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Theorem aannenlem1 26371
Description: Lemma for aannen 26374. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypothesis
Ref Expression
aannenlem.a 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})
Assertion
Ref Expression
aannenlem1 (𝐴 ∈ ℕ0 → (𝐻𝐴) ∈ Fin)
Distinct variable group:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒
Allowed substitution hints:   𝐻(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem aannenlem1
StepHypRef Expression
1 breq2 5146 . . . . . . 7 (𝑎 = 𝐴 → ((deg‘𝑑) ≤ 𝑎 ↔ (deg‘𝑑) ≤ 𝐴))
2 breq2 5146 . . . . . . . 8 (𝑎 = 𝐴 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
32ralbidv 3177 . . . . . . 7 (𝑎 = 𝐴 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴))
41, 33anbi23d 1440 . . . . . 6 (𝑎 = 𝐴 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎) ↔ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)))
54rabbidv 3443 . . . . 5 (𝑎 = 𝐴 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} = {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)})
65rexeqdv 3326 . . . 4 (𝑎 = 𝐴 → (∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0 ↔ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐𝑏) = 0))
76rabbidv 3443 . . 3 (𝑎 = 𝐴 → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐𝑏) = 0})
8 aannenlem.a . . 3 𝐻 = (𝑎 ∈ ℕ0 ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝑎 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝑎)} (𝑐𝑏) = 0})
9 cnex 11237 . . . 4 ℂ ∈ V
109rabex 5338 . . 3 {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐𝑏) = 0} ∈ V
117, 8, 10fvmpt 7015 . 2 (𝐴 ∈ ℕ0 → (𝐻𝐴) = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐𝑏) = 0})
12 iunrab 5051 . . 3 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} = {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐𝑏) = 0}
13 fzfi 14014 . . . . . . 7 (-𝐴...𝐴) ∈ Fin
14 fzfi 14014 . . . . . . 7 (0...𝐴) ∈ Fin
15 mapfi 9389 . . . . . . 7 (((-𝐴...𝐴) ∈ Fin ∧ (0...𝐴) ∈ Fin) → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
1613, 14, 15mp2an 692 . . . . . 6 ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin
1716a1i 11 . . . . 5 (𝐴 ∈ ℕ0 → ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin)
18 ovex 7465 . . . . . 6 ((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V
19 neeq1 3002 . . . . . . . . . . 11 (𝑑 = 𝑎 → (𝑑 ≠ 0𝑝𝑎 ≠ 0𝑝))
20 fveq2 6905 . . . . . . . . . . . 12 (𝑑 = 𝑎 → (deg‘𝑑) = (deg‘𝑎))
2120breq1d 5152 . . . . . . . . . . 11 (𝑑 = 𝑎 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑎) ≤ 𝐴))
22 fveq2 6905 . . . . . . . . . . . . . . 15 (𝑑 = 𝑎 → (coeff‘𝑑) = (coeff‘𝑎))
2322fveq1d 6907 . . . . . . . . . . . . . 14 (𝑑 = 𝑎 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑎)‘𝑒))
2423fveq2d 6909 . . . . . . . . . . . . 13 (𝑑 = 𝑎 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑎)‘𝑒)))
2524breq1d 5152 . . . . . . . . . . . 12 (𝑑 = 𝑎 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
2625ralbidv 3177 . . . . . . . . . . 11 (𝑑 = 𝑎 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
2719, 21, 263anbi123d 1437 . . . . . . . . . 10 (𝑑 = 𝑎 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)))
2827elrab 3691 . . . . . . . . 9 (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)))
29 simp3 1138 . . . . . . . . . 10 ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)
