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 37162
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 37153 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 id 22 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
3 zex 11383 . . . . . . 7 ℤ ∈ V
4 difexg 4806 . . . . . . 7 (ℤ ∈ V → (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ V)
53, 4mp1i 13 . . . . . 6 (𝑁 ∈ ℕ0 → (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ V)
6 ominf 8169 . . . . . . 7 ¬ ω ∈ Fin
7 nn0z 11397 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
8 lzenom 37159 . . . . . . . 8 (𝑁 ∈ ℤ → (ℤ ∖ (ℤ‘(𝑁 + 1))) ≈ ω)
9 enfi 8173 . . . . . . . 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 317 . . . . . 6 (𝑁 ∈ ℕ0 → ¬ (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ Fin)
12 fz1eqin 37158 . . . . . . 7 (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))
13 inss1 3831 . . . . . . 7 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1)))
1412, 13syl6eqss 3653 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1))))
15 eldioph2b 37152 . . . . . 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 1326 . . . . 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 11023 . . . . . . 7 ℕ ∈ V
1817a1i 11 . . . . . 6 (𝑁 ∈ ℕ0 → ℕ ∈ V)
19 1z 11404 . . . . . . 7 1 ∈ ℤ
20 nnuz 11720 . . . . . . . 8 ℕ = (ℤ‘1)
2120uzinf 12759 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
2219, 21mp1i 13 . . . . . 6 (𝑁 ∈ ℕ0 → ¬ ℕ ∈ Fin)
23 elfznn 12367 . . . . . . . 8 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
2423ssriv 3605 . . . . . . 7 (1...𝑁) ⊆ ℕ
2524a1i 11 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ)
26 eldioph2b 37152 . . . . . 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 1326 . . . . 5 (𝑁 ∈ ℕ0 → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}))
2816, 27anbi12d 747 . . . 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 3105 . . . . 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 3893 . . . . . . . . 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 3105 . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . . 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 801 . . . . . . . . . . . . . . . . . 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 2627 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ) = (1...𝑁))
3534reseq2d 5394 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
3635ad3antrrr 766 . . . . . . . . . . . . . . . . . . 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 5394 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ (1...𝑁)))
3837ad3antrrr 766 . . . . . . . . . . . . . . . . . . . 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 804 . . . . . . . . . . . . . . . . . . . 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 802 . . . . . . . . . . . . . . . . . . . 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 2661 . . . . . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . . . . . . 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 37160 . . . . . . . . . . . . . . . . . 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 1325 . . . . . . . . . . . . . . . . 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 3762 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1))
4619a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 1 ∈ ℤ)
47 nn0p1nn 11329 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
4847nnge1d 11060 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 1 ≤ (𝑁 + 1))
49 lzunuz 37157 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 1 ≤ (𝑁 + 1)) → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1)) = ℤ)
507, 46, 48, 49syl3anc 1325 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1)) = ℤ)
5145, 50syl5eq 2667 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ) = ℤ)
5251oveq2d 6663 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (ℕ0𝑚 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ)) = (ℕ0𝑚 ℤ))
5352ad3antrrr 766 . . . . . . . . . . . . . . . . 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 2702 . . . . . . . . . . . . . . . 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 3754 . . . . . . . . . . . . . . . . . . 19 (𝑐𝑐) = 𝑐
5640, 39uneq12d 3766 . . . . . . . . . . . . . . . . . . 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 2669 . . . . . . . . . . . . . . . . . 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 5409 . . . . . . . . . . . . . . . . . 18 ((𝑑𝑒) ↾ (1...𝑁)) = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁)))
5957, 58syl6eqr 2673 . . . . . . . . . . . . . . . . 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 3755 . . . . . . . . . . . . . . . . . . . . 21 (𝑑𝑒) = (𝑒𝑑)
6160reseq1i 5390 . . . . . . . . . . . . . . . . . . . 20 ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((𝑒𝑑) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))
62 incom 3803 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)
6362, 34syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = (1...𝑁))
6463reseq2d 5394 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
6564ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . 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 5394 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑑 ↾ (1...𝑁)))
