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Theorem vieta1lem2 26367
Description: Lemma for vieta1 26368: inductive step. Let 𝑧 be a root of 𝐹. Then 𝐹 = (Xp𝑧) · 𝑄 for some 𝑄 by the factor theorem, and 𝑄 is a degree- 𝐷 polynomial, so by the induction hypothesis Σ𝑥 ∈ (𝑄 “ 0)𝑥 = -(coeff‘𝑄)‘(𝐷 − 1) / (coeff‘𝑄)‘𝐷, so Σ𝑥𝑅𝑥 = 𝑧 − (coeff‘𝑄)‘ (𝐷 − 1) / (coeff‘𝑄)‘𝐷. Now the coefficients of 𝐹 are 𝐴‘(𝐷 + 1) = (coeff‘𝑄)‘𝐷 and 𝐴𝐷 = Σ𝑘 ∈ (0...𝐷)(coeff‘Xp𝑧)‘𝑘 · (coeff‘𝑄) ‘(𝐷𝑘), which works out to -𝑧 · (coeff‘𝑄)‘𝐷 + (coeff‘𝑄)‘(𝐷 − 1), so putting it all together we have Σ𝑥𝑅𝑥 = -𝐴𝐷 / 𝐴‘(𝐷 + 1) as we wanted to show. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
vieta1.1 𝐴 = (coeff‘𝐹)
vieta1.2 𝑁 = (deg‘𝐹)
vieta1.3 𝑅 = (𝐹 “ {0})
vieta1.4 (𝜑𝐹 ∈ (Poly‘𝑆))
vieta1.5 (𝜑 → (♯‘𝑅) = 𝑁)
vieta1lem.6 (𝜑𝐷 ∈ ℕ)
vieta1lem.7 (𝜑 → (𝐷 + 1) = 𝑁)
vieta1lem.8 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
vieta1lem.9 𝑄 = (𝐹 quot (Xpf − (ℂ × {𝑧})))
Assertion
Ref Expression
vieta1lem2 (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
Distinct variable groups:   𝐷,𝑓   𝑓,𝐹   𝑧,𝑓,𝑁   𝑥,𝑓,𝑄   𝑅,𝑓   𝑥,𝑧,𝑅   𝐴,𝑓,𝑧   𝜑,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑥)   𝐷(𝑥,𝑧)   𝑄(𝑧)   𝑆(𝑥,𝑧,𝑓)   𝐹(𝑥,𝑧)   𝑁(𝑥)

Proof of Theorem vieta1lem2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 vieta1.5 . . . . 5 (𝜑 → (♯‘𝑅) = 𝑁)
2 vieta1lem.7 . . . . . . 7 (𝜑 → (𝐷 + 1) = 𝑁)
3 vieta1lem.6 . . . . . . . 8 (𝜑𝐷 ∈ ℕ)
43peano2nnd 12280 . . . . . . 7 (𝜑 → (𝐷 + 1) ∈ ℕ)
52, 4eqeltrrd 2839 . . . . . 6 (𝜑𝑁 ∈ ℕ)
65nnne0d 12313 . . . . 5 (𝜑𝑁 ≠ 0)
71, 6eqnetrd 3005 . . . 4 (𝜑 → (♯‘𝑅) ≠ 0)
8 vieta1.4 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘𝑆))
9 vieta1.2 . . . . . . . . . 10 𝑁 = (deg‘𝐹)
109, 6eqnetrrid 3013 . . . . . . . . 9 (𝜑 → (deg‘𝐹) ≠ 0)
11 fveq2 6906 . . . . . . . . . . 11 (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝))
12 dgr0 26316 . . . . . . . . . . 11 (deg‘0𝑝) = 0
1311, 12eqtrdi 2790 . . . . . . . . . 10 (𝐹 = 0𝑝 → (deg‘𝐹) = 0)
1413necon3i 2970 . . . . . . . . 9 ((deg‘𝐹) ≠ 0 → 𝐹 ≠ 0𝑝)
1510, 14syl 17 . . . . . . . 8 (𝜑𝐹 ≠ 0𝑝)
16 vieta1.3 . . . . . . . . 9 𝑅 = (𝐹 “ {0})
1716fta1 26364 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
188, 15, 17syl2anc 584 . . . . . . 7 (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
1918simpld 494 . . . . . 6 (𝜑𝑅 ∈ Fin)
20 hasheq0 14398 . . . . . 6 (𝑅 ∈ Fin → ((♯‘𝑅) = 0 ↔ 𝑅 = ∅))
2119, 20syl 17 . . . . 5 (𝜑 → ((♯‘𝑅) = 0 ↔ 𝑅 = ∅))
2221necon3bid 2982 . . . 4 (𝜑 → ((♯‘𝑅) ≠ 0 ↔ 𝑅 ≠ ∅))
237, 22mpbid 232 . . 3 (𝜑𝑅 ≠ ∅)
24 n0 4358 . . 3 (𝑅 ≠ ∅ ↔ ∃𝑧 𝑧𝑅)
2523, 24sylib 218 . 2 (𝜑 → ∃𝑧 𝑧𝑅)
26 incom 4216 . . . . 5 ({𝑧} ∩ (𝑄 “ {0})) = ((𝑄 “ {0}) ∩ {𝑧})
27 vieta1.1 . . . . . . . . . . 11 𝐴 = (coeff‘𝐹)
28 vieta1lem.8 . . . . . . . . . . 11 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
29 vieta1lem.9 . . . . . . . . . . 11 𝑄 = (𝐹 quot (Xpf − (ℂ × {𝑧})))
3027, 9, 16, 8, 1, 3, 2, 28, 29vieta1lem1 26366 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄)))
3130simprd 495 . . . . . . . . 9 ((𝜑𝑧𝑅) → 𝐷 = (deg‘𝑄))
3230simpld 494 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → 𝑄 ∈ (Poly‘ℂ))
33 dgrcl 26286 . . . . . . . . . . 11 (𝑄 ∈ (Poly‘ℂ) → (deg‘𝑄) ∈ ℕ0)
3432, 33syl 17 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (deg‘𝑄) ∈ ℕ0)
3534nn0red 12585 . . . . . . . . 9 ((𝜑𝑧𝑅) → (deg‘𝑄) ∈ ℝ)
3631, 35eqeltrd 2838 . . . . . . . 8 ((𝜑𝑧𝑅) → 𝐷 ∈ ℝ)
3736ltp1d 12195 . . . . . . . 8 ((𝜑𝑧𝑅) → 𝐷 < (𝐷 + 1))
3836, 37gtned 11393 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐷 + 1) ≠ 𝐷)
39 snssi 4812 . . . . . . . . . . 11 (𝑧 ∈ (𝑄 “ {0}) → {𝑧} ⊆ (𝑄 “ {0}))
40 ssequn1 4195 . . . . . . . . . . 11 ({𝑧} ⊆ (𝑄 “ {0}) ↔ ({𝑧} ∪ (𝑄 “ {0})) = (𝑄 “ {0}))
4139, 40sylib 218 . . . . . . . . . 10 (𝑧 ∈ (𝑄 “ {0}) → ({𝑧} ∪ (𝑄 “ {0})) = (𝑄 “ {0}))
4241fveq2d 6910 . . . . . . . . 9 (𝑧 ∈ (𝑄 “ {0}) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) = (♯‘(𝑄 “ {0})))
438adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝐹 ∈ (Poly‘𝑆))
44 cnvimass 6101 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 “ {0}) ⊆ dom 𝐹
4516, 44eqsstri 4029 . . . . . . . . . . . . . . . . . . . 20 𝑅 ⊆ dom 𝐹
46 plyf 26251 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47 fdm 6745 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℂ⟶ℂ → dom 𝐹 = ℂ)
488, 46, 473syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = ℂ)
4945, 48sseqtrid 4047 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑅 ⊆ ℂ)
5049sselda 3994 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝑧 ∈ ℂ)
5116eleq2i 2830 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑅𝑧 ∈ (𝐹 “ {0}))
52 ffn 6736 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
53 fniniseg 7079 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn ℂ → (𝑧 ∈ (𝐹 “ {0}) ↔ (𝑧 ∈ ℂ ∧ (𝐹𝑧) = 0)))
548, 46, 52, 534syl 19 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑧 ∈ (𝐹 “ {0}) ↔ (𝑧 ∈ ℂ ∧ (𝐹𝑧) = 0)))
5551, 54bitrid 283 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑧𝑅 ↔ (𝑧 ∈ ℂ ∧ (𝐹𝑧) = 0)))
