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Theorem diophin 36244
 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 36235 . . 3 (𝐴 ∈ (Dioph‘𝑁) → 𝑁 ∈ ℕ0)
2 id 22 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
3 zex 11127 . . . . . . 7 ℤ ∈ V
4 difexg 4634 . . . . . . 7 (ℤ ∈ V → (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ V)
53, 4mp1i 13 . . . . . 6 (𝑁 ∈ ℕ0 → (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ V)
6 ominf 7933 . . . . . . 7 ¬ ω ∈ Fin
7 nn0z 11141 . . . . . . . 8 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
8 lzenom 36241 . . . . . . . 8 (𝑁 ∈ ℤ → (ℤ ∖ (ℤ‘(𝑁 + 1))) ≈ ω)
9 enfi 7937 . . . . . . . 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 315 . . . . . 6 (𝑁 ∈ ℕ0 → ¬ (ℤ ∖ (ℤ‘(𝑁 + 1))) ∈ Fin)
12 fz1eqin 36240 . . . . . . 7 (𝑁 ∈ ℕ0 → (1...𝑁) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ))
13 inss1 3698 . . . . . . 7 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1)))
1412, 13syl6eqss 3522 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ (ℤ ∖ (ℤ‘(𝑁 + 1))))
15 eldioph2b 36234 . . . . . 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 1318 . . . . 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 10781 . . . . . . 7 ℕ ∈ V
1817a1i 11 . . . . . 6 (𝑁 ∈ ℕ0 → ℕ ∈ V)
19 1z 11148 . . . . . . 7 1 ∈ ℤ
20 nnuz 11463 . . . . . . . 8 ℕ = (ℤ‘1)
2120uzinf 12494 . . . . . . 7 (1 ∈ ℤ → ¬ ℕ ∈ Fin)
2219, 21mp1i 13 . . . . . 6 (𝑁 ∈ ℕ0 → ¬ ℕ ∈ Fin)
23 elfznn 12109 . . . . . . . 8 (𝑎 ∈ (1...𝑁) → 𝑎 ∈ ℕ)
2423ssriv 3476 . . . . . . 7 (1...𝑁) ⊆ ℕ
2524a1i 11 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ⊆ ℕ)
26 eldioph2b 36234 . . . . . 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 1318 . . . . 5 (𝑁 ∈ ℕ0 → (𝐵 ∈ (Dioph‘𝑁) ↔ ∃𝑏 ∈ (mzPoly‘ℕ)𝐵 = {𝑐 ∣ ∃𝑒 ∈ (ℕ0𝑚 ℕ)(𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0)}))
2816, 27anbi12d 742 . . . 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 2990 . . . . 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 3757 . . . . . . . . 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 2990 . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . . . 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 796 . . . . . . . . . . . . . . . . . 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 2520 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ) = (1...𝑁))
3534reseq2d 5208 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → (𝑑 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑑 ↾ (1...𝑁)))
3635ad3antrrr 761 . . . . . . . . . . . . . . . . . . 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 5208 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → (𝑒 ↾ ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)) = (𝑒 ↾ (1...𝑁)))
3837ad3antrrr 761 . . . . . . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . . . . . 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 797 . . . . . . . . . . . . . . . . . . . 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 2554 . . . . . . . . . . . . . . . . . . 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 2551 . . . . . . . . . . . . . . . . . 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 36242 . . . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . . . 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 3630 . . . . . . . . . . . . . . . . . . . 20 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1))
4619a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 1 ∈ ℤ)
47 nn0p1nn 11087 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
4847nnge1d 10818 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 1 ≤ (𝑁 + 1))
49 lzunuz 36239 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℤ ∧ 1 ∈ ℤ ∧ 1 ≤ (𝑁 + 1)) → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1)) = ℤ)
507, 46, 48, 49syl3anc 1317 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ (ℤ‘1)) = ℤ)
5145, 50syl5eq 2560 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ0 → ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ) = ℤ)
5251oveq2d 6442 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (ℕ0𝑚 ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∪ ℕ)) = (ℕ0𝑚 ℤ))
5352ad3antrrr 761 . . . . . . . . . . . . . . . . 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 2594 . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . 19 (𝑐𝑐) = 𝑐
5640, 39uneq12d 3634 . . . . . . . . . . . . . . . . . . 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 2562 . . . . . . . . . . . . . . . . . 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 5222 . . . . . . . . . . . . . . . . . 18 ((𝑑𝑒) ↾ (1...𝑁)) = ((𝑑 ↾ (1...𝑁)) ∪ (𝑒 ↾ (1...𝑁)))
