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Theorem vieta1lem2 24360
Description: Lemma for vieta1 24361: 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 11295 . . . . . . 7 (𝜑 → (𝐷 + 1) ∈ ℕ)
52, 4eqeltrrd 2845 . . . . . 6 (𝜑𝑁 ∈ ℕ)
65nnne0d 11324 . . . . 5 (𝜑𝑁 ≠ 0)
71, 6eqnetrd 3004 . . . 4 (𝜑 → (♯‘𝑅) ≠ 0)
8 vieta1.4 . . . . . . . 8 (𝜑𝐹 ∈ (Poly‘𝑆))
9 vieta1.2 . . . . . . . . . 10 𝑁 = (deg‘𝐹)
109, 6syl5eqner 3012 . . . . . . . . 9 (𝜑 → (deg‘𝐹) ≠ 0)
11 fveq2 6377 . . . . . . . . . . 11 (𝐹 = 0𝑝 → (deg‘𝐹) = (deg‘0𝑝))
12 dgr0 24312 . . . . . . . . . . 11 (deg‘0𝑝) = 0
1311, 12syl6eq 2815 . . . . . . . . . 10 (𝐹 = 0𝑝 → (deg‘𝐹) = 0)
1413necon3i 2969 . . . . . . . . 9 ((deg‘𝐹) ≠ 0 → 𝐹 ≠ 0𝑝)
1510, 14syl 17 . . . . . . . 8 (𝜑𝐹 ≠ 0𝑝)
16 vieta1.3 . . . . . . . . 9 𝑅 = (𝐹 “ {0})
1716fta1 24357 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐹 ≠ 0𝑝) → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
188, 15, 17syl2anc 579 . . . . . . 7 (𝜑 → (𝑅 ∈ Fin ∧ (♯‘𝑅) ≤ (deg‘𝐹)))
1918simpld 488 . . . . . 6 (𝜑𝑅 ∈ Fin)
20 hasheq0 13359 . . . . . 6 (𝑅 ∈ Fin → ((♯‘𝑅) = 0 ↔ 𝑅 = ∅))
2119, 20syl 17 . . . . 5 (𝜑 → ((♯‘𝑅) = 0 ↔ 𝑅 = ∅))
2221necon3bid 2981 . . . 4 (𝜑 → ((♯‘𝑅) ≠ 0 ↔ 𝑅 ≠ ∅))
237, 22mpbid 223 . . 3 (𝜑𝑅 ≠ ∅)
24 n0 4097 . . 3 (𝑅 ≠ ∅ ↔ ∃𝑧 𝑧𝑅)
2523, 24sylib 209 . 2 (𝜑 → ∃𝑧 𝑧𝑅)
26 incom 3969 . . . . 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 24359 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (𝑄 ∈ (Poly‘ℂ) ∧ 𝐷 = (deg‘𝑄)))
3130simprd 489 . . . . . . . . 9 ((𝜑𝑧𝑅) → 𝐷 = (deg‘𝑄))
3230simpld 488 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → 𝑄 ∈ (Poly‘ℂ))
33 dgrcl 24283 . . . . . . . . . . 11 (𝑄 ∈ (Poly‘ℂ) → (deg‘𝑄) ∈ ℕ0)
3432, 33syl 17 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (deg‘𝑄) ∈ ℕ0)
3534nn0red 11601 . . . . . . . . 9 ((𝜑𝑧𝑅) → (deg‘𝑄) ∈ ℝ)
3631, 35eqeltrd 2844 . . . . . . . 8 ((𝜑𝑧𝑅) → 𝐷 ∈ ℝ)
3736ltp1d 11210 . . . . . . . 8 ((𝜑𝑧𝑅) → 𝐷 < (𝐷 + 1))
3836, 37gtned 10428 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐷 + 1) ≠ 𝐷)
39 snssi 4495 . . . . . . . . . . 11 (𝑧 ∈ (𝑄 “ {0}) → {𝑧} ⊆ (𝑄 “ {0}))
40 ssequn1 3947 . . . . . . . . . . 11 ({𝑧} ⊆ (𝑄 “ {0}) ↔ ({𝑧} ∪ (𝑄 “ {0})) = (𝑄 “ {0}))
4139, 40sylib 209 . . . . . . . . . 10 (𝑧 ∈ (𝑄 “ {0}) → ({𝑧} ∪ (𝑄 “ {0})) = (𝑄 “ {0}))
4241fveq2d 6381 . . . . . . . . 9 (𝑧 ∈ (𝑄 “ {0}) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) = (♯‘(𝑄 “ {0})))
438adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝐹 ∈ (Poly‘𝑆))
44 cnvimass 5669 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 “ {0}) ⊆ dom 𝐹
4516, 44eqsstri 3797 . . . . . . . . . . . . . . . . . . . 20 𝑅 ⊆ dom 𝐹
46 plyf 24248 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
47 fdm 6233 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℂ⟶ℂ → dom 𝐹 = ℂ)
488, 46, 473syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝐹 = ℂ)
4945, 48syl5sseq 3815 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑅 ⊆ ℂ)
5049sselda 3763 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝑧 ∈ ℂ)
5116eleq2i 2836 . . . . . . . . . . . . . . . . . . . 20 (𝑧𝑅𝑧 ∈ (𝐹 “ {0}))
52 ffn 6225 . . . . . . . . . . . . . . . . . . . . 21 (𝐹:ℂ⟶ℂ → 𝐹 Fn ℂ)
53 fniniseg 6530 . . . . . . . . . . . . . . . . . . . . 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, 54syl5bb 274 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑧𝑅 ↔ (𝑧 ∈ ℂ ∧ (𝐹𝑧) = 0)))
5655simplbda 493 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → (𝐹𝑧) = 0)
57 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (Xp𝑓 − (ℂ × {𝑧})) = (Xp𝑓 − (ℂ × {𝑧}))
5857facth 24355 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑧 ∈ ℂ ∧ (𝐹𝑧) = 0) → 𝐹 = ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝑧})))))
5943, 50, 56, 58syl3anc 1490 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → 𝐹 = ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝑧})))))
6029oveq2i 6855 . . . . . . . . . . . . . . . . 17 ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) = ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · (𝐹 quot (Xp𝑓 − (ℂ × {𝑧}))))
6159, 60syl6eqr 2817 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → 𝐹 = ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄))
6261cnveqd 5468 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → 𝐹 = ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄))
6362imaeq1d 5649 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (𝐹 “ {0}) = (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) “ {0}))
6416, 63syl5eq 2811 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → 𝑅 = (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) “ {0}))
65 cnex 10272 . . . . . . . . . . . . . . 15 ℂ ∈ V