5655simplbda 499 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → (𝐹𝑧) = 0)
57 eqid 2734 . . . . . . . . . . . . . . . . . . 19 (Xpf − (ℂ × {𝑧})) = (Xpf − (ℂ × {𝑧}))
5857facth 26362 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑧 ∈ ℂ ∧ (𝐹𝑧) = 0) → 𝐹 = ((Xpf − (ℂ × {𝑧})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝑧})))))
5943, 50, 56, 58syl3anc 1370 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → 𝐹 = ((Xpf − (ℂ × {𝑧})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝑧})))))
6029oveq2i 7441 . . . . . . . . . . . . . . . . 17 ((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) = ((Xpf − (ℂ × {𝑧})) ∘f · (𝐹 quot (Xpf − (ℂ × {𝑧}))))
6159, 60eqtr4di 2792 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → 𝐹 = ((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))
6261cnveqd 5888 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → 𝐹 = ((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))
6362imaeq1d 6078 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (𝐹 “ {0}) = (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) “ {0}))
6416, 63eqtrid 2786 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → 𝑅 = (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) “ {0}))
65 cnex 11233 . . . . . . . . . . . . . 14 ℂ ∈ V
6657plyremlem 26360 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℂ → ((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ (deg‘(Xpf − (ℂ × {𝑧}))) = 1 ∧ ((Xpf − (ℂ × {𝑧})) “ {0}) = {𝑧}))
6750, 66syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → ((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ (deg‘(Xpf − (ℂ × {𝑧}))) = 1 ∧ ((Xpf − (ℂ × {𝑧})) “ {0}) = {𝑧}))
6867simp1d 1141 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ))
69 plyf 26251 . . . . . . . . . . . . . . 15 ((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) → (Xpf − (ℂ × {𝑧})):ℂ⟶ℂ)
7068, 69syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (Xpf − (ℂ × {𝑧})):ℂ⟶ℂ)
71 plyf 26251 . . . . . . . . . . . . . . 15 (𝑄 ∈ (Poly‘ℂ) → 𝑄:ℂ⟶ℂ)
7232, 71syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → 𝑄:ℂ⟶ℂ)
73 ofmulrt 26337 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ (Xpf − (ℂ × {𝑧})):ℂ⟶ℂ ∧ 𝑄:ℂ⟶ℂ) → (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) “ {0}) = (((Xpf − (ℂ × {𝑧})) “ {0}) ∪ (𝑄 “ {0})))
7465, 70, 72, 73mp3an2i 1465 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) “ {0}) = (((Xpf − (ℂ × {𝑧})) “ {0}) ∪ (𝑄 “ {0})))
7567simp3d 1143 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ((Xpf − (ℂ × {𝑧})) “ {0}) = {𝑧})
7675uneq1d 4176 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((Xpf − (ℂ × {𝑧})) “ {0}) ∪ (𝑄 “ {0})) = ({𝑧} ∪ (𝑄 “ {0})))
7764, 74, 763eqtrd 2778 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → 𝑅 = ({𝑧} ∪ (𝑄 “ {0})))
7877fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (♯‘𝑅) = (♯‘({𝑧} ∪ (𝑄 “ {0}))))
791, 2eqtr4d 2777 . . . . . . . . . . . 12 (𝜑 → (♯‘𝑅) = (𝐷 + 1))
8079adantr 480 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (♯‘𝑅) = (𝐷 + 1))
8178, 80eqtr3d 2776 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) = (𝐷 + 1))
8215adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝐹 ≠ 0𝑝)
8361, 82eqnetrrd 3006 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → ((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) ≠ 0𝑝)
84 plymul0or 26336 . . . . . . . . . . . . . . . . . . 19 (((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑄 ∈ (Poly‘ℂ)) → (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) = 0𝑝 ↔ ((Xpf − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝)))
8568, 32, 84syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) = 0𝑝 ↔ ((Xpf − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝)))
8685necon3abid 2974 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (((Xpf − (ℂ × {𝑧})) ∘f · 𝑄) ≠ 0𝑝 ↔ ¬ ((Xpf − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝)))
8783, 86mpbid 232 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → ¬ ((Xpf − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝))
88 neanior 3032 . . . . . . . . . . . . . . . 16 (((Xpf − (ℂ × {𝑧})) ≠ 0𝑝𝑄 ≠ 0𝑝) ↔ ¬ ((Xpf − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝))
8987, 88sylibr 234 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ((Xpf − (ℂ × {𝑧})) ≠ 0𝑝𝑄 ≠ 0𝑝))
9089simprd 495 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → 𝑄 ≠ 0𝑝)
91 eqid 2734 . . . . . . . . . . . . . . 15 (𝑄 “ {0}) = (𝑄 “ {0})
9291fta1 26364 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Poly‘ℂ) ∧ 𝑄 ≠ 0𝑝) → ((𝑄 “ {0}) ∈ Fin ∧ (♯‘(𝑄 “ {0})) ≤ (deg‘𝑄)))
9332, 90, 92syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → ((𝑄 “ {0}) ∈ Fin ∧ (♯‘(𝑄 “ {0})) ≤ (deg‘𝑄)))
9493simprd 495 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ≤ (deg‘𝑄))
9594, 31breqtrrd 5175 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ≤ 𝐷)
96 snfi 9081 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
