Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diophin Structured version   Visualization version   GIF version

Theorem diophin 39560
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)) → (𝐴𝐵) ∈ (Dioph‘𝑁))

Proof of Theorem diophin
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldiophelnn0 39552 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 id 22 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
3 zex 11978 . . . . . . 7 ℤ ∈ V
4 difexg 5214 . . . . . . 7 (ℤ ∈ V → (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ V)
53, 4mp1i 13 . . . . . 6 (𝑁 ∈ ℕ0 → (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ V)
6 ominf 8716 . . . . . . 7 ¬ ω ∈ Fin
7 nn0z 11993 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
8 lzenom 39558 . . . . . . . 8 (𝑁 ∈ ℤ → (ℤ ∖ (ℤ‘(𝑁 + 1))) ≈ ω)
9 enfi 8720 . . . . . . . 8 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ≈ ω → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ Fin ↔ ω ∈ Fin))
107, 8, 93syl 18 . . . . . . 7 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ Fin ↔ ω ∈ Fin))
116, 10mtbiri 330 . . . . . 6 (𝑁 ∈ ℕ0 → ¬ (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ Fin)
12 fz1eqin 39557 . . . . . . 7 (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))
13 inss1 4188 . . . . . . 7 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1)))
1412, 13eqsstrdi 4005 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1))))
15 eldioph2b 39551 . . . . . 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 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)}))
162, 5, 11, 14, 15syl22anc 837 . . . . 5 (𝑁 ∈ ℕ0 → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)}))
17 nnex 11631 . . . . . . 7 ℕ ∈ V
1817a1i 11 . . . . . 6 (𝑁 ∈ ℕ0 → ℕ ∈ V)
19 1z 12000 . . . . . . 7 1 ∈ ℤ
20 nnuz 12269 . . . . . . . 8 ℕ = (ℤ‘1)
2120uzinf 13328 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
2219, 21mp1i 13 . . . . . 6 (𝑁 ∈ ℕ0 → ¬ ℕ ∈ Fin)
23 elfznn 12931 . . . . . . . 8 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
2423ssriv 3955 . . . . . . 7 (1...𝑁) ⊆ ℕ
2524a1i 11 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ)
26 eldioph2b 39551 . . . . . 6 (((𝑁 ∈ ℕ0 ∧ ℕ ∈ V) ∧ (¬ ℕ ∈ Fin ∧ (1...𝑁) ⊆ ℕ)) → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}))
272, 18, 22, 25, 26syl22anc 837 . . . . 5 (𝑁 ∈ ℕ0 → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}))
2816, 27anbi12d 633 . . . 4 (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) ↔ (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)})))
29 reeanv 3358 . . . . 5 (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑏 ∈ (mzPoly‘ℕ)(𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) ↔ (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}))
30 inab 4254 . . . . . . . . 9 ({𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) = {𝑐 ∣ (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))}
31 reeanv 3358 . . . . . . . . . . 11 (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) ↔ (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)))
32 simplrl 776 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → 𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
33 simplrr 777 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → 𝑒 ∈ (ℕ0m ℕ))
3412eqcomd 2830 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ) = (1...𝑁))
3534reseq2d 5836 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
3635ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
3734reseq2d 5836 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ (1...𝑁)))
3837ad3antrrr 729 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ (1...𝑁)))
39 simprrl 780 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → 𝑐 = (𝑒 ↾ (1...𝑁)))
40 simprll 778 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → 𝑐 = (𝑑 ↾ (1...𝑁)))
4138, 39, 403eqtr2d 2865 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
4236, 41eqtr4d 2862 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)))
43 elmapresaun 8429 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ) ∧ (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))) → (𝑑𝑒) ∈ (ℕ0m ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ)))
4432, 33, 42, 43syl3anc 1368 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑑𝑒) ∈ (ℕ0m ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ)))
4520uneq2i 4120 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1))
4619a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 1 ∈ ℤ)
47 nn0p1nn 11924 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
4847nnge1d 11673 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 1 ≤ (𝑁 + 1))
49 lzunuz 39556 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 1 ≤ (𝑁 + 1)) → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1)) = ℤ)
507, 46, 48, 49syl3anc 1368 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1)) = ℤ)
5145, 50syl5eq 2871 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ) = ℤ)
5251oveq2d 7156 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (ℕ0m ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ)) = (ℕ0m ℤ))
5352ad3antrrr 729 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (ℕ0m ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ)) = (ℕ0m ℤ))
5444, 53eleqtrd 2918 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑑𝑒) ∈ (ℕ0m ℤ))
55 unidm 4112 . . . . . . . . . . . . . . . . . . 19 (𝑐𝑐) = 𝑐
5640, 39uneq12d 4124 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑐𝑐) = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁))))
5755, 56syl5eqr 2873 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → 𝑐 = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁))))
58 resundir 5851 . . . . . . . . . . . . . . . . . 18 ((𝑑𝑒) ↾ (1...𝑁)) = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁)))
5957, 58syl6eqr 2877 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → 𝑐 = ((𝑑𝑒) ↾ (1...𝑁)))
60 uncom 4113 . . . . . . . . . . . . . . . . . . . . 21 (𝑑𝑒) = (𝑒𝑑)