6665a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ℂ ∈ V)
6757plyremlem 24353 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℂ → ((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ (deg‘(Xp𝑓 − (ℂ × {𝑧}))) = 1 ∧ ((Xp𝑓 − (ℂ × {𝑧})) “ {0}) = {𝑧}))
6850, 67syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → ((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ (deg‘(Xp𝑓 − (ℂ × {𝑧}))) = 1 ∧ ((Xp𝑓 − (ℂ × {𝑧})) “ {0}) = {𝑧}))
6968simp1d 1172 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ))
70 plyf 24248 . . . . . . . . . . . . . . 15 ((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) → (Xp𝑓 − (ℂ × {𝑧})):ℂ⟶ℂ)
7169, 70syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (Xp𝑓 − (ℂ × {𝑧})):ℂ⟶ℂ)
72 plyf 24248 . . . . . . . . . . . . . . 15 (𝑄 ∈ (Poly‘ℂ) → 𝑄:ℂ⟶ℂ)
7332, 72syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → 𝑄:ℂ⟶ℂ)
74 ofmulrt 24331 . . . . . . . . . . . . . 14 ((ℂ ∈ V ∧ (Xp𝑓 − (ℂ × {𝑧})):ℂ⟶ℂ ∧ 𝑄:ℂ⟶ℂ) → (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) “ {0}) = (((Xp𝑓 − (ℂ × {𝑧})) “ {0}) ∪ (𝑄 “ {0})))
7566, 71, 73, 74syl3anc 1490 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) “ {0}) = (((Xp𝑓 − (ℂ × {𝑧})) “ {0}) ∪ (𝑄 “ {0})))
7668simp3d 1174 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ((Xp𝑓 − (ℂ × {𝑧})) “ {0}) = {𝑧})
7776uneq1d 3930 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((Xp𝑓 − (ℂ × {𝑧})) “ {0}) ∪ (𝑄 “ {0})) = ({𝑧} ∪ (𝑄 “ {0})))
7864, 75, 773eqtrd 2803 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → 𝑅 = ({𝑧} ∪ (𝑄 “ {0})))
7978fveq2d 6381 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (♯‘𝑅) = (♯‘({𝑧} ∪ (𝑄 “ {0}))))
801, 2eqtr4d 2802 . . . . . . . . . . . 12 (𝜑 → (♯‘𝑅) = (𝐷 + 1))
8180adantr 472 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (♯‘𝑅) = (𝐷 + 1))
8279, 81eqtr3d 2801 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) = (𝐷 + 1))
8315adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝐹 ≠ 0𝑝)
8461, 83eqnetrrd 3005 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → ((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) ≠ 0𝑝)
85 plymul0or 24330 . . . . . . . . . . . . . . . . . . 19 (((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑄 ∈ (Poly‘ℂ)) → (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) = 0𝑝 ↔ ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝)))
8669, 32, 85syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) = 0𝑝 ↔ ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝)))
8786necon3abid 2973 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄) ≠ 0𝑝 ↔ ¬ ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝)))
8884, 87mpbid 223 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → ¬ ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝))
89 neanior 3029 . . . . . . . . . . . . . . . 16 (((Xp𝑓 − (ℂ × {𝑧})) ≠ 0𝑝𝑄 ≠ 0𝑝) ↔ ¬ ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝𝑄 = 0𝑝))
9088, 89sylibr 225 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ((Xp𝑓 − (ℂ × {𝑧})) ≠ 0𝑝𝑄 ≠ 0𝑝))
9190simprd 489 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → 𝑄 ≠ 0𝑝)
92 eqid 2765 . . . . . . . . . . . . . . 15 (𝑄 “ {0}) = (𝑄 “ {0})
9392fta1 24357 . . . . . . . . . . . . . 14 ((𝑄 ∈ (Poly‘ℂ) ∧ 𝑄 ≠ 0𝑝) → ((𝑄 “ {0}) ∈ Fin ∧ (♯‘(𝑄 “ {0})) ≤ (deg‘𝑄)))
9432, 91, 93syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → ((𝑄 “ {0}) ∈ Fin ∧ (♯‘(𝑄 “ {0})) ≤ (deg‘𝑄)))
9594simprd 489 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ≤ (deg‘𝑄))
9695, 31breqtrrd 4839 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ≤ 𝐷)
97 snfi 8247 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
9894simpld 488 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (𝑄 “ {0}) ∈ Fin)
99 hashun2 13377 . . . . . . . . . . . . . 14 (({𝑧} ∈ Fin ∧ (𝑄 “ {0}) ∈ Fin) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) ≤ ((♯‘{𝑧}) + (♯‘(𝑄 “ {0}))))
10097, 98, 99sylancr 581 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) ≤ ((♯‘{𝑧}) + (♯‘(𝑄 “ {0}))))
101 ax-1cn 10249 . . . . . . . . . . . . . . 15 1 ∈ ℂ
1023nncnd 11294 . . . . . . . . . . . . . . . 16 (𝜑𝐷 ∈ ℂ)
103102adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → 𝐷 ∈ ℂ)
104 addcom 10478 . . . . . . . . . . . . . . 15 ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ) → (1 + 𝐷) = (𝐷 + 1))
105101, 103, 104sylancr 581 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (1 + 𝐷) = (𝐷 + 1))
10682, 105eqtr4d 2802 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (♯‘({𝑧} ∪ (𝑄 “ {0}))) = (1 + 𝐷))
107 hashsng 13364 . . . . . . . . . . . . . . 15 (𝑧𝑅 → (♯‘{𝑧}) = 1)
108107adantl 473 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (♯‘{𝑧}) = 1)