9793simpld 494 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (𝑄 “ {0}) ∈ Fin)
98 hashun2 14418 . . . . . . . . . . . . . 14 (({𝑧} ∈ Fin ∧ (𝑄 “ {0}) ∈ Fin) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) ≤ ((♯‘{𝑧}) + (♯‘(𝑄 “ {0}))))
9996, 97, 98sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) ≤ ((♯‘{𝑧}) + (♯‘(𝑄 “ {0}))))
100 ax-1cn 11210 . . . . . . . . . . . . . . 15 1 ∈ ℂ
1013nncnd 12279 . . . . . . . . . . . . . . . 16 (𝜑𝐷 ∈ ℂ)
102101adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → 𝐷 ∈ ℂ)
103 addcom 11444 . . . . . . . . . . . . . . 15 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
104100, 102, 103sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (1 + 𝐷) = (𝐷 + 1))
10581, 104eqtr4d 2777 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) = (1 + 𝐷))
106 hashsng 14404 . . . . . . . . . . . . . . 15 (𝑧𝑅 → (♯‘{𝑧}) = 1)
107106adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (♯‘{𝑧}) = 1)
108107oveq1d 7445 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → ((♯‘{𝑧}) + (♯‘(𝑄 “ {0}))) = (1 + (♯‘(𝑄 “ {0}))))
10999, 105, 1083brtr3d 5178 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (1 + 𝐷) ≤ (1 + (♯‘(𝑄 “ {0}))))
110 hashcl 14391 . . . . . . . . . . . . . . 15 ((𝑄 “ {0}) ∈ Fin → (♯‘(𝑄 “ {0})) ∈ ℕ0)
11197, 110syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ∈ ℕ0)
112111nn0red 12585 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ∈ ℝ)
113 1red 11259 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → 1 ∈ ℝ)
11436, 112, 113leadd2d 11855 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (𝐷 ≤ (♯‘(𝑄 “ {0})) ↔ (1 + 𝐷) ≤ (1 + (♯‘(𝑄 “ {0})))))
115109, 114mpbird 257 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → 𝐷 ≤ (♯‘(𝑄 “ {0})))
116112, 36letri3d 11400 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → ((♯‘(𝑄 “ {0})) = 𝐷 ↔ ((♯‘(𝑄 “ {0})) ≤ 𝐷𝐷 ≤ (♯‘(𝑄 “ {0})))))
11795, 115, 116mpbir2and 713 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) = 𝐷)
11881, 117eqeq12d 2750 . . . . . . . . 9 ((𝜑𝑧𝑅) → ((♯‘({𝑧} ∪ (𝑄 “ {0}))) = (♯‘(𝑄 “ {0})) ↔ (𝐷 + 1) = 𝐷))
11942, 118imbitrid 244 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝑧 ∈ (𝑄 “ {0}) → (𝐷 + 1) = 𝐷))
120119necon3ad 2950 . . . . . . 7 ((𝜑𝑧𝑅) → ((𝐷 + 1) ≠ 𝐷 → ¬ 𝑧 ∈ (𝑄 “ {0})))
12138, 120mpd 15 . . . . . 6 ((𝜑𝑧𝑅) → ¬ 𝑧 ∈ (𝑄 “ {0}))
122 disjsn 4715 . . . . . 6 (((𝑄 “ {0}) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑄 “ {0}))
123121, 122sylibr 234 . . . . 5 ((𝜑𝑧𝑅) → ((𝑄 “ {0}) ∩ {𝑧}) = ∅)
12426, 123eqtrid 2786 . . . 4 ((𝜑𝑧𝑅) → ({𝑧} ∩ (𝑄 “ {0})) = ∅)
12519adantr 480 . . . 4 ((𝜑𝑧𝑅) → 𝑅 ∈ Fin)
12649adantr 480 . . . . 5 ((𝜑𝑧𝑅) → 𝑅 ⊆ ℂ)
127126sselda 3994 . . . 4 (((𝜑𝑧𝑅) ∧ 𝑥𝑅) → 𝑥 ∈ ℂ)
128124, 77, 125, 127fsumsplit 15773 . . 3 ((𝜑𝑧𝑅) → Σ𝑥𝑅 𝑥 = (Σ𝑥 ∈ {𝑧}𝑥 + Σ𝑥 ∈ (𝑄 “ {0})𝑥))
129 id 22 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
130129sumsn 15778 . . . . . 6 ((𝑧 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑥 ∈ {𝑧}𝑥 = 𝑧)
13150, 50, 130syl2anc 584 . . . . 5 ((𝜑𝑧𝑅) → Σ𝑥 ∈ {𝑧}𝑥 = 𝑧)
13250negnegd 11608 . . . . 5 ((𝜑𝑧𝑅) → --𝑧 = 𝑧)
133131, 132eqtr4d 2777 . . . 4 ((𝜑𝑧𝑅) → Σ𝑥 ∈ {𝑧}𝑥 = --𝑧)
134117, 31eqtrd 2774 . . . . . 6 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) = (deg‘𝑄))
135 fveq2 6906 . . . . . . . . . 10 (𝑓 = 𝑄 → (deg‘𝑓) = (deg‘𝑄))
136135eqeq2d 2745 . . . . . . . . 9 (𝑓 = 𝑄 → (𝐷 = (deg‘𝑓) ↔ 𝐷 = (deg‘𝑄)))
137 cnveq 5886 . . . . . . . . . . . 12 (𝑓 = 𝑄𝑓 = 𝑄)
138137imaeq1d 6078 . . . . . . . . . . 11 (𝑓 = 𝑄 → (𝑓 “ {0}) = (𝑄 “ {0}))
139138fveq2d 6910 . . . . . . . . . 10 (𝑓 = 𝑄 → (♯‘(𝑓 “ {0})) = (♯‘(𝑄 “ {0})))
140139, 135eqeq12d 2750 . . . . . . . . 9 (𝑓 = 𝑄 → ((♯‘(𝑓 “ {0})) = (deg‘𝑓) ↔ (♯‘(𝑄 “ {0})) = (deg‘𝑄)))
141136, 140anbi12d 632 . . . . . . . 8 (𝑓 = 𝑄 → ((𝐷 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝐷 = (deg‘𝑄) ∧ (♯‘(𝑄 “ {0})) = (deg‘𝑄))))
142138sumeq1d 15732 . . . . . . . . 9 (𝑓 = 𝑄 → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = Σ𝑥 ∈ (𝑄 “ {0})𝑥)
143 fveq2 6906 . . . . . . . . . . . 12 (𝑓 = 𝑄 → (coeff‘𝑓) = (coeff‘𝑄))
144135oveq1d 7445 . . . . . . . . . . . 12 (𝑓 = 𝑄 → ((deg‘𝑓) − 1) = ((deg‘𝑄) − 1))
145143, 144fveq12d 6913 . . . . . . . . . . 11 (𝑓 = 𝑄 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = ((coeff‘𝑄)‘((deg‘𝑄) − 1)))
146143, 135fveq12d 6913 . . . . . . . . . . 11 (𝑓 = 𝑄 → ((coeff‘𝑓)‘(deg‘𝑓)) = ((coeff‘𝑄)‘(deg‘𝑄)))
147145, 146oveq12d 7448 . . . . . . . . . 10 (𝑓 = 𝑄 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = (((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
148147negeqd 11499 . . . . . . . . 9 (𝑓 = 𝑄 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
149142, 148eqeq12d 2750 . . . . . . . 8 (𝑓 = 𝑄 → (Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄)))))
150141, 149imbi12d 344 . . . . . . 7 (𝑓 = 𝑄 → (((𝐷 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝐷 = (deg‘𝑄) ∧ (♯‘(𝑄 “ {0})) = (deg‘𝑄)) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))))