3029anim2i 617 . . . . . . . . 9 ((𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
3128, 30sylbi 217 . . . . . . . 8 (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴))
32 0z 12626 . . . . . . . . . . . . . . 15 0 ∈ ℤ
33 eqid 2736 . . . . . . . . . . . . . . . 16 (coeff‘𝑎) = (coeff‘𝑎)
3433coef2 26271 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (Poly‘ℤ) ∧ 0 ∈ ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
3532, 34mpan2 691 . . . . . . . . . . . . . 14 (𝑎 ∈ (Poly‘ℤ) → (coeff‘𝑎):ℕ0⟶ℤ)
3635ad2antrl 728 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶ℤ)
3736ffnd 6736 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎) Fn ℕ0)
3835adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) → (coeff‘𝑎):ℕ0⟶ℤ)
3938ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℤ)
4039zred 12724 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((coeff‘𝑎)‘𝑒) ∈ ℝ)
41 nn0re 12537 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ ℕ0𝐴 ∈ ℝ)
4241ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℝ)
4340, 42absled 15470 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
44 nn0z 12640 . . . . . . . . . . . . . . . . . . . 20 (𝐴 ∈ ℕ0𝐴 ∈ ℤ)
4544ad2antrr 726 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → 𝐴 ∈ ℤ)
4645znegcld 12726 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → -𝐴 ∈ ℤ)
47 elfz 13554 . . . . . . . . . . . . . . . . . 18 ((((coeff‘𝑎)‘𝑒) ∈ ℤ ∧ -𝐴 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴) ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
4839, 46, 45, 47syl3anc 1372 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → (((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴) ↔ (-𝐴 ≤ ((coeff‘𝑎)‘𝑒) ∧ ((coeff‘𝑎)‘𝑒) ≤ 𝐴)))
4943, 48bitr4d 282 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 ↔ ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
5049biimpd 229 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) ∧ 𝑒 ∈ ℕ0) → ((abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
5150ralimdva 3166 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ0𝑎 ∈ (Poly‘ℤ)) → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴 → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)))
5251impr 454 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴))
53 fnfvrnss 7140 . . . . . . . . . . . . 13 (((coeff‘𝑎) Fn ℕ0 ∧ ∀𝑒 ∈ ℕ0 ((coeff‘𝑎)‘𝑒) ∈ (-𝐴...𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
5437, 52, 53syl2anc 584 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ran (coeff‘𝑎) ⊆ (-𝐴...𝐴))
55 df-f 6564 . . . . . . . . . . . 12 ((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ↔ ((coeff‘𝑎) Fn ℕ0 ∧ ran (coeff‘𝑎) ⊆ (-𝐴...𝐴)))
5637, 54, 55sylanbrc 583 . . . . . . . . . . 11 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (coeff‘𝑎):ℕ0⟶(-𝐴...𝐴))
57 fz0ssnn0 13663 . . . . . . . . . . 11 (0...𝐴) ⊆ ℕ0
58 fssres 6773 . . . . . . . . . . 11 (((coeff‘𝑎):ℕ0⟶(-𝐴...𝐴) ∧ (0...𝐴) ⊆ ℕ0) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
5956, 57, 58sylancl 586 . . . . . . . . . 10 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
60 ovex 7465 . . . . . . . . . . 11 (-𝐴...𝐴) ∈ V
61 ovex 7465 . . . . . . . . . . 11 (0...𝐴) ∈ V
6260, 61elmap 8912 . . . . . . . . . 10 (((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)) ↔ ((coeff‘𝑎) ↾ (0...𝐴)):(0...𝐴)⟶(-𝐴...𝐴))
6359, 62sylibr 234 . . . . . . . . 9 ((𝐴 ∈ ℕ0 ∧ (𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴)))