109108oveq1d 6859 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → ((♯‘{𝑧}) + (♯‘(𝑄 “ {0}))) = (1 + (♯‘(𝑄 “ {0}))))
110100, 106, 1093brtr3d 4842 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (1 + 𝐷) ≤ (1 + (♯‘(𝑄 “ {0}))))
111 hashcl 13352 . . . . . . . . . . . . . . 15 ((𝑄 “ {0}) ∈ Fin → (♯‘(𝑄 “ {0})) ∈ ℕ0)
11298, 111syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ∈ ℕ0)
113112nn0red 11601 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) ∈ ℝ)
114 1red 10296 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → 1 ∈ ℝ)
11536, 113, 114leadd2d 10878 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (𝐷 ≤ (♯‘(𝑄 “ {0})) ↔ (1 + 𝐷) ≤ (1 + (♯‘(𝑄 “ {0})))))
116110, 115mpbird 248 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → 𝐷 ≤ (♯‘(𝑄 “ {0})))
117113, 36letri3d 10435 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → ((♯‘(𝑄 “ {0})) = 𝐷 ↔ ((♯‘(𝑄 “ {0})) ≤ 𝐷𝐷 ≤ (♯‘(𝑄 “ {0})))))
11896, 116, 117mpbir2and 704 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) = 𝐷)
11982, 118eqeq12d 2780 . . . . . . . . 9 ((𝜑𝑧𝑅) → ((♯‘({𝑧} ∪ (𝑄 “ {0}))) = (♯‘(𝑄 “ {0})) ↔ (𝐷 + 1) = 𝐷))
12042, 119syl5ib 235 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝑧 ∈ (𝑄 “ {0}) → (𝐷 + 1) = 𝐷))
121120necon3ad 2950 . . . . . . 7 ((𝜑𝑧𝑅) → ((𝐷 + 1) ≠ 𝐷 → ¬ 𝑧 ∈ (𝑄 “ {0})))
12238, 121mpd 15 . . . . . 6 ((𝜑𝑧𝑅) → ¬ 𝑧 ∈ (𝑄 “ {0}))
123 disjsn 4404 . . . . . 6 (((𝑄 “ {0}) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑄 “ {0}))
124122, 123sylibr 225 . . . . 5 ((𝜑𝑧𝑅) → ((𝑄 “ {0}) ∩ {𝑧}) = ∅)
12526, 124syl5eq 2811 . . . 4 ((𝜑𝑧𝑅) → ({𝑧} ∩ (𝑄 “ {0})) = ∅)
12619adantr 472 . . . 4 ((𝜑𝑧𝑅) → 𝑅 ∈ Fin)
12749adantr 472 . . . . 5 ((𝜑𝑧𝑅) → 𝑅 ⊆ ℂ)
128127sselda 3763 . . . 4 (((𝜑𝑧𝑅) ∧ 𝑥𝑅) → 𝑥 ∈ ℂ)
129125, 78, 126, 128fsumsplit 14759 . . 3 ((𝜑𝑧𝑅) → Σ𝑥𝑅 𝑥 = (Σ𝑥 ∈ {𝑧}𝑥 + Σ𝑥 ∈ (𝑄 “ {0})𝑥))
130 id 22 . . . . . . 7 (𝑥 = 𝑧𝑥 = 𝑧)
131130sumsn 14763 . . . . . 6 ((𝑧 ∈ ℂ ∧ 𝑧 ∈ ℂ) → Σ𝑥 ∈ {𝑧}𝑥 = 𝑧)
13250, 50, 131syl2anc 579 . . . . 5 ((𝜑𝑧𝑅) → Σ𝑥 ∈ {𝑧}𝑥 = 𝑧)
13350negnegd 10639 . . . . 5 ((𝜑𝑧𝑅) → --𝑧 = 𝑧)
134132, 133eqtr4d 2802 . . . 4 ((𝜑𝑧𝑅) → Σ𝑥 ∈ {𝑧}𝑥 = --𝑧)
135118, 31eqtrd 2799 . . . . . 6 ((𝜑𝑧𝑅) → (♯‘(𝑄 “ {0})) = (deg‘𝑄))
136 fveq2 6377 . . . . . . . . . 10 (𝑓 = 𝑄 → (deg‘𝑓) = (deg‘𝑄))
137136eqeq2d 2775 . . . . . . . . 9 (𝑓 = 𝑄 → (𝐷 = (deg‘𝑓) ↔ 𝐷 = (deg‘𝑄)))
138 cnveq 5466 . . . . . . . . . . . 12 (𝑓 = 𝑄𝑓 = 𝑄)
139138imaeq1d 5649 . . . . . . . . . . 11 (𝑓 = 𝑄 → (𝑓 “ {0}) = (𝑄 “ {0}))
140139fveq2d 6381 . . . . . . . . . 10 (𝑓 = 𝑄 → (♯‘(𝑓 “ {0})) = (♯‘(𝑄 “ {0})))
141140, 136eqeq12d 2780 . . . . . . . . 9 (𝑓 = 𝑄 → ((♯‘(𝑓 “ {0})) = (deg‘𝑓) ↔ (♯‘(𝑄 “ {0})) = (deg‘𝑄)))
142137, 141anbi12d 624 . . . . . . . 8 (𝑓 = 𝑄 → ((𝐷 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝐷 = (deg‘𝑄) ∧ (♯‘(𝑄 “ {0})) = (deg‘𝑄))))
143139sumeq1d 14719 . . . . . . . . 9 (𝑓 = 𝑄 → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = Σ𝑥 ∈ (𝑄 “ {0})𝑥)
144 fveq2 6377 . . . . . . . . . . . 12 (𝑓 = 𝑄 → (coeff‘𝑓) = (coeff‘𝑄))
145136oveq1d 6859 . . . . . . . . . . . 12 (𝑓 = 𝑄 → ((deg‘𝑓) − 1) = ((deg‘𝑄) − 1))
146144, 145fveq12d 6384 . . . . . . . . . . 11 (𝑓 = 𝑄 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = ((coeff‘𝑄)‘((deg‘𝑄) − 1)))
147144, 136fveq12d 6384 . . . . . . . . . . 11 (𝑓 = 𝑄 → ((coeff‘𝑓)‘(deg‘𝑓)) = ((coeff‘𝑄)‘(deg‘𝑄)))
148146, 147oveq12d 6862 . . . . . . . . . 10 (𝑓 = 𝑄 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = (((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
149148negeqd 10531 . . . . . . . . 9 (𝑓 = 𝑄 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
150143, 149eqeq12d 2780 . . . . . . . 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)))))
151142, 150imbi12d 335 . . . . . . 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))))))
15228adantr 472 . . . . . . 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)))))
153151, 152, 32rspcdva 3468 . . . . . 6 ((𝜑𝑧𝑅) → ((𝐷 = (deg‘𝑄) ∧ (♯‘(𝑄 “ {0})) = (deg‘𝑄)) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄)))))
15431, 135, 153mp2and 690 . . . . 5 ((𝜑𝑧𝑅) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
15531fvoveq1d 6866 . . . . . . 7 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘(𝐷 − 1)) = ((coeff‘𝑄)‘((deg‘𝑄) − 1)))
15661fveq2d 6381 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (coeff‘𝐹) = (coeff‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄)))
15727, 156syl5eq 2811 . . . . . . . . 9 ((𝜑𝑧𝑅) → 𝐴 = (coeff‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄)))
15861fveq2d 6381 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (deg‘𝐹) = (deg‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄)))
15968simp2d 1173 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (deg‘(Xp𝑓 − (ℂ × {𝑧}))) = 1)
160 ax-1ne0 10260 . . . . . . . . . . . . . . 15 1 ≠ 0