15128adantr 480 . . . . . . 7 ((𝜑𝑧𝑅) → ∀𝑓 ∈ (Poly‘ℂ)((𝐷 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
152150, 151, 32rspcdva 3622 . . . . . 6 ((𝜑𝑧𝑅) → ((𝐷 = (deg‘𝑄) ∧ (♯‘(𝑄 “ {0})) = (deg‘𝑄)) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄)))))
15331, 134, 152mp2and 699 . . . . 5 ((𝜑𝑧𝑅) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
15431fvoveq1d 7452 . . . . . . 7 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘(𝐷 − 1)) = ((coeff‘𝑄)‘((deg‘𝑄) − 1)))
15561fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (coeff‘𝐹) = (coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄)))
15627, 155eqtrid 2786 . . . . . . . . 9 ((𝜑𝑧𝑅) → 𝐴 = (coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄)))
15761fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (deg‘𝐹) = (deg‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄)))
15867simp2d 1142 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (deg‘(Xpf − (ℂ × {𝑧}))) = 1)
159 ax-1ne0 11221 . . . . . . . . . . . . . . 15 1 ≠ 0
160159a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → 1 ≠ 0)
161158, 160eqnetrd 3005 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (deg‘(Xpf − (ℂ × {𝑧}))) ≠ 0)
162 fveq2 6906 . . . . . . . . . . . . . . 15 ((Xpf − (ℂ × {𝑧})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝑧}))) = (deg‘0𝑝))
163162, 12eqtrdi 2790 . . . . . . . . . . . . . 14 ((Xpf − (ℂ × {𝑧})) = 0𝑝 → (deg‘(Xpf − (ℂ × {𝑧}))) = 0)
164163necon3i 2970 . . . . . . . . . . . . 13 ((deg‘(Xpf − (ℂ × {𝑧}))) ≠ 0 → (Xpf − (ℂ × {𝑧})) ≠ 0𝑝)
165161, 164syl 17 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (Xpf − (ℂ × {𝑧})) ≠ 0𝑝)
166 eqid 2734 . . . . . . . . . . . . 13 (deg‘(Xpf − (ℂ × {𝑧}))) = (deg‘(Xpf − (ℂ × {𝑧})))
167 eqid 2734 . . . . . . . . . . . . 13 (deg‘𝑄) = (deg‘𝑄)
168166, 167dgrmul 26324 . . . . . . . . . . . 12 ((((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ (Xpf − (ℂ × {𝑧})) ≠ 0𝑝) ∧ (𝑄 ∈ (Poly‘ℂ) ∧ 𝑄 ≠ 0𝑝)) → (deg‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄)) = ((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄)))
16968, 165, 32, 90, 168syl22anc 839 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (deg‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄)) = ((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄)))
170157, 169eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (deg‘𝐹) = ((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄)))
1719, 170eqtrid 2786 . . . . . . . . 9 ((𝜑𝑧𝑅) → 𝑁 = ((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄)))
172156, 171fveq12d 6913 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝐴𝑁) = ((coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))‘((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄))))
173 eqid 2734 . . . . . . . . . 10 (coeff‘(Xpf − (ℂ × {𝑧}))) = (coeff‘(Xpf − (ℂ × {𝑧})))
174 eqid 2734 . . . . . . . . . 10 (coeff‘𝑄) = (coeff‘𝑄)
175173, 174, 166, 167coemulhi 26307 . . . . . . . . 9 (((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑄 ∈ (Poly‘ℂ)) → ((coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))‘((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄))) = (((coeff‘(Xpf − (ℂ × {𝑧})))‘(deg‘(Xpf − (ℂ × {𝑧})))) · ((coeff‘𝑄)‘(deg‘𝑄))))
17668, 32, 175syl2anc 584 . . . . . . . 8 ((𝜑𝑧𝑅) → ((coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))‘((deg‘(Xpf − (ℂ × {𝑧}))) + (deg‘𝑄))) = (((coeff‘(Xpf − (ℂ × {𝑧})))‘(deg‘(Xpf − (ℂ × {𝑧})))) · ((coeff‘𝑄)‘(deg‘𝑄))))
177158fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘(deg‘(Xpf − (ℂ × {𝑧})))) = ((coeff‘(Xpf − (ℂ × {𝑧})))‘1))
178 ssid 4017 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
179 plyid 26262 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
180178, 100, 179mp2an 692 . . . . . . . . . . . . . 14 Xp ∈ (Poly‘ℂ)
181 plyconst 26259 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ 𝑧 ∈ ℂ) → (ℂ × {𝑧}) ∈ (Poly‘ℂ))
182178, 50, 181sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (ℂ × {𝑧}) ∈ (Poly‘ℂ))
183 eqid 2734 . . . . . . . . . . . . . . 15 (coeff‘Xp) = (coeff‘Xp)
184 eqid 2734 . . . . . . . . . . . . . . 15 (coeff‘(ℂ × {𝑧})) = (coeff‘(ℂ × {𝑧}))
185183, 184coesub 26310 . . . . . . . . . . . . . 14 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝑧}) ∈ (Poly‘ℂ)) → (coeff‘(Xpf − (ℂ × {𝑧}))) = ((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧}))))
186180, 182, 185sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (coeff‘(Xpf − (ℂ × {𝑧}))) = ((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧}))))
187186fveq1d 6908 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘1) = (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘1))
188 1nn0 12539 . . . . . . . . . . . . . 14 1 ∈ ℕ0
189183coef3 26285 . . . . . . . . . . . . . . . . 17 (Xp ∈ (Poly‘ℂ) → (coeff‘Xp):ℕ0⟶ℂ)
190 ffn 6736 . . . . . . . . . . . . . . . . 17 ((coeff‘Xp):ℕ0⟶ℂ → (coeff‘Xp) Fn ℕ0)
191180, 189, 190mp2b 10 . . . . . . . . . . . . . . . 16 (coeff‘Xp) Fn ℕ0