5957, 58syl6eqr 2566 . . . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . . . . . . . . . 21 (𝑑𝑒) = (𝑒𝑑)
6160reseq1i 5204 . . . . . . . . . . . . . . . . . . . 20 ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((𝑒𝑑) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))
62 incom 3670 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((ℤ ∖ (ℤ‘(𝑁 + 1))) ∩ ℕ)
6362, 34syl5eq 2560 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = (1...𝑁))
6463reseq2d 5208 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ0 → (𝑒 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑒 ↾ (1...𝑁)))
6564ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . 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 5208 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ0 → (𝑑 ↾ (ℕ ∩ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑑 ↾ (1...𝑁)))
6766ad3antrrr 761 . . . . . . . . . . . . . . . . . . . . . . 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 2554 . . . . . . . . . . . . . . . . . . . . . 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 2551 . . . . . . . . . . . . . . . . . . . . 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 36243 . . . . . . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . . . . . . 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 2560 . . . . . . . . . . . . . . . . . . 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 5991 . . . . . . . . . . . . . . . . . 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 798 . . . . . . . . . . . . . . . . . 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 2548 . . . . . . . . . . . . . . . . 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 36243 . . . . . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . . . . . 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 5991 . . . . . . . . . . . . . . . . . 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 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))) → (𝑏𝑒) = 0)
8078, 79eqtrd 2548 . . . . . . . . . . . . . . . . 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 555 . . . . . . . . . . . . . . . 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 5202 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → (𝑓 ↾ (1...𝑁)) = ((𝑑𝑒) ↾ (1...𝑁)))
8382eqeq2d 2524 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝑑𝑒) → (𝑐 = (𝑓 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑑𝑒) ↾ (1...𝑁))))
84 reseq1 5202 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑑𝑒) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = ((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))
8584fveq2d 5991 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑑𝑒) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
8685eqeq1d 2516 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0 ↔ (𝑎‘((𝑑𝑒) ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
87 reseq1 5202 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝑑𝑒) → (𝑓 ↾ ℕ) = ((𝑑𝑒) ↾ ℕ))
8887fveq2d 5991 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝑑𝑒) → (𝑏‘(𝑓 ↾ ℕ)) = (𝑏‘((𝑑𝑒) ↾ ℕ)))
8988eqeq1d 2516 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝑑𝑒) → ((𝑏‘(𝑓 ↾ ℕ)) = 0 ↔ (𝑏‘((𝑑𝑒) ↾ ℕ)) = 0))
9086, 89anbi12d 742 . . . . . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . 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 2924 . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑓 ∈ (ℕ0𝑚 ℤ))
97 difss 3603 . . . . . . . . . . . . . . . . . 18 (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ
98 elmapssres 7644 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (ℕ0𝑚 ℤ) ∧ (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℕ0𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
9996, 97, 98sylancl 692 . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 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 11138 . . . . . . . . . . . . . . . . . 18 ℕ ⊆ ℤ
102 elmapssres 7644 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (ℕ0𝑚 ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℕ0𝑚 ℕ))
10396, 101, 102sylancl 692 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑓 ↾ ℕ) ∈ (ℕ0𝑚 ℕ))
104103adantr 479 . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . 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 5239 . . . . . . . . . . . . . . . . . . 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 2551 . . . . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . . . . 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 552 . . . . . . . . . . . . . . . . 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 5238 . . . . . . . . . . . . . . . . . . 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 2551 . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . 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 555 . . . . . . . . . . . . . . . 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 5202 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑑 ↾ (1...𝑁)) = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁)))