6463ex 412 . . . . . . . 8 (𝐴 ∈ ℕ0 → ((𝑎 ∈ (Poly‘ℤ) ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴))))
6531, 64syl5 34 . . . . . . 7 (𝐴 ∈ ℕ0 → (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → ((coeff‘𝑎) ↾ (0...𝐴)) ∈ ((-𝐴...𝐴) ↑m (0...𝐴))))
66 simp2 1137 . . . . . . . . . 10 ((𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴) → (deg‘𝑎) ≤ 𝐴)
6766anim2i 617 . . . . . . . . 9 ((𝑎 ∈ (Poly‘ℤ) ∧ (𝑎 ≠ 0𝑝 ∧ (deg‘𝑎) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑎)‘𝑒)) ≤ 𝐴)) → (𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴))
6828, 67sylbi 217 . . . . . . . 8 (𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴))
69 neeq1 3002 . . . . . . . . . . 11 (𝑑 = 𝑏 → (𝑑 ≠ 0𝑝𝑏 ≠ 0𝑝))
70 fveq2 6905 . . . . . . . . . . . 12 (𝑑 = 𝑏 → (deg‘𝑑) = (deg‘𝑏))
7170breq1d 5152 . . . . . . . . . . 11 (𝑑 = 𝑏 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑏) ≤ 𝐴))
72 fveq2 6905 . . . . . . . . . . . . . . 15 (𝑑 = 𝑏 → (coeff‘𝑑) = (coeff‘𝑏))
7372fveq1d 6907 . . . . . . . . . . . . . 14 (𝑑 = 𝑏 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑏)‘𝑒))
7473fveq2d 6909 . . . . . . . . . . . . 13 (𝑑 = 𝑏 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑏)‘𝑒)))
7574breq1d 5152 . . . . . . . . . . . 12 (𝑑 = 𝑏 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
7675ralbidv 3177 . . . . . . . . . . 11 (𝑑 = 𝑏 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴))
7769, 71, 763anbi123d 1437 . . . . . . . . . 10 (𝑑 = 𝑏 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)))
7877elrab 3691 . . . . . . . . 9 (𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑏 ∈ (Poly‘ℤ) ∧ (𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)))
79 simp2 1137 . . . . . . . . . 10 ((𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴) → (deg‘𝑏) ≤ 𝐴)
8079anim2i 617 . . . . . . . . 9 ((𝑏 ∈ (Poly‘ℤ) ∧ (𝑏 ≠ 0𝑝 ∧ (deg‘𝑏) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑏)‘𝑒)) ≤ 𝐴)) → (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴))
8178, 80sylbi 217 . . . . . . . 8 (𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴))
82 simplll 774 . . . . . . . . . . . . 13 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑎 ∈ (Poly‘ℤ))
83 plyf 26238 . . . . . . . . . . . . 13 (𝑎 ∈ (Poly‘ℤ) → 𝑎:ℂ⟶ℂ)
84 ffn 6735 . . . . . . . . . . . . 13 (𝑎:ℂ⟶ℂ → 𝑎 Fn ℂ)
8582, 83, 843syl 18 . . . . . . . . . . . 12 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑎 Fn ℂ)
86 simplrl 776 . . . . . . . . . . . . 13 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑏 ∈ (Poly‘ℤ))
87 plyf 26238 . . . . . . . . . . . . 13 (𝑏 ∈ (Poly‘ℤ) → 𝑏:ℂ⟶ℂ)
88 ffn 6735 . . . . . . . . . . . . 13 (𝑏:ℂ⟶ℂ → 𝑏 Fn ℂ)
8986, 87, 883syl 18 . . . . . . . . . . . 12 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑏 Fn ℂ)
90 simplrr 777 . . . . . . . . . . . . . . . . . 18 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
9190adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
9291fveq1d 6907 . . . . . . . . . . . . . . . 16 ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑))
93 fvres 6924 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (0...𝐴) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑))
9493adantl 481 . . . . . . . . . . . . . . . 16 ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑎)‘𝑑))
95 fvres 6924 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ (0...𝐴) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑))