192191a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (coeff‘Xp) Fn ℕ0)
193184coef3 26285 . . . . . . . . . . . . . . . 16 ((ℂ × {𝑧}) ∈ (Poly‘ℂ) → (coeff‘(ℂ × {𝑧})):ℕ0⟶ℂ)
194 ffn 6736 . . . . . . . . . . . . . . . 16 ((coeff‘(ℂ × {𝑧})):ℕ0⟶ℂ → (coeff‘(ℂ × {𝑧})) Fn ℕ0)
195182, 193, 1943syl 18 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (coeff‘(ℂ × {𝑧})) Fn ℕ0)
196 nn0ex 12529 . . . . . . . . . . . . . . . 16 0 ∈ V
197196a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ℕ0 ∈ V)
198 inidm 4234 . . . . . . . . . . . . . . 15 (ℕ0 ∩ ℕ0) = ℕ0
199 coeidp 26317 . . . . . . . . . . . . . . . . 17 (1 ∈ ℕ0 → ((coeff‘Xp)‘1) = if(1 = 1, 1, 0))
200199adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ((coeff‘Xp)‘1) = if(1 = 1, 1, 0))
201 eqid 2734 . . . . . . . . . . . . . . . . 17 1 = 1
202201iftruei 4537 . . . . . . . . . . . . . . . 16 if(1 = 1, 1, 0) = 1
203200, 202eqtrdi 2790 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ((coeff‘Xp)‘1) = 1)
204 0lt1 11782 . . . . . . . . . . . . . . . . . 18 0 < 1
205 0re 11260 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ
206 1re 11258 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
207205, 206ltnlei 11379 . . . . . . . . . . . . . . . . . 18 (0 < 1 ↔ ¬ 1 ≤ 0)
208204, 207mpbi 230 . . . . . . . . . . . . . . . . 17 ¬ 1 ≤ 0
20950adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → 𝑧 ∈ ℂ)
210 0dgr 26298 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℂ → (deg‘(ℂ × {𝑧})) = 0)
211209, 210syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (deg‘(ℂ × {𝑧})) = 0)
212211breq2d 5159 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (1 ≤ (deg‘(ℂ × {𝑧})) ↔ 1 ≤ 0))
213208, 212mtbiri 327 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ¬ 1 ≤ (deg‘(ℂ × {𝑧})))
214 eqid 2734 . . . . . . . . . . . . . . . . . . . 20 (deg‘(ℂ × {𝑧})) = (deg‘(ℂ × {𝑧}))
215184, 214dgrub 26287 . . . . . . . . . . . . . . . . . . 19 (((ℂ × {𝑧}) ∈ (Poly‘ℂ) ∧ 1 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝑧}))‘1) ≠ 0) → 1 ≤ (deg‘(ℂ × {𝑧})))
2162153expia 1120 . . . . . . . . . . . . . . . . . 18 (((ℂ × {𝑧}) ∈ (Poly‘ℂ) ∧ 1 ∈ ℕ0) → (((coeff‘(ℂ × {𝑧}))‘1) ≠ 0 → 1 ≤ (deg‘(ℂ × {𝑧}))))
217182, 216sylan 580 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (((coeff‘(ℂ × {𝑧}))‘1) ≠ 0 → 1 ≤ (deg‘(ℂ × {𝑧}))))
218217necon1bd 2955 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (¬ 1 ≤ (deg‘(ℂ × {𝑧})) → ((coeff‘(ℂ × {𝑧}))‘1) = 0))
219213, 218mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ((coeff‘(ℂ × {𝑧}))‘1) = 0)
220192, 195, 197, 197, 198, 203, 219ofval 7707 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘1) = (1 − 0))
221188, 220mpan2 691 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘1) = (1 − 0))
222 1m0e1 12384 . . . . . . . . . . . . 13 (1 − 0) = 1
223221, 222eqtrdi 2790 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘1) = 1)
224187, 223eqtrd 2774 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘1) = 1)
225177, 224eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑧𝑅) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘(deg‘(Xpf − (ℂ × {𝑧})))) = 1)
226225oveq1d 7445 . . . . . . . . 9 ((𝜑𝑧𝑅) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘(deg‘(Xpf − (ℂ × {𝑧})))) · ((coeff‘𝑄)‘(deg‘𝑄))) = (1 · ((coeff‘𝑄)‘(deg‘𝑄))))
227174coef3 26285 . . . . . . . . . . . 12 (𝑄 ∈ (Poly‘ℂ) → (coeff‘𝑄):ℕ0⟶ℂ)
22832, 227syl 17 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (coeff‘𝑄):ℕ0⟶ℂ)
229228, 34ffvelcdmd 7104 . . . . . . . . . 10 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘(deg‘𝑄)) ∈ ℂ)
230229mullidd 11276 . . . . . . . . 9 ((𝜑𝑧𝑅) → (1 · ((coeff‘𝑄)‘(deg‘𝑄))) = ((coeff‘𝑄)‘(deg‘𝑄)))
231226, 230eqtrd 2774 . . . . . . . 8 ((𝜑𝑧𝑅) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘(deg‘(Xpf − (ℂ × {𝑧})))) · ((coeff‘𝑄)‘(deg‘𝑄))) = ((coeff‘𝑄)‘(deg‘𝑄)))
232172, 176, 2313eqtrd 2778 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐴𝑁) = ((coeff‘𝑄)‘(deg‘𝑄)))
233154, 232oveq12d 7448 . . . . . 6 ((𝜑𝑧𝑅) → (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)) = (((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
234233negeqd 11499 . . . . 5 ((𝜑𝑧𝑅) → -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)) = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
235153, 234eqtr4d 2777 . . . 4 ((𝜑𝑧𝑅) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)))
236133, 235oveq12d 7448 . . 3 ((𝜑𝑧𝑅) → (Σ𝑥 ∈ {𝑧}𝑥 + Σ𝑥 ∈ (𝑄 “ {0})𝑥) = (--𝑧 + -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
23750negcld 11604 . . . . 5 ((𝜑𝑧𝑅) → -𝑧 ∈ ℂ)
238 nnm1nn0 12564 . . . . . . . . 9 (𝐷 ∈ ℕ → (𝐷 − 1) ∈ ℕ0)
2393, 238syl 17 . . . . . . . 8 (𝜑 → (𝐷 − 1) ∈ ℕ0)
240239adantr 480 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐷 − 1) ∈ ℕ0)
241228, 240ffvelcdmd 7104 . . . . . 6 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘(𝐷 − 1)) ∈ ℂ)
242232, 229eqeltrd 2838 . . . . . 6 ((𝜑𝑧𝑅) → (𝐴𝑁) ∈ ℂ)
2439, 27dgreq0 26319 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴𝑁) = 0))