6766ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . 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 2661 . . . . . . . . . . . . . . . . . . . . . 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 2658 . . . . . . . . . . . . . . . . . . . . 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 37161 . . . . . . . . . . . . . . . . . . . . 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 1325 . . . . . . . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . . . . . . . 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 6193 . . . . . . . . . . . . . . . . . 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 803 . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . 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 37161 . . . . . . . . . . . . . . . . . . . 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 1325 . . . . . . . . . . . . . . . . . . 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 6193 . . . . . . . . . . . . . . . . . 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 805 . . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . . . 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 558 . . . . . . . . . . . . . . . 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 5388 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → (𝑓 ↾ (1...𝑁)) = ((𝑑𝑒) ↾ (1...𝑁)))
8382eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑑𝑒) → (𝑐 = (𝑓 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑑𝑒) ↾ (1...𝑁))))
84 reseq1 5388 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑑𝑒) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))
8584fveq2d 6193 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑑𝑒) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
8685eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ↔ (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
87 reseq1 5388 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑑𝑒) → (𝑓 ↾ ℕ) = ((𝑑𝑒) ↾ ℕ))
8887fveq2d 6193 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑑𝑒) → (𝑏‘(𝑓 ↾ ℕ)) = (𝑏‘((𝑑𝑒) ↾ ℕ)))
8988eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → ((𝑏‘(𝑓 ↾ ℕ)) = 0 ↔ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0))
9086, 89anbi12d 747 . . . . . . . . . . . . . . . . . 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)))
9183, 90anbi12d 747 . . . . . . . . . . . . . . . . 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))))
9291rspcev 3307 . . . . . . . . . . . . . . . 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)))
9354, 81, 92syl2anc 693 . . . . . . . . . . . . . . 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)))
9493ex 450 . . . . . . . . . . . . . 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))))
9594rexlimdvva 3036 . . . . . . . . . . . . 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))))
96 simpr 477 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑓 ∈ (ℕ0𝑚 ℤ))
97 difss 3735 . . . . . . . . . . . . . . . . . 18 (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ
98 elmapssres 7879 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (ℕ0𝑚 ℤ) ∧ (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
9996, 97, 98sylancl 694 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
10099adantr 481 . . . . . . . . . . . . . . . 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))) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
101 nnssz 11394 . . . . . . . . . . . . . . . . . 18 ℕ ⊆ ℤ
102 elmapssres 7879 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (ℕ0𝑚 ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℕ0𝑚 ℕ))
10396, 101, 102sylancl 694 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑓 ↾ ℕ) ∈ (ℕ0𝑚 ℕ))
104103adantr 481 . . . . . . . . . . . . . . . 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))) → (𝑓 ↾ ℕ) ∈ (ℕ0𝑚 ℕ))
105 simprl 794 . . . . . . . . . . . . . . . . . . 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))) → 𝑐 = (𝑓 ↾ (1...𝑁)))
10614ad3antrrr 766 . . . . . . . . . . . . . . . . . . . 20 ((((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))))
107106resabs1d 5426 . . . . . . . . . . . . . . . . . . 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))) → ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
108105, 107eqtr4d 2658 . . . . . . . . . . . . . . . . . 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 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)))
109 simprrl 804 . . . . . . . . . . . . . . . . . 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))) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0)
110108, 109jca 554 . . . . . . . . . . . . . . . . 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)) ∧ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
111 resabs1 5425 . . . . . . . . . . . . . . . . . . 19 ((1...𝑁) ⊆ ℕ → ((𝑓 ↾ ℕ) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
11224, 111mp1i 13 . . . . . . . . . . . . . . . . . 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 ↾ ℕ) ↾ (1...𝑁)) = (𝑓 ↾ (1...𝑁)))
113105, 112eqtr4d 2658 . . . . . . . . . . . . . . . . 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 ↾ ℕ) ↾ (1...𝑁)))