6160reseq1i 5832 . . . . . . . . . . . . . . . . . . . 20 ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((𝑒𝑑) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))
62 incom 4161 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)
6362, 34syl5eq 2871 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = (1...𝑁))
6463reseq2d 5836 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
6564ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
6663reseq2d 5836 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑑 ↾ (1...𝑁)))
6766ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑑 ↾ (1...𝑁)))
6867, 40, 393eqtr2d 2865 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
6965, 68eqtr4d 2862 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
70 elmapresaunres2 39559 . . . . . . . . . . . . . . . . . . . . 21 ((𝑒 ∈ (ℕ0m ℕ) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))))) → ((𝑒𝑑) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = 𝑑)
7133, 32, 69, 70syl3anc 1368 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → ((𝑒𝑑) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = 𝑑)
7261, 71syl5eq 2871 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = 𝑑)
7372fveq2d 6657 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑎𝑑))
74 simprlr 779 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑎𝑑) = 0)
7573, 74eqtrd 2859 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0)
76 elmapresaunres2 39559 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ) ∧ (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))) → ((𝑑𝑒) ↾ ℕ) = 𝑒)
7732, 33, 42, 76syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → ((𝑑𝑒) ↾ ℕ) = 𝑒)
7877fveq2d 6657 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑏‘((𝑑𝑒) ↾ ℕ)) = (𝑏𝑒))
79 simprrr 781 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑏𝑒) = 0)
8078, 79eqtrd 2859 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0)
8159, 75, 80jca32 519 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → (𝑐 = ((𝑑𝑒) ↾ (1...𝑁)) ∧ ((𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0)))
82 reseq1 5830 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → (𝑓 ↾ (1...𝑁)) = ((𝑑𝑒) ↾ (1...𝑁)))
8382eqeq2d 2835 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑑𝑒) → (𝑐 = (𝑓 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑑𝑒) ↾ (1...𝑁))))
84 reseq1 5830 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑑𝑒) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))
8584fveqeq2d 6661 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ↔ (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
86 reseq1 5830 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑑𝑒) → (𝑓 ↾ ℕ) = ((𝑑𝑒) ↾ ℕ))
8786fveqeq2d 6661 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → ((𝑏‘(𝑓 ↾ ℕ)) = 0 ↔ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0))
8885, 87anbi12d 633 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑑𝑒) → (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0) ↔ ((𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0)))
8983, 88anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑓 = (𝑑𝑒) → ((𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) ↔ (𝑐 = ((𝑑𝑒) ↾ (1...𝑁)) ∧ ((𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0))))
9089rspcev 3608 . . . . . . . . . . . . . . . 16 (((𝑑𝑒) ∈ (ℕ0m ℤ) ∧ (𝑐 = ((𝑑𝑒) ↾ (1...𝑁)) ∧ ((𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0))) → ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
9154, 81, 90syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) ∧ ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))) → ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
9291ex 416 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ (𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑒 ∈ (ℕ0m ℕ))) → (((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) → ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
9392rexlimdvva 3286 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) → ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
94 simpr 488 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → 𝑓 ∈ (ℕ0m ℤ))
95 difss 4092 . . . . . . . . . . . . . . . . 17 (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ
96 elmapssres 8416 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (ℕ0m ℤ) ∧ (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
9794, 95, 96sylancl 589 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
9897adantr 484 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
99 nnssz 11990 . . . . . . . . . . . . . . . . 17 ℕ ⊆ ℤ
100 elmapssres 8416 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (ℕ0m ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℕ0m ℕ))
10194, 99, 100sylancl 589 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑓 ↾ ℕ) ∈ (ℕ0m ℕ))
102101adantr 484 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑓 ↾ ℕ) ∈ (ℕ0m ℕ))
103 simprl 770 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → 𝑐 = (𝑓 ↾ (1...𝑁)))
10414ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (1...𝑁) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1))))
105104resabs1d 5867 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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)))
106103, 105eqtr4d 2862 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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 780 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0)
108106, 107jca 515 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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 5866 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) ⊆ ℕ → ((𝑓 ↾ ℕ) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
11024, 109mp1i 13 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ((𝑓 ↾ ℕ) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
111103, 110eqtr4d 2862 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → 𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)))