24443, 243syl 17 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝐹 = 0𝑝 ↔ (𝐴𝑁) = 0))
245244necon3bid 2982 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐹 ≠ 0𝑝 ↔ (𝐴𝑁) ≠ 0))
24682, 245mpbid 232 . . . . . 6 ((𝜑𝑧𝑅) → (𝐴𝑁) ≠ 0)
247241, 242, 246divcld 12040 . . . . 5 ((𝜑𝑧𝑅) → (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)) ∈ ℂ)
248237, 247negdid 11630 . . . 4 ((𝜑𝑧𝑅) → -(-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = (--𝑧 + -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
249237, 242mulcld 11278 . . . . . . 7 ((𝜑𝑧𝑅) → (-𝑧 · (𝐴𝑁)) ∈ ℂ)
250249, 241, 242, 246divdird 12078 . . . . . 6 ((𝜑𝑧𝑅) → (((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) / (𝐴𝑁)) = (((-𝑧 · (𝐴𝑁)) / (𝐴𝑁)) + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
251 nnm1nn0 12564 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
2525, 251syl 17 . . . . . . . . . 10 (𝜑 → (𝑁 − 1) ∈ ℕ0)
253252adantr 480 . . . . . . . . 9 ((𝜑𝑧𝑅) → (𝑁 − 1) ∈ ℕ0)
254173, 174coemul 26305 . . . . . . . . 9 (((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑄 ∈ (Poly‘ℂ) ∧ (𝑁 − 1) ∈ ℕ0) → ((coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))‘(𝑁 − 1)) = Σ𝑘 ∈ (0...(𝑁 − 1))(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
25568, 32, 253, 254syl3anc 1370 . . . . . . . 8 ((𝜑𝑧𝑅) → ((coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))‘(𝑁 − 1)) = Σ𝑘 ∈ (0...(𝑁 − 1))(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
256156fveq1d 6908 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝐴‘(𝑁 − 1)) = ((coeff‘((Xpf − (ℂ × {𝑧})) ∘f · 𝑄))‘(𝑁 − 1)))
257 1e0p1 12772 . . . . . . . . . . . 12 1 = (0 + 1)
258257oveq2i 7441 . . . . . . . . . . 11 (0...1) = (0...(0 + 1))
259258sumeq1i 15729 . . . . . . . . . 10 Σ𝑘 ∈ (0...1)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = Σ𝑘 ∈ (0...(0 + 1))(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)))
260 0nn0 12538 . . . . . . . . . . . . 13 0 ∈ ℕ0
261 nn0uz 12917 . . . . . . . . . . . . 13 0 = (ℤ‘0)
262260, 261eleqtri 2836 . . . . . . . . . . . 12 0 ∈ (ℤ‘0)
263262a1i 11 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → 0 ∈ (ℤ‘0))
264258eleq2i 2830 . . . . . . . . . . . 12 (𝑘 ∈ (0...1) ↔ 𝑘 ∈ (0...(0 + 1)))
265173coef3 26285 . . . . . . . . . . . . . . 15 ((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) → (coeff‘(Xpf − (ℂ × {𝑧}))):ℕ0⟶ℂ)
26668, 265syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (coeff‘(Xpf − (ℂ × {𝑧}))):ℕ0⟶ℂ)
267 elfznn0 13656 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ0)
268 ffvelcdm 7100 . . . . . . . . . . . . . 14 (((coeff‘(Xpf − (ℂ × {𝑧}))):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) ∈ ℂ)
269266, 267, 268syl2an 596 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) ∈ ℂ)
2702oveq1d 7445 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐷 + 1) − 1) = (𝑁 − 1))
271 pncan 11511 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐷 + 1) − 1) = 𝐷)
272101, 100, 271sylancl 586 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐷 + 1) − 1) = 𝐷)
273270, 272eqtr3d 2776 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) = 𝐷)
274273adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → (𝑁 − 1) = 𝐷)
2753adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝐷 ∈ ℕ)
276274, 275eqeltrd 2838 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (𝑁 − 1) ∈ ℕ)
277 nnuz 12918 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
278276, 277eleqtrdi 2848 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → (𝑁 − 1) ∈ (ℤ‘1))
279 fzss2 13600 . . . . . . . . . . . . . . . 16 ((𝑁 − 1) ∈ (ℤ‘1) → (0...1) ⊆ (0...(𝑁 − 1)))
280278, 279syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (0...1) ⊆ (0...(𝑁 − 1)))
281280sselda 3994 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → 𝑘 ∈ (0...(𝑁 − 1)))
282 fznn0sub 13592 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) − 𝑘) ∈ ℕ0)
283 ffvelcdm 7100 . . . . . . . . . . . . . . 15 (((coeff‘𝑄):ℕ0⟶ℂ ∧ ((𝑁 − 1) − 𝑘) ∈ ℕ0) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
284228, 282, 283syl2an 596 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
285281, 284syldan 591 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
286269, 285mulcld 11278 . . . . . . . . . . . 12 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) ∈ ℂ)
287264, 286sylan2br 595 . . . . . . . . . . 11 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...(0 + 1))) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) ∈ ℂ)
288 id 22 . . . . . . . . . . . . . 14 (𝑘 = (0 + 1) → 𝑘 = (0 + 1))
289288, 257eqtr4di 2792 . . . . . . . . . . . . 13 (𝑘 = (0 + 1) → 𝑘 = 1)
290289fveq2d 6910 . . . . . . . . . . . 12 (𝑘 = (0 + 1) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) = ((coeff‘(Xpf − (ℂ × {𝑧})))‘1))
291289oveq2d 7446 . . . . . . . . . . . . 13 (𝑘 = (0 + 1) → ((𝑁 − 1) − 𝑘) = ((𝑁 − 1) − 1))
292291fveq2d 6910 . . . . . . . . . . . 12 (𝑘 = (0 + 1) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) = ((coeff‘𝑄)‘((𝑁 − 1) − 1)))