9695adantl 481 . . . . . . . . . . . . . . . 16 ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑏) ↾ (0...𝐴))‘𝑑) = ((coeff‘𝑏)‘𝑑))
9792, 94, 963eqtr3d 2784 . . . . . . . . . . . . . . 15 ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → ((coeff‘𝑎)‘𝑑) = ((coeff‘𝑏)‘𝑑))
9897oveq1d 7447 . . . . . . . . . . . . . 14 ((((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) ∧ 𝑑 ∈ (0...𝐴)) → (((coeff‘𝑎)‘𝑑) · (𝑐𝑑)) = (((coeff‘𝑏)‘𝑑) · (𝑐𝑑)))
9998sumeq2dv 15739 . . . . . . . . . . . . 13 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐𝑑)) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐𝑑)))
100 simp-4l 782 . . . . . . . . . . . . . 14 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑎 ∈ (Poly‘ℤ))
101 simp-4r 783 . . . . . . . . . . . . . . 15 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑎) ≤ 𝐴)
102 dgrcl 26273 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (Poly‘ℤ) → (deg‘𝑎) ∈ ℕ0)
103 nn0z 12640 . . . . . . . . . . . . . . . . 17 ((deg‘𝑎) ∈ ℕ0 → (deg‘𝑎) ∈ ℤ)
104100, 102, 1033syl 18 . . . . . . . . . . . . . . . 16 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑎) ∈ ℤ)
105 simplrl 776 . . . . . . . . . . . . . . . . 17 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ ℕ0)
106105nn0zd 12641 . . . . . . . . . . . . . . . 16 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ ℤ)
107 eluz 12893 . . . . . . . . . . . . . . . 16 (((deg‘𝑎) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
108104, 106, 107syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝐴 ∈ (ℤ‘(deg‘𝑎)) ↔ (deg‘𝑎) ≤ 𝐴))
109101, 108mpbird 257 . . . . . . . . . . . . . 14 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ (ℤ‘(deg‘𝑎)))
110 simpr 484 . . . . . . . . . . . . . 14 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
111 eqid 2736 . . . . . . . . . . . . . . 15 (deg‘𝑎) = (deg‘𝑎)
11233, 111coeid3 26280 . . . . . . . . . . . . . 14 ((𝑎 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ (ℤ‘(deg‘𝑎)) ∧ 𝑐 ∈ ℂ) → (𝑎𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐𝑑)))
113100, 109, 110, 112syl3anc 1372 . . . . . . . . . . . . 13 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑎𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑎)‘𝑑) · (𝑐𝑑)))
114 simp1rl 1238 . . . . . . . . . . . . . . 15 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴))) ∧ 𝑐 ∈ ℂ) → 𝑏 ∈ (Poly‘ℤ))
1151143expa 1118 . . . . . . . . . . . . . 14 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝑏 ∈ (Poly‘ℤ))
116 simplrr 777 . . . . . . . . . . . . . . . 16 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → (deg‘𝑏) ≤ 𝐴)
117116adantr 480 . . . . . . . . . . . . . . 15 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑏) ≤ 𝐴)
118 dgrcl 26273 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ (Poly‘ℤ) → (deg‘𝑏) ∈ ℕ0)
119 nn0z 12640 . . . . . . . . . . . . . . . . 17 ((deg‘𝑏) ∈ ℕ0 → (deg‘𝑏) ∈ ℤ)
120115, 118, 1193syl 18 . . . . . . . . . . . . . . . 16 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (deg‘𝑏) ∈ ℤ)
121 eluz 12893 . . . . . . . . . . . . . . . 16 (((deg‘𝑏) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
122120, 106, 121syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝐴 ∈ (ℤ‘(deg‘𝑏)) ↔ (deg‘𝑏) ≤ 𝐴))
123117, 122mpbird 257 . . . . . . . . . . . . . 14 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → 𝐴 ∈ (ℤ‘(deg‘𝑏)))