114 simprrr 805 . . . . . . . . . . . . . . . . 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))) → (𝑏‘(𝑓 ↾ ℕ)) = 0)
115110, 113, 114jca32 558 . . . . . . . . . . . . . . . 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) ∧ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
116 reseq1 5388 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑑 ↾ (1...𝑁)) = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)))
117116eqeq2d 2631 . . . . . . . . . . . . . . . . . . 19 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑐 = (𝑑 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁))))
118 fveq2 6189 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑎𝑑) = (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
119118eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → ((𝑎𝑑) = 0 ↔ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
120117, 119anbi12d 747 . . . . . . . . . . . . . . . . . 18 (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)))
121120anbi1d 741 . . . . . . . . . . . . . . . . 17 (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))))
122 reseq1 5388 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝑓 ↾ ℕ) → (𝑒 ↾ (1...𝑁)) = ((𝑓 ↾ ℕ) ↾ (1...𝑁)))
123122eqeq2d 2631 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝑓 ↾ ℕ) → (𝑐 = (𝑒 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁))))
124 fveq2 6189 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝑓 ↾ ℕ) → (𝑏𝑒) = (𝑏‘(𝑓 ↾ ℕ)))
125124eqeq1d 2623 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝑓 ↾ ℕ) → ((𝑏𝑒) = 0 ↔ (𝑏‘(𝑓 ↾ ℕ)) = 0))
126123, 125anbi12d 747 . . . . . . . . . . . . . . . . . 18 (𝑒 = (𝑓 ↾ ℕ) → ((𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0) ↔ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
127126anbi2d 740 . . . . . . . . . . . . . . . . 17 (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))))
128121, 127rspc2ev 3322 . . . . . . . . . . . . . . . 16 (((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)))
129100, 104, 115, 128syl3anc 1325 . . . . . . . . . . . . . . 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))) → ∃𝑑 ∈ (ℕ0𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1))))∃𝑒 ∈ (ℕ0𝑚 ℕ)((𝑐 = (𝑑 ↾ (1...𝑁)) ∧ (𝑎𝑑) = 0) ∧ (𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)))
130129ex 450 . . . . . . . . . . . . . 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))))
131130rexlimdva 3029 . . . . . . . . . . . . 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))))
13295, 131impbid 202 . . . . . . . . . . . 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))))
133 simplrl 800 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))))
134 mzpf 37125 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) → 𝑎:(ℤ ↑𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1))))⟶ℤ)
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑎:(ℤ ↑𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1))))⟶ℤ)
136 nn0ssz 11395 . . . . . . . . . . . . . . . . . . . . . 22 0 ⊆ ℤ
137 mapss 7897 . . . . . . . . . . . . . . . . . . . . . 22 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 ℤ) ⊆ (ℤ ↑𝑚 ℤ))
1383, 136, 137mp2an 708 . . . . . . . . . . . . . . . . . . . . 21 (ℕ0𝑚 ℤ) ⊆ (ℤ ↑𝑚 ℤ)
139138sseli 3597 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (ℕ0𝑚 ℤ) → 𝑓 ∈ (ℤ ↑𝑚 ℤ))
140 elmapssres 7879 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (ℤ ↑𝑚 ℤ) ∧ (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
141139, 97, 140sylancl 694 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (ℕ0𝑚 ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
142141adantl 482 . . . . . . . . . . . . . . . . . 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)))))
143135, 142ffvelrnd 6358 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) ∈ ℤ)
144143zred 11479 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) ∈ ℝ)
145 simplrr 801 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑏 ∈ (mzPoly‘ℕ))
146 mzpf 37125 . . . . . . . . . . . . . . . . . . 19 (𝑏 ∈ (mzPoly‘ℕ) → 𝑏:(ℤ ↑𝑚 ℕ)⟶ℤ)
147145, 146syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑏:(ℤ ↑𝑚 ℕ)⟶ℤ)
148 elmapssres 7879 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (ℤ ↑𝑚 ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
149139, 101, 148sylancl 694 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (ℕ0𝑚 ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
150149adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
151147, 150ffvelrnd 6358 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℤ)
152151zred 11479 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℝ)
153 sumsqeq0 12937 . . . . . . . . . . . . . . . 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))
154144, 152, 153syl2anc 693 . . . . . . . . . . . . . . 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))
155139adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑓 ∈ (ℤ ↑𝑚 ℤ))
156 reseq1 5388 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))
157156fveq2d 6193 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
158157oveq1d 6662 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) = ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2))
159 reseq1 5388 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (𝑔 ↾ ℕ) = (𝑓 ↾ ℕ))