293290, 292oveq12d 7448 . . . . . . . . . . 11 (𝑘 = (0 + 1) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (((coeff‘(Xpf − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1))))
294263, 287, 293fsump1 15788 . . . . . . . . . 10 ((𝜑𝑧𝑅) → Σ𝑘 ∈ (0...(0 + 1))(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (Σ𝑘 ∈ (0...0)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) + (((coeff‘(Xpf − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1)))))
295259, 294eqtrid 2786 . . . . . . . . 9 ((𝜑𝑧𝑅) → Σ𝑘 ∈ (0...1)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (Σ𝑘 ∈ (0...0)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) + (((coeff‘(Xpf − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1)))))
296 eldifn 4141 . . . . . . . . . . . . . 14 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → ¬ 𝑘 ∈ (0...1))
297296adantl 481 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → ¬ 𝑘 ∈ (0...1))
298 eldifi 4140 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → 𝑘 ∈ (0...(𝑁 − 1)))
299 elfznn0 13656 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
300298, 299syl 17 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → 𝑘 ∈ ℕ0)
301173, 166dgrub 26287 . . . . . . . . . . . . . . . . 17 (((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(Xpf − (ℂ × {𝑧}))))
3023013expia 1120 . . . . . . . . . . . . . . . 16 (((Xpf − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(Xpf − (ℂ × {𝑧})))))
30368, 300, 302syl2an 596 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(Xpf − (ℂ × {𝑧})))))
304 elfzuz 13556 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ (ℤ‘0))
305298, 304syl 17 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → 𝑘 ∈ (ℤ‘0))
306305adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → 𝑘 ∈ (ℤ‘0))
307 1z 12644 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
308 elfz5 13552 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ (ℤ‘0) ∧ 1 ∈ ℤ) → (𝑘 ∈ (0...1) ↔ 𝑘 ≤ 1))
309306, 307, 308sylancl 586 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (𝑘 ∈ (0...1) ↔ 𝑘 ≤ 1))
310158breq2d 5159 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (𝑘 ≤ (deg‘(Xpf − (ℂ × {𝑧}))) ↔ 𝑘 ≤ 1))
311310adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (𝑘 ≤ (deg‘(Xpf − (ℂ × {𝑧}))) ↔ 𝑘 ≤ 1))
312309, 311bitr4d 282 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (𝑘 ∈ (0...1) ↔ 𝑘 ≤ (deg‘(Xpf − (ℂ × {𝑧})))))
313303, 312sylibrd 259 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) ≠ 0 → 𝑘 ∈ (0...1)))
314313necon1bd 2955 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (¬ 𝑘 ∈ (0...1) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) = 0))
315297, 314mpd 15 . . . . . . . . . . . 12 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) = 0)
316315oveq1d 7445 . . . . . . . . . . 11 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (0 · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
317298, 284sylan2 593 . . . . . . . . . . . 12 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
318317mul02d 11456 . . . . . . . . . . 11 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (0 · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = 0)
319316, 318eqtrd 2774 . . . . . . . . . 10 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = 0)
320 fzfid 14010 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (0...(𝑁 − 1)) ∈ Fin)
321280, 286, 319, 320fsumss 15757 . . . . . . . . 9 ((𝜑𝑧𝑅) → Σ𝑘 ∈ (0...1)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = Σ𝑘 ∈ (0...(𝑁 − 1))(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
322 0z 12621 . . . . . . . . . . . 12 0 ∈ ℤ
323186fveq1d 6908 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘0) = (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘0))
324 coeidp 26317 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℕ0 → ((coeff‘Xp)‘0) = if(0 = 1, 1, 0))
325159nesymi 2995 . . . . . . . . . . . . . . . . . . . . 21 ¬ 0 = 1
326325iffalsei 4540 . . . . . . . . . . . . . . . . . . . 20 if(0 = 1, 1, 0) = 0
327324, 326eqtrdi 2790 . . . . . . . . . . . . . . . . . . 19 (0 ∈ ℕ0 → ((coeff‘Xp)‘0) = 0)
328327adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑅) ∧ 0 ∈ ℕ0) → ((coeff‘Xp)‘0) = 0)
329184coefv0 26301 . . . . . . . . . . . . . . . . . . . . 21 ((ℂ × {𝑧}) ∈ (Poly‘ℂ) → ((ℂ × {𝑧})‘0) = ((coeff‘(ℂ × {𝑧}))‘0))
330182, 329syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑅) → ((ℂ × {𝑧})‘0) = ((coeff‘(ℂ × {𝑧}))‘0))
331 0cn 11250 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℂ
332 vex 3481 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
333332fvconst2 7223 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ ℂ → ((ℂ × {𝑧})‘0) = 𝑧)
334331, 333ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((ℂ × {𝑧})‘0) = 𝑧
335330, 334eqtr3di 2789 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑅) → ((coeff‘(ℂ × {𝑧}))‘0) = 𝑧)