161160a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → 1 ≠ 0)
162159, 161eqnetrd 3004 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (deg‘(Xp𝑓 − (ℂ × {𝑧}))) ≠ 0)
163 fveq2 6377 . . . . . . . . . . . . . . 15 ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝑧}))) = (deg‘0𝑝))
164163, 12syl6eq 2815 . . . . . . . . . . . . . 14 ((Xp𝑓 − (ℂ × {𝑧})) = 0𝑝 → (deg‘(Xp𝑓 − (ℂ × {𝑧}))) = 0)
165164necon3i 2969 . . . . . . . . . . . . 13 ((deg‘(Xp𝑓 − (ℂ × {𝑧}))) ≠ 0 → (Xp𝑓 − (ℂ × {𝑧})) ≠ 0𝑝)
166162, 165syl 17 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (Xp𝑓 − (ℂ × {𝑧})) ≠ 0𝑝)
167 eqid 2765 . . . . . . . . . . . . 13 (deg‘(Xp𝑓 − (ℂ × {𝑧}))) = (deg‘(Xp𝑓 − (ℂ × {𝑧})))
168 eqid 2765 . . . . . . . . . . . . 13 (deg‘𝑄) = (deg‘𝑄)
169167, 168dgrmul 24320 . . . . . . . . . . . 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)))
17069, 166, 32, 91, 169syl22anc 867 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (deg‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄)) = ((deg‘(Xp𝑓 − (ℂ × {𝑧}))) + (deg‘𝑄)))
171158, 170eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (deg‘𝐹) = ((deg‘(Xp𝑓 − (ℂ × {𝑧}))) + (deg‘𝑄)))
1729, 171syl5eq 2811 . . . . . . . . 9 ((𝜑𝑧𝑅) → 𝑁 = ((deg‘(Xp𝑓 − (ℂ × {𝑧}))) + (deg‘𝑄)))
173157, 172fveq12d 6384 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝐴𝑁) = ((coeff‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄))‘((deg‘(Xp𝑓 − (ℂ × {𝑧}))) + (deg‘𝑄))))
174 eqid 2765 . . . . . . . . . 10 (coeff‘(Xp𝑓 − (ℂ × {𝑧}))) = (coeff‘(Xp𝑓 − (ℂ × {𝑧})))
175 eqid 2765 . . . . . . . . . 10 (coeff‘𝑄) = (coeff‘𝑄)
176174, 175, 167, 168coemulhi 24304 . . . . . . . . 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))))
17769, 32, 176syl2anc 579 . . . . . . . 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))))
178159fveq2d 6381 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘(deg‘(Xp𝑓 − (ℂ × {𝑧})))) = ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1))
179 ssid 3785 . . . . . . . . . . . . . . 15 ℂ ⊆ ℂ
180 plyid 24259 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ 1 ∈ ℂ) → Xp ∈ (Poly‘ℂ))
181179, 101, 180mp2an 683 . . . . . . . . . . . . . 14 Xp ∈ (Poly‘ℂ)
182 plyconst 24256 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ 𝑧 ∈ ℂ) → (ℂ × {𝑧}) ∈ (Poly‘ℂ))
183179, 50, 182sylancr 581 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (ℂ × {𝑧}) ∈ (Poly‘ℂ))
184 eqid 2765 . . . . . . . . . . . . . . 15 (coeff‘Xp) = (coeff‘Xp)
185 eqid 2765 . . . . . . . . . . . . . . 15 (coeff‘(ℂ × {𝑧})) = (coeff‘(ℂ × {𝑧}))
186184, 185coesub 24307 . . . . . . . . . . . . . 14 ((Xp ∈ (Poly‘ℂ) ∧ (ℂ × {𝑧}) ∈ (Poly‘ℂ)) → (coeff‘(Xp𝑓 − (ℂ × {𝑧}))) = ((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧}))))
187181, 183, 186sylancr 581 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (coeff‘(Xp𝑓 − (ℂ × {𝑧}))) = ((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧}))))
188187fveq1d 6379 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1) = (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘1))
189 1nn0 11558 . . . . . . . . . . . . . 14 1 ∈ ℕ0
190184coef3 24282 . . . . . . . . . . . . . . . . 17 (Xp ∈ (Poly‘ℂ) → (coeff‘Xp):ℕ0⟶ℂ)
191 ffn 6225 . . . . . . . . . . . . . . . . 17 ((coeff‘Xp):ℕ0⟶ℂ → (coeff‘Xp) Fn ℕ0)
192181, 190, 191mp2b 10 . . . . . . . . . . . . . . . 16 (coeff‘Xp) Fn ℕ0
193192a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (coeff‘Xp) Fn ℕ0)
194185coef3 24282 . . . . . . . . . . . . . . . 16 ((ℂ × {𝑧}) ∈ (Poly‘ℂ) → (coeff‘(ℂ × {𝑧})):ℕ0⟶ℂ)
195 ffn 6225 . . . . . . . . . . . . . . . 16 ((coeff‘(ℂ × {𝑧})):ℕ0⟶ℂ → (coeff‘(ℂ × {𝑧})) Fn ℕ0)
196183, 194, 1953syl 18 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (coeff‘(ℂ × {𝑧})) Fn ℕ0)
197 nn0ex 11547 . . . . . . . . . . . . . . . 16 0 ∈ V
198197a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ℕ0 ∈ V)
199 inidm 3984 . . . . . . . . . . . . . . 15 (ℕ0 ∩ ℕ0) = ℕ0
200 coeidp 24313 . . . . . . . . . . . . . . . . 17 (1 ∈ ℕ0 → ((coeff‘Xp)‘1) = if(1 = 1, 1, 0))
201200adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ((coeff‘Xp)‘1) = if(1 = 1, 1, 0))
202 eqid 2765 . . . . . . . . . . . . . . . . 17 1 = 1
203202iftruei 4252 . . . . . . . . . . . . . . . 16 if(1 = 1, 1, 0) = 1
204201, 203syl6eq 2815 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ((coeff‘Xp)‘1) = 1)
205 0lt1 10806 . . . . . . . . . . . . . . . . . 18 0 < 1
206 0re 10297 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℝ
207 1re 10295 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
208206, 207ltnlei 10414 . . . . . . . . . . . . . . . . . 18 (0 < 1 ↔ ¬ 1 ≤ 0)
209205, 208mpbi 221 . . . . . . . . . . . . . . . . 17 ¬ 1 ≤ 0