112 simprrr 781 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → (𝑏‘(𝑓 ↾ ℕ)) = 0)
113108, 111, 112jca32 519 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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 5830 . . . . . . . . . . . . . . . . . . 19 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑑 ↾ (1...𝑁)) = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)))
115114eqeq2d 2835 . . . . . . . . . . . . . . . . . 18 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑐 = (𝑑 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁))))
116 fveqeq2 6662 . . . . . . . . . . . . . . . . . 18 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → ((𝑎𝑑) = 0 ↔ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
117115, 116anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → ((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ↔ (𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0)))
118117anbi1d 632 . . . . . . . . . . . . . . . 16 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) ↔ ((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))))
119 reseq1 5830 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝑓 ↾ ℕ) → (𝑒 ↾ (1...𝑁)) = ((𝑓 ↾ ℕ) ↾ (1...𝑁)))
120119eqeq2d 2835 . . . . . . . . . . . . . . . . . 18 (𝑒 = (𝑓 ↾ ℕ) → (𝑐 = (𝑒 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁))))
121 fveqeq2 6662 . . . . . . . . . . . . . . . . . 18 (𝑒 = (𝑓 ↾ ℕ) → ((𝑏𝑒) = 0 ↔ (𝑏‘(𝑓 ↾ ℕ)) = 0))
122120, 121anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑒 = (𝑓 ↾ ℕ) → ((𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0) ↔ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
123122anbi2d 631 . . . . . . . . . . . . . . . 16 (𝑒 = (𝑓 ↾ ℕ) → (((𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 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))))
124118, 123rspc2ev 3620 . . . . . . . . . . . . . . 15 (((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ (𝑓 ↾ ℕ) ∈ (ℕ0m 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 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)))
12598, 102, 113, 124syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) ∧ (𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))) → ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)))
126125rexlimdva2 3279 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) → ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))))
12793, 126impbid 215 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) ↔ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0))))
128 simplrl 776 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))))
129 mzpf 39524 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) → 𝑎:(ℤ ↑m (ℤ ∖ (ℤ‘(𝑁 + 1))))⟶ℤ)
130128, 129syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → 𝑎:(ℤ ↑m (ℤ ∖ (ℤ‘(𝑁 + 1))))⟶ℤ)
131 nn0ssz 11991 . . . . . . . . . . . . . . . . . . . . . 22 0 ⊆ ℤ
132 mapss 8438 . . . . . . . . . . . . . . . . . . . . . 22 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ℤ) ⊆ (ℤ ↑m ℤ))
1333, 131, 132mp2an 691 . . . . . . . . . . . . . . . . . . . . 21 (ℕ0m ℤ) ⊆ (ℤ ↑m ℤ)
134133sseli 3947 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (ℕ0m ℤ) → 𝑓 ∈ (ℤ ↑m ℤ))
135 elmapssres 8416 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (ℤ ↑m ℤ) ∧ (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
136134, 95, 135sylancl 589 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (ℕ0m ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
137136adantl 485 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑m (ℤ ∖ (ℤ‘(𝑁 + 1)))))
138130, 137ffvelrnd 6835 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) ∈ ℤ)
139138zred 12075 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) ∈ ℝ)
140 simplrr 777 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → 𝑏 ∈ (mzPoly‘ℕ))
141 mzpf 39524 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (mzPoly‘ℕ) → 𝑏:(ℤ ↑m ℕ)⟶ℤ)
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → 𝑏:(ℤ ↑m ℕ)⟶ℤ)
143 elmapssres 8416 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (ℤ ↑m ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑m ℕ))
144134, 99, 143sylancl 589 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (ℕ0m ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑m ℕ))
145144adantl 485 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑓 ↾ ℕ) ∈ (ℤ ↑m ℕ))
146142, 145ffvelrnd 6835 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℤ)
147146zred 12075 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℝ)
148 sumsqeq0 13538 . . . . . . . . . . . . . . . 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))
149139, 147, 148syl2anc 587 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0) ↔ (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) = 0))
150134adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m ℤ)) → 𝑓 ∈ (ℤ ↑m ℤ))
151 reseq1 5830 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))
152151fveq2d 6657 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
153152oveq1d 7155 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) = ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2))
154 reseq1 5830 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (𝑔 ↾ ℕ) = (𝑓 ↾ ℕ))
155154fveq2d 6657 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑏‘(𝑔 ↾ ℕ)) = (𝑏‘(𝑓 ↾ ℕ)))
156155oveq1d 7155 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑏‘(𝑔 ↾ ℕ))↑2) = ((𝑏‘(𝑓 ↾ ℕ))↑2))
157153, 156oveq12d 7158 . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . 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 7173 . . . . . . . . . . . . . . . . . 18 (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) ∈ V
160157, 158, 159fvmpt 6751 . . . . . . . . . . . . . . . . 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)))
161150, 160syl 17 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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)))
162161eqeq1d 2826 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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))