117116eqeq2d 2524 . . . . . . . . . . . . . . . . . . 19 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑐 = (𝑑 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ↾ (1...𝑁))))
118 fveq2 5987 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → (𝑎𝑑) = (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
119118eqeq1d 2516 . . . . . . . . . . . . . . . . . . 19 (𝑑 = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) → ((𝑎𝑑) = 0 ↔ (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = 0))
120117, 119anbi12d 742 . . . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . . 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 5202 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝑓 ↾ ℕ) → (𝑒 ↾ (1...𝑁)) = ((𝑓 ↾ ℕ) ↾ (1...𝑁)))
123122eqeq2d 2524 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝑓 ↾ ℕ) → (𝑐 = (𝑒 ↾ (1...𝑁)) ↔ 𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁))))
124 fveq2 5987 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝑓 ↾ ℕ) → (𝑏𝑒) = (𝑏‘(𝑓 ↾ ℕ)))
125124eqeq1d 2516 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝑓 ↾ ℕ) → ((𝑏𝑒) = 0 ↔ (𝑏‘(𝑓 ↾ ℕ)) = 0))
126123, 125anbi12d 742 . . . . . . . . . . . . . . . . . 18 (𝑒 = (𝑓 ↾ ℕ) → ((𝑐 = (𝑒 ↾ (1...𝑁)) ∧ (𝑏𝑒) = 0) ↔ (𝑐 = ((𝑓 ↾ ℕ) ↾ (1...𝑁)) ∧ (𝑏‘(𝑓 ↾ ℕ)) = 0)))
127126anbi2d 735 . . . . . . . . . . . . . . . . 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 3199 . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . 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 2917 . . . . . . . . . . . . 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 200 . . . . . . . . . . . 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 795 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))))
134 mzpf 36207 . . . . . . . . . . . . . . . . . . 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 11139 . . . . . . . . . . . . . . . . . . . . . 22 0 ⊆ ℤ
137 mapss 7662 . . . . . . . . . . . . . . . . . . . . . 22 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0𝑚 ℤ) ⊆ (ℤ ↑𝑚 ℤ))
1383, 136, 137mp2an 703 . . . . . . . . . . . . . . . . . . . . 21 (ℕ0𝑚 ℤ) ⊆ (ℤ ↑𝑚 ℤ)
139138sseli 3468 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (ℕ0𝑚 ℤ) → 𝑓 ∈ (ℤ ↑𝑚 ℤ))
140 elmapssres 7644 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (ℤ ↑𝑚 ℤ) ∧ (ℤ ∖ (ℤ‘(𝑁 + 1))) ⊆ ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
141139, 97, 140sylancl 692 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (ℕ0𝑚 ℤ) → (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) ∈ (ℤ ↑𝑚 (ℤ ∖ (ℤ‘(𝑁 + 1)))))
142141adantl 480 . . . . . . . . . . . . . . . . . 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 6152 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) ∈ ℤ)
144143zred 11222 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) ∈ ℝ)
145 simplrr 796 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑏 ∈ (mzPoly‘ℕ))
146 mzpf 36207 . . . . . . . . . . . . . . . . . . 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 7644 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (ℤ ↑𝑚 ℤ) ∧ ℕ ⊆ ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
149139, 101, 148sylancl 692 . . . . . . . . . . . . . . . . . . 19 (𝑓 ∈ (ℕ0𝑚 ℤ) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
150149adantl 480 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑓 ↾ ℕ) ∈ (ℤ ↑𝑚 ℕ))
151147, 150ffvelrnd 6152 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℤ)
152151zred 11222 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → (𝑏‘(𝑓 ↾ ℕ)) ∈ ℝ)
153 sumsqeq0 12672 . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) ∧ 𝑓 ∈ (ℕ0𝑚 ℤ)) → 𝑓 ∈ (ℤ ↑𝑚 ℤ))
156 reseq1 5202 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))) = (𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))
157156fveq2d 5991 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))) = (𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1))))))
158157oveq1d 6441 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑎‘(𝑔 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) = ((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2))
159 reseq1 5202 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝑓 → (𝑔 ↾ ℕ) = (𝑓 ↾ ℕ))