21050adantr 472 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → 𝑧 ∈ ℂ)
211 0dgr 24295 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℂ → (deg‘(ℂ × {𝑧})) = 0)
212210, 211syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (deg‘(ℂ × {𝑧})) = 0)
213212breq2d 4823 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (1 ≤ (deg‘(ℂ × {𝑧})) ↔ 1 ≤ 0))
214209, 213mtbiri 318 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ¬ 1 ≤ (deg‘(ℂ × {𝑧})))
215 eqid 2765 . . . . . . . . . . . . . . . . . . . 20 (deg‘(ℂ × {𝑧})) = (deg‘(ℂ × {𝑧}))
216185, 215dgrub 24284 . . . . . . . . . . . . . . . . . . 19 (((ℂ × {𝑧}) ∈ (Poly‘ℂ) ∧ 1 ∈ ℕ0 ∧ ((coeff‘(ℂ × {𝑧}))‘1) ≠ 0) → 1 ≤ (deg‘(ℂ × {𝑧})))
2172163expia 1150 . . . . . . . . . . . . . . . . . 18 (((ℂ × {𝑧}) ∈ (Poly‘ℂ) ∧ 1 ∈ ℕ0) → (((coeff‘(ℂ × {𝑧}))‘1) ≠ 0 → 1 ≤ (deg‘(ℂ × {𝑧}))))
218183, 217sylan 575 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (((coeff‘(ℂ × {𝑧}))‘1) ≠ 0 → 1 ≤ (deg‘(ℂ × {𝑧}))))
219218necon1bd 2955 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (¬ 1 ≤ (deg‘(ℂ × {𝑧})) → ((coeff‘(ℂ × {𝑧}))‘1) = 0))
220214, 219mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → ((coeff‘(ℂ × {𝑧}))‘1) = 0)
221193, 196, 198, 198, 199, 204, 220ofval 7106 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 1 ∈ ℕ0) → (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘1) = (1 − 0))
222189, 221mpan2 682 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘1) = (1 − 0))
223 1m0e1 11402 . . . . . . . . . . . . 13 (1 − 0) = 1
224222, 223syl6eq 2815 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘1) = 1)
225188, 224eqtrd 2799 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1) = 1)
226178, 225eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑧𝑅) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘(deg‘(Xp𝑓 − (ℂ × {𝑧})))) = 1)
227226oveq1d 6859 . . . . . . . . 9 ((𝜑𝑧𝑅) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘(deg‘(Xp𝑓 − (ℂ × {𝑧})))) · ((coeff‘𝑄)‘(deg‘𝑄))) = (1 · ((coeff‘𝑄)‘(deg‘𝑄))))
228175coef3 24282 . . . . . . . . . . . 12 (𝑄 ∈ (Poly‘ℂ) → (coeff‘𝑄):ℕ0⟶ℂ)
22932, 228syl 17 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (coeff‘𝑄):ℕ0⟶ℂ)
230229, 34ffvelrnd 6552 . . . . . . . . . 10 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘(deg‘𝑄)) ∈ ℂ)
231230mulid2d 10314 . . . . . . . . 9 ((𝜑𝑧𝑅) → (1 · ((coeff‘𝑄)‘(deg‘𝑄))) = ((coeff‘𝑄)‘(deg‘𝑄)))
232227, 231eqtrd 2799 . . . . . . . 8 ((𝜑𝑧𝑅) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘(deg‘(Xp𝑓 − (ℂ × {𝑧})))) · ((coeff‘𝑄)‘(deg‘𝑄))) = ((coeff‘𝑄)‘(deg‘𝑄)))
233173, 177, 2323eqtrd 2803 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐴𝑁) = ((coeff‘𝑄)‘(deg‘𝑄)))
234155, 233oveq12d 6862 . . . . . 6 ((𝜑𝑧𝑅) → (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)) = (((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
235234negeqd 10531 . . . . 5 ((𝜑𝑧𝑅) → -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)) = -(((coeff‘𝑄)‘((deg‘𝑄) − 1)) / ((coeff‘𝑄)‘(deg‘𝑄))))
236154, 235eqtr4d 2802 . . . 4 ((𝜑𝑧𝑅) → Σ𝑥 ∈ (𝑄 “ {0})𝑥 = -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)))
237134, 236oveq12d 6862 . . 3 ((𝜑𝑧𝑅) → (Σ𝑥 ∈ {𝑧}𝑥 + Σ𝑥 ∈ (𝑄 “ {0})𝑥) = (--𝑧 + -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
23850negcld 10635 . . . . 5 ((𝜑𝑧𝑅) → -𝑧 ∈ ℂ)
239 nnm1nn0 11583 . . . . . . . . 9 (𝐷 ∈ ℕ → (𝐷 − 1) ∈ ℕ0)
2403, 239syl 17 . . . . . . . 8 (𝜑 → (𝐷 − 1) ∈ ℕ0)
241240adantr 472 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐷 − 1) ∈ ℕ0)
242229, 241ffvelrnd 6552 . . . . . 6 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘(𝐷 − 1)) ∈ ℂ)
243233, 230eqeltrd 2844 . . . . . 6 ((𝜑𝑧𝑅) → (𝐴𝑁) ∈ ℂ)
2449, 27dgreq0 24315 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴𝑁) = 0))
24543, 244syl 17 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝐹 = 0𝑝 ↔ (𝐴𝑁) = 0))
246245necon3bid 2981 . . . . . . 7 ((𝜑𝑧𝑅) → (𝐹 ≠ 0𝑝 ↔ (𝐴𝑁) ≠ 0))
24783, 246mpbid 223 . . . . . 6 ((𝜑𝑧𝑅) → (𝐴𝑁) ≠ 0)
248242, 243, 247divcld 11057 . . . . 5 ((𝜑𝑧𝑅) → (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁)) ∈ ℂ)
249238, 248negdid 10661 . . . 4 ((𝜑𝑧𝑅) → -(-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = (--𝑧 + -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
250238, 243mulcld 10316 . . . . . . 7 ((𝜑𝑧𝑅) → (-𝑧 · (𝐴𝑁)) ∈ ℂ)
251250, 242, 243, 247divdird 11095 . . . . . 6 ((𝜑𝑧𝑅) → (((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) / (𝐴𝑁)) = (((-𝑧 · (𝐴𝑁)) / (𝐴𝑁)) + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
252 nnm1nn0 11583 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
2535, 252syl 17 . . . . . . . . . 10 (𝜑 → (𝑁 − 1) ∈ ℕ0)
254253adantr 472 . . . . . . . . 9 ((𝜑𝑧𝑅) → (𝑁 − 1) ∈ ℕ0)
