑐 ∣ ∃𝑓 ∈ (ℕ0 ↑𝑚 ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ≥‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)} ∈ (Dioph‘𝑁))
194	169, 193	eqeltrd 2859	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ({𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) ∈ (Dioph‘𝑁))
195		ineq12 4032	. . . . . . . 8 ⊢ ((𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) → (𝐴 ∩ 𝐵) = ({𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}))
196	195	eleq1d 2844	. . . . . . 7 ⊢ ((𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) → ((𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁) ↔ ({𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) ∈ (Dioph‘𝑁)))
197	194, 196	syl5ibrcom 239	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ((𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)))
198	197	rexlimdvva 3221	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))∃𝑏 ∈ (mzPoly‘ℕ)(𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)))
199	29, 198	syl5bir 235	. . . 4 ⊢ (𝑁 ∈ ℕ0 → ((∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ≥‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0 ↑𝑚 (ℤ ∖ (ℤ≥‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎‘𝑑) = 0)} ∧ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0 ↑𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏‘𝑒) = 0)}) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)))
200	28, 199	sylbid 232	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)))
201	1, 200	syl 17	. 2 ⊢ (𝐴 ∈ (Dioph‘𝑁) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁)))
202	201	anabsi5 659	1 ⊢ ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴 ∩ 𝐵) ∈ (Dioph‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {cab 2763  ∃wrex 3091  Vcvv 3398   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792   class class class wbr 4886   ↦ cmpt 4965   ↾ cres 5357  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922  ωcom 7343   ↑_𝑚 cmap 8140   ≈ cen 8238  Fincfn 8241  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   ≤ cle 10412  ℕcn 11374  2c2 11430  ℕ₀cn0 11642  ℤcz 11728  ℤ_≥cuz 11992  ...cfz 12643  ↑cexp 13178  mzPolycmzp 38245  Diophcdioph 38278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-seq 13120  df-exp 13179  df-hash 13436  df-mzpcl 38246  df-mzp 38247  df-dioph 38279

This theorem is referenced by:  anrabdioph  38304