160159fveq2d 5991 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝑓 → (𝑏‘(𝑔 ↾ ℕ)) = (𝑏‘(𝑓 ↾ ℕ)))
161160oveq1d 6441 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑓 → ((𝑏‘(𝑔 ↾ ℕ))↑2) = ((𝑏‘(𝑓 ↾ ℕ))↑2))
162158, 161oveq12d 6444 . . . . . . . . . . . . . . . . . 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 2514 . . . . . . . . . . . . . . . . . 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 6454 . . . . . . . . . . . . . . . . . 18 (((𝑎‘(𝑓 ↾ (ℤ ∖ (ℤ‘(𝑁 + 1)))))↑2) + ((𝑏‘(𝑓 ↾ ℕ))↑2)) ∈ V
165162, 163, 164fvmpt 6075 . . . . . . . . . . . . . . . . 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 2516 . . . . . . . . . . . . . . 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 269 . . . . . . . . . . . . . 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 735 . . . . . . . . . . . . 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 2935 . . . . . . . . . . . 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 266 . . . . . . . . . . 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 272 . . . . . . . . . 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 2632 . . . . . . . . 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 2560 . . . . . . . 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 471 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑁 ∈ ℕ0)
176 fzssuz 12121 . . . . . . . . . . . 12 (1...𝑁) ⊆ (ℤ‘1)
177 uzssz 11447 . . . . . . . . . . . 12 (ℤ‘1) ⊆ ℤ
178176, 177sstri 3481 . . . . . . . . . . 11 (1...𝑁) ⊆ ℤ
1793, 178pm3.2i 469 . . . . . . . . . 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 789 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))))
184 mzpresrename 36221 . . . . . . . . . . . 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 1317 . . . . . . . . . . 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 11064 . . . . . . . . . . 11 2 ∈ ℕ0
187 mzpexpmpt 36216 . . . . . . . . . . 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 692 . . . . . . . . . 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 791 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → 𝑏 ∈ (mzPoly‘ℕ))
191 mzpresrename 36221 . . . . . . . . . . . 12 ((ℤ ∈ V ∧ ℕ ⊆ ℤ ∧ 𝑏 ∈ (mzPoly‘ℕ)) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
192181, 189, 190, 191syl3anc 1317 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ (𝑏‘(𝑔 ↾ ℕ))) ∈ (mzPoly‘ℤ))
193 mzpexpmpt 36216 . . . . . . . . . . 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 692 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑎 ∈ (mzPoly‘(ℤ ∖ (ℤ‘(𝑁 + 1)))) ∧ 𝑏 ∈ (mzPoly‘ℕ))) → (𝑔 ∈ (ℤ ↑𝑚 ℤ) ↦ ((𝑏‘(𝑔 ↾ ℕ))↑2)) ∈ (mzPoly‘ℤ))
195 mzpaddmpt 36212 . . . . . . . . . 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 690 . . . . . . . . 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 36233 . . . . . . . . 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 1317 . . . . . . . 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 2592 . . . . . . 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 3674 . . . . . . . 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 2576 . . . . . . 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 235 . . . . . 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 2924 . . . . 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 231 . . . 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 228 . . 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 853 1 ((𝐴 ∈ (Dioph‘𝑁) ∧ 𝐵 ∈ (Dioph‘𝑁)) → (𝐴𝐵) ∈ (Dioph‘𝑁))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 194   ∧ wa 382   = wceq 1474   ∈ wcel 1938  {cab 2500  ∃wrex 2801  Vcvv 3077   ∖ cdif 3441   ∪ cun 3442   ∩ cin 3443   ⊆ wss 3444   class class class wbr 4481   ↦ cmpt 4541   ↾ cres 4934  ⟶wf 5685  ‘cfv 5689  (class class class)co 6426  ωcom 6833   ↑𝑚 cmap 7620   ≈ cen 7714  Fincfn 7717  ℝcr 9690  0cc0 9691  1c1 9692   + caddc 9694   ≤ cle 9830  ℕcn 10775  2c2 10825  ℕ0cn0 11047  ℤcz 11118  ℤ≥cuz 11427  ...cfz 12065  ↑cexp 12590  mzPolycmzp 36193  Diophcdioph 36226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-inf2 8297  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-of 6671  df-om 6834  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-1o 7323  df-oadd 7327  df-er 7505  df-map 7622  df-en 7718  df-dom 7719  df-sdom 7720  df-fin 7721  df-card 8524  df-cda 8749  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-2 10834  df-n0 11048  df-z 11119  df-uz 11428  df-fz 12066  df-seq 12532  df-exp 12591  df-hash 12848  df-mzpcl 36194  df-mzp 36195  df-dioph 36227 This theorem is referenced by:  anrabdioph  36252
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