336335adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑅) ∧ 0 ∈ ℕ0) → ((coeff‘(ℂ × {𝑧}))‘0) = 𝑧)
337192, 195, 197, 197, 198, 328, 336ofval 7707 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 0 ∈ ℕ0) → (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘0) = (0 − 𝑧))
338260, 337mpan2 691 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘0) = (0 − 𝑧))
339 df-neg 11492 . . . . . . . . . . . . . . . 16 -𝑧 = (0 − 𝑧)
340338, 339eqtr4di 2792 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘f − (coeff‘(ℂ × {𝑧})))‘0) = -𝑧)
341323, 340eqtrd 2774 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ((coeff‘(Xpf − (ℂ × {𝑧})))‘0) = -𝑧)
342274oveq1d 7445 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → ((𝑁 − 1) − 0) = (𝐷 − 0))
343102subid1d 11606 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (𝐷 − 0) = 𝐷)
344342, 343, 313eqtrd 2778 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → ((𝑁 − 1) − 0) = (deg‘𝑄))
345344fveq2d 6910 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘((𝑁 − 1) − 0)) = ((coeff‘𝑄)‘(deg‘𝑄)))
346345, 232eqtr4d 2777 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘((𝑁 − 1) − 0)) = (𝐴𝑁))
347341, 346oveq12d 7448 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))) = (-𝑧 · (𝐴𝑁)))
348347, 249eqeltrd 2838 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))) ∈ ℂ)
349 fveq2 6906 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) = ((coeff‘(Xpf − (ℂ × {𝑧})))‘0))
350 oveq2 7438 . . . . . . . . . . . . . . 15 (𝑘 = 0 → ((𝑁 − 1) − 𝑘) = ((𝑁 − 1) − 0))
351350fveq2d 6910 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) = ((coeff‘𝑄)‘((𝑁 − 1) − 0)))
352349, 351oveq12d 7448 . . . . . . . . . . . . 13 (𝑘 = 0 → (((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (((coeff‘(Xpf − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))))
353352fsum1 15779 . . . . . . . . . . . 12 ((0 ∈ ℤ ∧ (((coeff‘(Xpf − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (((coeff‘(Xpf − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))))
354322, 348, 353sylancr 587 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → Σ𝑘 ∈ (0...0)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (((coeff‘(Xpf − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))))
355354, 347eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑧𝑅) → Σ𝑘 ∈ (0...0)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (-𝑧 · (𝐴𝑁)))
356274fvoveq1d 7452 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘((𝑁 − 1) − 1)) = ((coeff‘𝑄)‘(𝐷 − 1)))
357224, 356oveq12d 7448 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1))) = (1 · ((coeff‘𝑄)‘(𝐷 − 1))))
358241mullidd 11276 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (1 · ((coeff‘𝑄)‘(𝐷 − 1))) = ((coeff‘𝑄)‘(𝐷 − 1)))
359357, 358eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (((coeff‘(Xpf − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1))) = ((coeff‘𝑄)‘(𝐷 − 1)))
360355, 359oveq12d 7448 . . . . . . . . 9 ((𝜑𝑧𝑅) → (Σ𝑘 ∈ (0...0)(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) + (((coeff‘(Xpf − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1)))) = ((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))))
361295, 321, 3603eqtr3rd 2783 . . . . . . . 8 ((𝜑𝑧𝑅) → ((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))(((coeff‘(Xpf − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
362255, 256, 3613eqtr4rd 2785 . . . . . . 7 ((𝜑𝑧𝑅) → ((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) = (𝐴‘(𝑁 − 1)))
363362oveq1d 7445 . . . . . 6 ((𝜑𝑧𝑅) → (((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) / (𝐴𝑁)) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
364237, 242, 246divcan4d 12046 . . . . . . 7 ((𝜑𝑧𝑅) → ((-𝑧 · (𝐴𝑁)) / (𝐴𝑁)) = -𝑧)
365364oveq1d 7445 . . . . . 6 ((𝜑𝑧𝑅) → (((-𝑧 · (𝐴𝑁)) / (𝐴𝑁)) + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = (-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
366250, 363, 3653eqtr3rd 2783 . . . . 5 ((𝜑𝑧𝑅) → (-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
367366negeqd 11499 . . . 4 ((𝜑𝑧𝑅) → -(-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
368248, 367eqtr3d 2776 . . 3 ((𝜑𝑧𝑅) → (--𝑧 + -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
369128, 236, 3683eqtrd 2778 . 2 ((𝜑𝑧𝑅) → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
37025, 369exlimddv 1932 1 (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  ifcif 4530  {csn 4630   class class class wbr 5147   × cxp 5686  ccnv 5687  dom cdm 5688  cima 5691   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cmin 11489  -cneg 11490   / cdiv 11917  cn 12263  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  chash 14365  Σcsu 15718  0𝑝c0p 25717  Polycply 26237  Xpcidp 26238  coeffccoe 26239  degcdgr 26240   quot cquot 26346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-0p 25718  df-ply 26241  df-idp 26242  df-coe 26243  df-dgr 26244  df-quot 26347
This theorem is referenced by:  vieta1  26368
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