255174, 175coemul 24302 . . . . . . . . 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))))
25669, 32, 254, 255syl3anc 1490 . . . . . . . 8 ((𝜑𝑧𝑅) → ((coeff‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄))‘(𝑁 − 1)) = Σ𝑘 ∈ (0...(𝑁 − 1))(((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
257157fveq1d 6379 . . . . . . . 8 ((𝜑𝑧𝑅) → (𝐴‘(𝑁 − 1)) = ((coeff‘((Xp𝑓 − (ℂ × {𝑧})) ∘𝑓 · 𝑄))‘(𝑁 − 1)))
258 1e0p1 11786 . . . . . . . . . . . 12 1 = (0 + 1)
259258oveq2i 6855 . . . . . . . . . . 11 (0...1) = (0...(0 + 1))
260259sumeq1i 14716 . . . . . . . . . 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)))
261 0nn0 11557 . . . . . . . . . . . . 13 0 ∈ ℕ0
262 nn0uz 11925 . . . . . . . . . . . . 13 0 = (ℤ‘0)
263261, 262eleqtri 2842 . . . . . . . . . . . 12 0 ∈ (ℤ‘0)
264263a1i 11 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → 0 ∈ (ℤ‘0))
265259eleq2i 2836 . . . . . . . . . . . 12 (𝑘 ∈ (0...1) ↔ 𝑘 ∈ (0...(0 + 1)))
266174coef3 24282 . . . . . . . . . . . . . . 15 ((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) → (coeff‘(Xp𝑓 − (ℂ × {𝑧}))):ℕ0⟶ℂ)
26769, 266syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → (coeff‘(Xp𝑓 − (ℂ × {𝑧}))):ℕ0⟶ℂ)
268 elfznn0 12643 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...1) → 𝑘 ∈ ℕ0)
269 ffvelrn 6549 . . . . . . . . . . . . . 14 (((coeff‘(Xp𝑓 − (ℂ × {𝑧}))):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) ∈ ℂ)
270267, 268, 269syl2an 589 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) ∈ ℂ)
2712oveq1d 6859 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐷 + 1) − 1) = (𝑁 − 1))
272 pncan 10543 . . . . . . . . . . . . . . . . . . . . 21 ((𝐷 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐷 + 1) − 1) = 𝐷)
273102, 101, 272sylancl 580 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝐷 + 1) − 1) = 𝐷)
274271, 273eqtr3d 2801 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 − 1) = 𝐷)
275274adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → (𝑁 − 1) = 𝐷)
2763adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑧𝑅) → 𝐷 ∈ ℕ)
277275, 276eqeltrd 2844 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (𝑁 − 1) ∈ ℕ)
278 nnuz 11926 . . . . . . . . . . . . . . . . 17 ℕ = (ℤ‘1)
279277, 278syl6eleq 2854 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → (𝑁 − 1) ∈ (ℤ‘1))
280 fzss2 12591 . . . . . . . . . . . . . . . 16 ((𝑁 − 1) ∈ (ℤ‘1) → (0...1) ⊆ (0...(𝑁 − 1)))
281279, 280syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (0...1) ⊆ (0...(𝑁 − 1)))
282281sselda 3763 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → 𝑘 ∈ (0...(𝑁 − 1)))
283 fznn0sub 12583 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) − 𝑘) ∈ ℕ0)
284 ffvelrn 6549 . . . . . . . . . . . . . . 15 (((coeff‘𝑄):ℕ0⟶ℂ ∧ ((𝑁 − 1) − 𝑘) ∈ ℕ0) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
285229, 283, 284syl2an 589 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
286282, 285syldan 585 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
287270, 286mulcld 10316 . . . . . . . . . . . 12 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...1)) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) ∈ ℂ)
288265, 287sylan2br 588 . . . . . . . . . . 11 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ (0...(0 + 1))) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) ∈ ℂ)
289 id 22 . . . . . . . . . . . . . 14 (𝑘 = (0 + 1) → 𝑘 = (0 + 1))
290289, 258syl6eqr 2817 . . . . . . . . . . . . 13 (𝑘 = (0 + 1) → 𝑘 = 1)
291290fveq2d 6381 . . . . . . . . . . . 12 (𝑘 = (0 + 1) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) = ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1))
292290oveq2d 6860 . . . . . . . . . . . . 13 (𝑘 = (0 + 1) → ((𝑁 − 1) − 𝑘) = ((𝑁 − 1) − 1))
293292fveq2d 6381 . . . . . . . . . . . 12 (𝑘 = (0 + 1) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) = ((coeff‘𝑄)‘((𝑁 − 1) − 1)))
294291, 293oveq12d 6862 . . . . . . . . . . 11 (𝑘 = (0 + 1) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1))))
295264, 288, 294fsump1 14775 . . . . . . . . . 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)))))
296260, 295syl5eq 2811 . . . . . . . . 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)))))
297 eldifn 3897 . . . . . . . . . . . . . 14 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → ¬ 𝑘 ∈ (0...1))
298297adantl 473 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → ¬ 𝑘 ∈ (0...1))
299 eldifi 3896 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → 𝑘 ∈ (0...(𝑁 − 1)))
300 elfznn0 12643 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
301299, 300syl 17 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → 𝑘 ∈ ℕ0)