163149, 162bitr4d 285 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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))
164163anbi2d 631 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0m 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)))
165164rexbidva 3288 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)) ↔ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
166127, 165bitrd 282 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0m ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) ↔ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
16731, 166bitr3id 288 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ((∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)) ↔ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)))
168167abbidv 2888 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → {𝑐 ∣ (∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0))} = {𝑐 ∣ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)})
16930, 168syl5eq 2871 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ({𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) = {𝑐 ∣ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)})
170 simpl 486 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
171 fzssuz 12943 . . . . . . . . . . . 12 (1...𝑁) ⊆ (ℤ‘1)
172 uzssz 12252 . . . . . . . . . . . 12 (ℤ‘1) ⊆ ℤ
173171, 172sstri 3960 . . . . . . . . . . 11 (1...𝑁) ⊆ ℤ
1743, 173pm3.2i 474 . . . . . . . . . 10 (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ)
175174a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ))
1763a1i 11 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ℤ ∈ V)
17795a1i 11 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ)
178 simprl 770 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))))
179 mzpresrename 39538 . . . . . . . . . . . 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))
180176, 177, 178, 179syl3anc 1368 . . . . . . . . . . 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 11902 . . . . . . . . . . 11 2 ∈ ℕ0
182 mzpexpmpt 39533 . . . . . . . . . . 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))
183180, 181, 182sylancl 589 . . . . . . . . . 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))
18499a1i 11 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ℕ ⊆ ℤ)
185 simprr 772 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑏 ∈ (mzPoly‘ℕ))
186 mzpresrename 39538 . . . . . . . . . . . 12 ((ℤ ∈ V ∧ ℕ ⊆ ℤ ∧ 𝑏 ∈ (mzPoly‘ℕ)) → (𝑔 ∈ (ℤ ↑m ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
187176, 184, 185, 186syl3anc 1368 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑m ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
188 mzpexpmpt 39533 . . . . . . . . . . 11 (((𝑔 ∈ (ℤ ↑m ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ) ∧ 2 ∈ ℕ0) → (𝑔 ∈ (ℤ ↑m ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
189187, 181, 188sylancl 589 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑m ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
190 mzpaddmpt 39529 . . . . . . . . . 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))
191183, 189, 190syl2anc 587 . . . . . . . . 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 39550 . . . . . . . . 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 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)} ∈ (Dioph‘𝑁))
193170, 175, 191, 192syl3anc 1368 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → {𝑐 ∣ ∃𝑓 ∈ (ℕ0m ℤ)(𝑐 = (𝑓 ↾ (1...𝑁)) ∧ ((𝑔 ∈ (ℤ ↑m ℤ) ↦ (((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑔 ↾ ℕ))↑2)))‘𝑓) = 0)} ∈ (Dioph‘𝑁))
194169, 193eqeltrd 2916 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ({𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) ∈ (Dioph‘𝑁))
195 ineq12 4167 . . . . . . . 8 ((𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) → (𝐴𝐵) = ({𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}))
196195eleq1d 2900 . . . . . . 7 ((𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) → ((𝐴𝐵) ∈ (Dioph‘𝑁) ↔ ({𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∩ {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) ∈ (Dioph‘𝑁)))
197194, 196syl5ibrcom 250 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ((𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
198197rexlimdvva 3286 . . . . 5 (𝑁 ∈ ℕ0 → (∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑏 ∈ (mzPoly‘ℕ)(𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ 𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
19929, 198syl5bir 246 . . . 4 (𝑁 ∈ ℕ0 → ((∃𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1))))𝐴 = {𝑐 ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ (ℤ‘(𝑁 + 1))))(𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0)} ∧ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
20028, 199sylbid 243 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
2011, 200syl 17 . 2 (𝐴 ∈ (Dioph‘𝑁) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
202201anabsi5 668 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wrex 3133  Vcvv 3479  cdif 3915  cun 3916  cin 3917  wss 3918   class class class wbr 5049  cmpt 5129  cres 5540  wf 6334  cfv 6338  (class class class)co 7140  ωcom 7565  m cmap 8391  cen 8491  Fincfn 8494  cr 10523  0cc0 10524  1c1 10525   + caddc 10527  cle 10663  cn 11625  2c2 11680  0cn0 11885  cz 11969  cuz 12231  ...cfz 12885  cexp 13425  mzPolycmzp 39510  Diophcdioph 39543
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-inf2 9090  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-of 7394  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-map 8393  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-dju 9316  df-card 9354  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-seq 13365  df-exp 13426  df-hash 13687  df-mzpcl 39511  df-mzp 39512  df-dioph 39544
This theorem is referenced by:  anrabdioph  39568
  Copyright terms: Public domain W3C validator