160159fveq2d 6193 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑏‘(𝑔 ↾ ℕ)) = (𝑏‘(𝑓 ↾ ℕ)))
161160oveq1d 6662 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑏‘(𝑔 ↾ ℕ))↑2) = ((𝑏‘(𝑓 ↾ ℕ))↑2))
162158, 161oveq12d 6665 . . . . . . . . . . . . . . . . . 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)))
163 eqid 2621 . . . . . . . . . . . . . . . . . 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)))
164 ovex 6675 . . . . . . . . . . . . . . . . . 18 (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) ∈ V
165162, 163, 164fvmpt 6280 . . . . . . . . . . . . . . . . 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)))
166155, 165syl 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)))
167166eqeq1d 2623 . . . . . . . . . . . . . . 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))
168154, 167bitr4d 271 . . . . . . . . . . . . . 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))
169168anbi2d 740 . . . . . . . . . . . . 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)))
170169rexbidva 3047 . . . . . . . . . . . 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)))
171132, 170bitrd 268 . . . . . . . . . . 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)))
17231, 171syl5bbr 274 . . . . . . . . . 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)))
173172abbidv 2740 . . . . . . . . 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)})
17430, 173syl5eq 2667 . . . . . . . 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)})
175 simpl 473 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
176 fzssuz 12379 . . . . . . . . . . . 12 (1...𝑁) ⊆ (ℤ‘1)
177 uzssz 11704 . . . . . . . . . . . 12 (ℤ‘1) ⊆ ℤ
178176, 177sstri 3610 . . . . . . . . . . 11 (1...𝑁) ⊆ ℤ
1793, 178pm3.2i 471 . . . . . . . . . 10 (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ)
180179a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (ℤ ∈ V ∧ (1...𝑁) ⊆ ℤ))
1813a1i 11 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ℤ ∈ V)
18297a1i 11 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ)
183 simprl 794 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))))
184 mzpresrename 37139 . . . . . . . . . . . 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))
185181, 182, 183, 184syl3anc 1325 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))) ∈ (mzPoly‘ℤ))
186 2nn0 11306 . . . . . . . . . . 11 2 ∈ ℕ0
187 mzpexpmpt 37134 . . . . . . . . . . 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))
188185, 186, 187sylancl 694 . . . . . . . . . 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))
189101a1i 11 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → ℕ ⊆ ℤ)
190 simprr 796 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑏 ∈ (mzPoly‘ℕ))
191 mzpresrename 37139 . . . . . . . . . . . 12 ((ℤ ∈ V ∧ ℕ ⊆ ℤ ∧ 𝑏 ∈ (mzPoly‘ℕ)) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
192181, 189, 190, 191syl3anc 1325 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
193 mzpexpmpt 37134 . . . . . . . . . . 11 (((𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ) ∧ 2 ∈ ℕ0) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
194192, 186, 193sylancl 694 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
195 mzpaddmpt 37130 . . . . . . . . . 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))
196188, 194, 195syl2anc 693 . . . . . . . . 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))
197 eldioph2 37151 . . . . . . . . 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))
198175, 180, 196, 197syl3anc 1325 . . . . . . . 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))
199174, 198eqeltrd 2700 . . . . . . 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))
200 ineq12 3807 . . . . . . . 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)}))
201200eleq1d 2685 . . . . . . 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)))
202199, 201syl5ibrcom 237 . . . . . 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)))
203202rexlimdvva 3036 . . . . 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)))
20429, 203syl5bir 233 . . . 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)))
20528, 204sylbid 230 . . 3 (𝑁 ∈ ℕ0 → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
2061, 205syl 17 . 2 (𝐴 ∈ (Dioph‘𝑁) → ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁)))
207206anabsi5 858 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  {cab 2607  wrex 2912  Vcvv 3198  cdif 3569  cun 3570  cin 3571  wss 3572   class class class wbr 4651  cmpt 4727  cres 5114  wf 5882  cfv 5886  (class class class)co 6647  ωcom 7062  𝑚 cmap 7854  cen 7949  Fincfn 7952  cr 9932  0cc0 9933  1c1 9934   + caddc 9936  cle 10072  cn 11017  2c2 11067  0cn0 11289  cz 11374  cuz 11684  ...cfz 12323  cexp 12855  mzPolycmzp 37111  Diophcdioph 37144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-card 8762  df-cda 8987  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-n0 11290  df-z 11375  df-uz 11685  df-fz 12324  df-seq 12797  df-exp 12856  df-hash 13113  df-mzpcl 37112  df-mzp 37113  df-dioph 37145
This theorem is referenced by:  anrabdioph  37170
  Copyright terms: Public domain W3C validator