302174, 167dgrub 24284 . . . . . . . . . . . . . . . . 17 (((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0 ∧ ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) ≠ 0) → 𝑘 ≤ (deg‘(Xp𝑓 − (ℂ × {𝑧}))))
3033023expia 1150 . . . . . . . . . . . . . . . 16 (((Xp𝑓 − (ℂ × {𝑧})) ∈ (Poly‘ℂ) ∧ 𝑘 ∈ ℕ0) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(Xp𝑓 − (ℂ × {𝑧})))))
30469, 301, 303syl2an 589 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘(Xp𝑓 − (ℂ × {𝑧})))))
305 elfzuz 12548 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ (ℤ‘0))
306299, 305syl 17 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1)) → 𝑘 ∈ (ℤ‘0))
307306adantl 473 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → 𝑘 ∈ (ℤ‘0))
308 1z 11657 . . . . . . . . . . . . . . . . 17 1 ∈ ℤ
309 elfz5 12544 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ (ℤ‘0) ∧ 1 ∈ ℤ) → (𝑘 ∈ (0...1) ↔ 𝑘 ≤ 1))
310307, 308, 309sylancl 580 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (𝑘 ∈ (0...1) ↔ 𝑘 ≤ 1))
311159breq2d 4823 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (𝑘 ≤ (deg‘(Xp𝑓 − (ℂ × {𝑧}))) ↔ 𝑘 ≤ 1))
312311adantr 472 . . . . . . . . . . . . . . . 16 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (𝑘 ≤ (deg‘(Xp𝑓 − (ℂ × {𝑧}))) ↔ 𝑘 ≤ 1))
313310, 312bitr4d 273 . . . . . . . . . . . . . . 15 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (𝑘 ∈ (0...1) ↔ 𝑘 ≤ (deg‘(Xp𝑓 − (ℂ × {𝑧})))))
314304, 313sylibrd 250 . . . . . . . . . . . . . 14 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) ≠ 0 → 𝑘 ∈ (0...1)))
315314necon1bd 2955 . . . . . . . . . . . . 13 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (¬ 𝑘 ∈ (0...1) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) = 0))
316298, 315mpd 15 . . . . . . . . . . . 12 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) = 0)
317316oveq1d 6859 . . . . . . . . . . 11 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (0 · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
318299, 285sylan2 586 . . . . . . . . . . . 12 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) ∈ ℂ)
319318mul02d 10490 . . . . . . . . . . 11 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (0 · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = 0)
320317, 319eqtrd 2799 . . . . . . . . . 10 (((𝜑𝑧𝑅) ∧ 𝑘 ∈ ((0...(𝑁 − 1)) ∖ (0...1))) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = 0)
321 fzfid 12983 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (0...(𝑁 − 1)) ∈ Fin)
322281, 287, 320, 321fsumss 14744 . . . . . . . . 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))))
323 0z 11637 . . . . . . . . . . . 12 0 ∈ ℤ
324187fveq1d 6379 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘0) = (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘0))
325 coeidp 24313 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℕ0 → ((coeff‘Xp)‘0) = if(0 = 1, 1, 0))
326160nesymi 2994 . . . . . . . . . . . . . . . . . . . . 21 ¬ 0 = 1
327326iffalsei 4255 . . . . . . . . . . . . . . . . . . . 20 if(0 = 1, 1, 0) = 0
328325, 327syl6eq 2815 . . . . . . . . . . . . . . . . . . 19 (0 ∈ ℕ0 → ((coeff‘Xp)‘0) = 0)
329328adantl 473 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑅) ∧ 0 ∈ ℕ0) → ((coeff‘Xp)‘0) = 0)
330 0cn 10287 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ ℂ
331 vex 3353 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ V
332331fvconst2 6664 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ ℂ → ((ℂ × {𝑧})‘0) = 𝑧)
333330, 332ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((ℂ × {𝑧})‘0) = 𝑧
334185coefv0 24298 . . . . . . . . . . . . . . . . . . . . 21 ((ℂ × {𝑧}) ∈ (Poly‘ℂ) → ((ℂ × {𝑧})‘0) = ((coeff‘(ℂ × {𝑧}))‘0))
335183, 334syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑧𝑅) → ((ℂ × {𝑧})‘0) = ((coeff‘(ℂ × {𝑧}))‘0))
336333, 335syl5reqr 2814 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑧𝑅) → ((coeff‘(ℂ × {𝑧}))‘0) = 𝑧)
337336adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑧𝑅) ∧ 0 ∈ ℕ0) → ((coeff‘(ℂ × {𝑧}))‘0) = 𝑧)
338193, 196, 198, 198, 199, 329, 337ofval 7106 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑅) ∧ 0 ∈ ℕ0) → (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘0) = (0 − 𝑧))
339261, 338mpan2 682 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘0) = (0 − 𝑧))
340 df-neg 10525 . . . . . . . . . . . . . . . 16 -𝑧 = (0 − 𝑧)
341339, 340syl6eqr 2817 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → (((coeff‘Xp) ∘𝑓 − (coeff‘(ℂ × {𝑧})))‘0) = -𝑧)
342324, 341eqtrd 2799 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘0) = -𝑧)
343275oveq1d 6859 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → ((𝑁 − 1) − 0) = (𝐷 − 0))
344103subid1d 10637 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑅) → (𝐷 − 0) = 𝐷)