124 eqid 2736 . . . . . . . . . . . . . . 15 (coeff‘𝑏) = (coeff‘𝑏)
125 eqid 2736 . . . . . . . . . . . . . . 15 (deg‘𝑏) = (deg‘𝑏)
126124, 125coeid3 26280 . . . . . . . . . . . . . 14 ((𝑏 ∈ (Poly‘ℤ) ∧ 𝐴 ∈ (ℤ‘(deg‘𝑏)) ∧ 𝑐 ∈ ℂ) → (𝑏𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐𝑑)))
127115, 123, 110, 126syl3anc 1372 . . . . . . . . . . . . 13 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑏𝑐) = Σ𝑑 ∈ (0...𝐴)(((coeff‘𝑏)‘𝑑) · (𝑐𝑑)))
12899, 113, 1273eqtr4d 2786 . . . . . . . . . . . 12 (((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) ∧ 𝑐 ∈ ℂ) → (𝑎𝑐) = (𝑏𝑐))
12985, 89, 128eqfnfvd 7053 . . . . . . . . . . 11 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ (𝐴 ∈ ℕ0 ∧ ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))) → 𝑎 = 𝑏)
130129expr 456 . . . . . . . . . 10 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ 𝐴 ∈ ℕ0) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) → 𝑎 = 𝑏))
131 fveq2 6905 . . . . . . . . . . 11 (𝑎 = 𝑏 → (coeff‘𝑎) = (coeff‘𝑏))
132131reseq1d 5995 . . . . . . . . . 10 (𝑎 = 𝑏 → ((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)))
133130, 132impbid1 225 . . . . . . . . 9 ((((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) ∧ 𝐴 ∈ ℕ0) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏))
134133expcom 413 . . . . . . . 8 (𝐴 ∈ ℕ0 → (((𝑎 ∈ (Poly‘ℤ) ∧ (deg‘𝑎) ≤ 𝐴) ∧ (𝑏 ∈ (Poly‘ℤ) ∧ (deg‘𝑏) ≤ 𝐴)) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏)))
13568, 81, 134syl2ani 607 . . . . . . 7 (𝐴 ∈ ℕ0 → ((𝑎 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∧ 𝑏 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)}) → (((coeff‘𝑎) ↾ (0...𝐴)) = ((coeff‘𝑏) ↾ (0...𝐴)) ↔ 𝑎 = 𝑏)))
13665, 135dom2d 9034 . . . . . 6 (𝐴 ∈ ℕ0 → (((-𝐴...𝐴) ↑m (0...𝐴)) ∈ V → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴))))
13718, 136mpi 20 . . . . 5 (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴)))
138 domfi 9230 . . . . 5 ((((-𝐴...𝐴) ↑m (0...𝐴)) ∈ Fin ∧ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ≼ ((-𝐴...𝐴) ↑m (0...𝐴))) → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
13917, 137, 138syl2anc 584 . . . 4 (𝐴 ∈ ℕ0 → {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin)
140 neeq1 3002 . . . . . . . . 9 (𝑑 = 𝑐 → (𝑑 ≠ 0𝑝𝑐 ≠ 0𝑝))
141 fveq2 6905 . . . . . . . . . 10 (𝑑 = 𝑐 → (deg‘𝑑) = (deg‘𝑐))
142141breq1d 5152 . . . . . . . . 9 (𝑑 = 𝑐 → ((deg‘𝑑) ≤ 𝐴 ↔ (deg‘𝑐) ≤ 𝐴))
143 fveq2 6905 . . . . . . . . . . . . 13 (𝑑 = 𝑐 → (coeff‘𝑑) = (coeff‘𝑐))
144143fveq1d 6907 . . . . . . . . . . . 12 (𝑑 = 𝑐 → ((coeff‘𝑑)‘𝑒) = ((coeff‘𝑐)‘𝑒))
145144fveq2d 6909 . . . . . . . . . . 11 (𝑑 = 𝑐 → (abs‘((coeff‘𝑑)‘𝑒)) = (abs‘((coeff‘𝑐)‘𝑒)))
146145breq1d 5152 . . . . . . . . . 10 (𝑑 = 𝑐 → ((abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
147146ralbidv 3177 . . . . . . . . 9 (𝑑 = 𝑐 → (∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴 ↔ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴))
148140, 142, 1473anbi123d 1437 . . . . . . . 8 (𝑑 = 𝑐 → ((𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴) ↔ (𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)))
149148elrab 3691 . . . . . . 7 (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ↔ (𝑐 ∈ (Poly‘ℤ) ∧ (𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)))
150 simp1 1136 . . . . . . . 8 ((𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴) → 𝑐 ≠ 0𝑝)