345343, 344, 313eqtrd 2803 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑅) → ((𝑁 − 1) − 0) = (deg‘𝑄))
346345fveq2d 6381 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘((𝑁 − 1) − 0)) = ((coeff‘𝑄)‘(deg‘𝑄)))
347346, 233eqtr4d 2802 . . . . . . . . . . . . . 14 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘((𝑁 − 1) − 0)) = (𝐴𝑁))
348342, 347oveq12d 6862 . . . . . . . . . . . . 13 ((𝜑𝑧𝑅) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))) = (-𝑧 · (𝐴𝑁)))
349348, 250eqeltrd 2844 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))) ∈ ℂ)
350 fveq2 6377 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) = ((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘0))
351 oveq2 6852 . . . . . . . . . . . . . . 15 (𝑘 = 0 → ((𝑁 − 1) − 𝑘) = ((𝑁 − 1) − 0))
352351fveq2d 6381 . . . . . . . . . . . . . 14 (𝑘 = 0 → ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘)) = ((coeff‘𝑄)‘((𝑁 − 1) − 0)))
353350, 352oveq12d 6862 . . . . . . . . . . . . 13 (𝑘 = 0 → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘0) · ((coeff‘𝑄)‘((𝑁 − 1) − 0))))
354353fsum1 14764 . . . . . . . . . . . 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))))
355323, 349, 354sylancr 581 . . . . . . . . . . 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))))
356355, 348eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑧𝑅) → Σ𝑘 ∈ (0...0)(((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))) = (-𝑧 · (𝐴𝑁)))
357275fvoveq1d 6866 . . . . . . . . . . . 12 ((𝜑𝑧𝑅) → ((coeff‘𝑄)‘((𝑁 − 1) − 1)) = ((coeff‘𝑄)‘(𝐷 − 1)))
358225, 357oveq12d 6862 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1))) = (1 · ((coeff‘𝑄)‘(𝐷 − 1))))
359242mulid2d 10314 . . . . . . . . . . 11 ((𝜑𝑧𝑅) → (1 · ((coeff‘𝑄)‘(𝐷 − 1))) = ((coeff‘𝑄)‘(𝐷 − 1)))
360358, 359eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑧𝑅) → (((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘1) · ((coeff‘𝑄)‘((𝑁 − 1) − 1))) = ((coeff‘𝑄)‘(𝐷 − 1)))
361356, 360oveq12d 6862 . . . . . . . . 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))))
362296, 322, 3613eqtr3rd 2808 . . . . . . . 8 ((𝜑𝑧𝑅) → ((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))(((coeff‘(Xp𝑓 − (ℂ × {𝑧})))‘𝑘) · ((coeff‘𝑄)‘((𝑁 − 1) − 𝑘))))
363256, 257, 3623eqtr4rd 2810 . . . . . . 7 ((𝜑𝑧𝑅) → ((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) = (𝐴‘(𝑁 − 1)))
364363oveq1d 6859 . . . . . 6 ((𝜑𝑧𝑅) → (((-𝑧 · (𝐴𝑁)) + ((coeff‘𝑄)‘(𝐷 − 1))) / (𝐴𝑁)) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
365238, 243, 247divcan4d 11063 . . . . . . 7 ((𝜑𝑧𝑅) → ((-𝑧 · (𝐴𝑁)) / (𝐴𝑁)) = -𝑧)
366365oveq1d 6859 . . . . . 6 ((𝜑𝑧𝑅) → (((-𝑧 · (𝐴𝑁)) / (𝐴𝑁)) + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = (-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))))
367251, 364, 3663eqtr3rd 2808 . . . . 5 ((𝜑𝑧𝑅) → (-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
368367negeqd 10531 . . . 4 ((𝜑𝑧𝑅) → -(-𝑧 + (((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
369249, 368eqtr3d 2801 . . 3 ((𝜑𝑧𝑅) → (--𝑧 + -(((coeff‘𝑄)‘(𝐷 − 1)) / (𝐴𝑁))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
370129, 237, 3693eqtrd 2803 . 2 ((𝜑𝑧𝑅) → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
37125, 370exlimddv 2030 1 (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3731  cun 3732  cin 3733  wss 3734  c0 4081  ifcif 4245  {csn 4336   class class class wbr 4811   × cxp 5277  ccnv 5278  dom cdm 5279  cima 5282   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  𝑓 cof 7095  Fincfn 8162  cc 10189  cr 10190  0cc0 10191  1c1 10192   + caddc 10194   · cmul 10196   < clt 10330  cle 10331  cmin 10522  -cneg 10523   / cdiv 10940  cn 11276  0cn0 11540  cz 11626  cuz 11889  ...cfz 12536  chash 13324  Σcsu 14704  0𝑝c0p 23730  Polycply 24234  Xpcidp 24235  coeffccoe 24236  degcdgr 24237   quot cquot 24339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-inf2 8755  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268  ax-pre-sup 10269  ax-addf 10270
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-1o 7766  df-oadd 7770  df-er 7949  df-map 8064  df-pm 8065  df-en 8163  df-dom 8164  df-sdom 8165  df-fin 8166  df-sup 8557  df-inf 8558  df-oi 8624  df-card 9018  df-cda 9245  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-xnn0 11613  df-z 11627  df-uz 11890  df-rp 12032  df-fz 12537  df-fzo 12677  df-fl 12804  df-seq 13012  df-exp 13071  df-hash 13325  df-cj 14127  df-re 14128  df-im 14129  df-sqrt 14263  df-abs 14264  df-clim 14507  df-rlim 14508  df-sum 14705  df-0p 23731  df-ply 24238  df-idp 24239  df-coe 24240  df-dgr 24241  df-quot 24340
This theorem is referenced by:  vieta1  24361
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