151150anim2i 617 . . . . . . 7 ((𝑐 ∈ (Poly‘ℤ) ∧ (𝑐 ≠ 0𝑝 ∧ (deg‘𝑐) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑐)‘𝑒)) ≤ 𝐴)) → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
152149, 151sylbi 217 . . . . . 6 (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → (𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝))
153 fveqeq2 6914 . . . . . . . . . . 11 (𝑏 = 𝑎 → ((𝑐𝑏) = 0 ↔ (𝑐𝑎) = 0))
154153elrab 3691 . . . . . . . . . 10 (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ↔ (𝑎 ∈ ℂ ∧ (𝑐𝑎) = 0))
155 plyf 26238 . . . . . . . . . . . . 13 (𝑐 ∈ (Poly‘ℤ) → 𝑐:ℂ⟶ℂ)
156155ffnd 6736 . . . . . . . . . . . 12 (𝑐 ∈ (Poly‘ℤ) → 𝑐 Fn ℂ)
157156adantr 480 . . . . . . . . . . 11 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → 𝑐 Fn ℂ)
158 fniniseg 7079 . . . . . . . . . . 11 (𝑐 Fn ℂ → (𝑎 ∈ (𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐𝑎) = 0)))
159157, 158syl 17 . . . . . . . . . 10 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ (𝑐 “ {0}) ↔ (𝑎 ∈ ℂ ∧ (𝑐𝑎) = 0)))
160154, 159bitr4id 290 . . . . . . . . 9 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑎 ∈ {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ↔ 𝑎 ∈ (𝑐 “ {0})))
161160eqrdv 2734 . . . . . . . 8 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} = (𝑐 “ {0}))
162 eqid 2736 . . . . . . . . . 10 (𝑐 “ {0}) = (𝑐 “ {0})
163162fta1 26351 . . . . . . . . 9 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → ((𝑐 “ {0}) ∈ Fin ∧ (♯‘(𝑐 “ {0})) ≤ (deg‘𝑐)))
164163simpld 494 . . . . . . . 8 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → (𝑐 “ {0}) ∈ Fin)
165161, 164eqeltrd 2840 . . . . . . 7 ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin)
166165a1i 11 . . . . . 6 (𝐴 ∈ ℕ0 → ((𝑐 ∈ (Poly‘ℤ) ∧ 𝑐 ≠ 0𝑝) → {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin))
167152, 166syl5 34 . . . . 5 (𝐴 ∈ ℕ0 → (𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} → {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin))
168167ralrimiv 3144 . . . 4 (𝐴 ∈ ℕ0 → ∀𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin)
169 iunfi 9384 . . . 4 (({𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} ∈ Fin ∧ ∀𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin) → 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin)
170139, 168, 169syl2anc 584 . . 3 (𝐴 ∈ ℕ0 𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} {𝑏 ∈ ℂ ∣ (𝑐𝑏) = 0} ∈ Fin)
17112, 170eqeltrrid 2845 . 2 (𝐴 ∈ ℕ0 → {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ {𝑑 ∈ (Poly‘ℤ) ∣ (𝑑 ≠ 0𝑝 ∧ (deg‘𝑑) ≤ 𝐴 ∧ ∀𝑒 ∈ ℕ0 (abs‘((coeff‘𝑑)‘𝑒)) ≤ 𝐴)} (𝑐𝑏) = 0} ∈ Fin)
17211, 171eqeltrd 2840 1 (𝐴 ∈ ℕ0 → (𝐻𝐴) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  wss 3950  {csn 4625   ciun 4990   class class class wbr 5142  cmpt 5224  ccnv 5683  ran crn 5685  cres 5686  cima 5687   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  cdom 8984  Fincfn 8986  cc 11154  cr 11155  0cc0 11156   · cmul 11161  cle 11297  -cneg 11494  0cn0 12528  cz 12615  cuz 12879  ...cfz 13548  cexp 14103  chash 14370  abscabs 15274  Σcsu 15723  0𝑝c0p 25705  Polycply 26224  coeffccoe 26226  degcdgr 26227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-rlim 15526  df-sum 15724  df-0p 25706  df-ply 26228  df-idp 26229  df-coe 26230  df-dgr 26231  df-quot 26334
This theorem is referenced by:  aannenlem3  26373
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