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Theorem vieta1 24515
Description: The first-order Vieta's formula (see http://en.wikipedia.org/wiki/Vieta%27s_formulas). If a polynomial of degree 𝑁 has 𝑁 distinct roots, then the sum over these roots can be calculated as -𝐴(𝑁 − 1) / 𝐴(𝑁). (If the roots are not distinct, then this formula is still true but must double-count some of the roots according to their multiplicities.) (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)
vieta1.6 (𝜑𝑁 ∈ ℕ)
Assertion
Ref Expression
vieta1 (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
Distinct variable groups:   𝑥,𝑅   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥)   𝐹(𝑥)   𝑁(𝑥)

Proof of Theorem vieta1
Dummy variables 𝑓 𝑘 𝑦 𝑧 𝑑 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vieta1.5 . 2 (𝜑 → (♯‘𝑅) = 𝑁)
2 fveq2 6448 . . . . . . 7 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
32eqeq2d 2788 . . . . . 6 (𝑓 = 𝐹 → (𝑁 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝐹)))
4 cnveq 5543 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
54imaeq1d 5721 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
6 vieta1.3 . . . . . . . . 9 𝑅 = (𝐹 “ {0})
75, 6syl6eqr 2832 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
87fveq2d 6452 . . . . . . 7 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
9 vieta1.2 . . . . . . . 8 𝑁 = (deg‘𝐹)
102, 9syl6eqr 2832 . . . . . . 7 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑁)
118, 10eqeq12d 2793 . . . . . 6 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) = (deg‘𝑓) ↔ (♯‘𝑅) = 𝑁))
123, 11anbi12d 624 . . . . 5 (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝐹) ∧ (♯‘𝑅) = 𝑁)))
139biantrur 526 . . . . 5 ((♯‘𝑅) = 𝑁 ↔ (𝑁 = (deg‘𝐹) ∧ (♯‘𝑅) = 𝑁))
1412, 13syl6bbr 281 . . . 4 (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (♯‘𝑅) = 𝑁))
157sumeq1d 14848 . . . . 5 (𝑓 = 𝐹 → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = Σ𝑥𝑅 𝑥)
16 fveq2 6448 . . . . . . . . 9 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
17 vieta1.1 . . . . . . . . 9 𝐴 = (coeff‘𝐹)
1816, 17syl6eqr 2832 . . . . . . . 8 (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴)
1910oveq1d 6939 . . . . . . . 8 (𝑓 = 𝐹 → ((deg‘𝑓) − 1) = (𝑁 − 1))
2018, 19fveq12d 6455 . . . . . . 7 (𝑓 = 𝐹 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = (𝐴‘(𝑁 − 1)))
2118, 10fveq12d 6455 . . . . . . 7 (𝑓 = 𝐹 → ((coeff‘𝑓)‘(deg‘𝑓)) = (𝐴𝑁))
2220, 21oveq12d 6942 . . . . . 6 (𝑓 = 𝐹 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
2322negeqd 10618 . . . . 5 (𝑓 = 𝐹 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
2415, 23eqeq12d 2793 . . . 4 (𝑓 = 𝐹 → (Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁))))
2514, 24imbi12d 336 . . 3 (𝑓 = 𝐹 → (((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((♯‘𝑅) = 𝑁 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))))
26 vieta1.6 . . . 4 (𝜑𝑁 ∈ ℕ)
27 eqeq1 2782 . . . . . . . 8 (𝑦 = 1 → (𝑦 = (deg‘𝑓) ↔ 1 = (deg‘𝑓)))
2827anbi1d 623 . . . . . . 7 (𝑦 = 1 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
2928imbi1d 333 . . . . . 6 (𝑦 = 1 → (((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
3029ralbidv 3168 . . . . 5 (𝑦 = 1 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
31 eqeq1 2782 . . . . . . . 8 (𝑦 = 𝑑 → (𝑦 = (deg‘𝑓) ↔ 𝑑 = (deg‘𝑓)))
3231anbi1d 623 . . . . . . 7 (𝑦 = 𝑑 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
3332imbi1d 333 . . . . . 6 (𝑦 = 𝑑 → (((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
3433ralbidv 3168 . . . . 5 (𝑦 = 𝑑 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
35 eqeq1 2782 . . . . . . . 8 (𝑦 = (𝑑 + 1) → (𝑦 = (deg‘𝑓) ↔ (𝑑 + 1) = (deg‘𝑓)))
3635anbi1d 623 . . . . . . 7 (𝑦 = (𝑑 + 1) → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
3736imbi1d 333 . . . . . 6 (𝑦 = (𝑑 + 1) → (((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ (((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
3837ralbidv 3168 . . . . 5 (𝑦 = (𝑑 + 1) → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
39 eqeq1 2782 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝑓)))
4039anbi1d 623 . . . . . . 7 (𝑦 = 𝑁 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
4140imbi1d 333 . . . . . 6 (𝑦 = 𝑁 → (((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
4241ralbidv 3168 . . . . 5 (𝑦 = 𝑁 → (∀𝑓 ∈ (Poly‘ℂ)((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
43 eqid 2778 . . . . . . . . . . . . . . 15 (coeff‘𝑓) = (coeff‘𝑓)
4443coef3 24436 . . . . . . . . . . . . . 14 (𝑓 ∈ (Poly‘ℂ) → (coeff‘𝑓):ℕ0⟶ℂ)
4544adantr 474 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (coeff‘𝑓):ℕ0⟶ℂ)
46 0nn0 11664 . . . . . . . . . . . . 13 0 ∈ ℕ0
47 ffvelrn 6623 . . . . . . . . . . . . 13 (((coeff‘𝑓):ℕ0⟶ℂ ∧ 0 ∈ ℕ0) → ((coeff‘𝑓)‘0) ∈ ℂ)
4845, 46, 47sylancl 580 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) ∈ ℂ)
49 1nn0 11665 . . . . . . . . . . . . 13 1 ∈ ℕ0
50 ffvelrn 6623 . . . . . . . . . . . . 13 (((coeff‘𝑓):ℕ0⟶ℂ ∧ 1 ∈ ℕ0) → ((coeff‘𝑓)‘1) ∈ ℂ)
5145, 49, 50sylancl 580 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ∈ ℂ)
52 simpr 479 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 = (deg‘𝑓))
5352fveq2d 6452 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) = ((coeff‘𝑓)‘(deg‘𝑓)))
54 ax-1ne0 10343 . . . . . . . . . . . . . . . . 17 1 ≠ 0
5554a1i 11 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 ≠ 0)
5652, 55eqnetrrd 3037 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (deg‘𝑓) ≠ 0)
57 fveq2 6448 . . . . . . . . . . . . . . . . 17 (𝑓 = 0𝑝 → (deg‘𝑓) = (deg‘0𝑝))
58 dgr0 24466 . . . . . . . . . . . . . . . . 17 (deg‘0𝑝) = 0
5957, 58syl6eq 2830 . . . . . . . . . . . . . . . 16 (𝑓 = 0𝑝 → (deg‘𝑓) = 0)
6059necon3i 3001 . . . . . . . . . . . . . . 15 ((deg‘𝑓) ≠ 0 → 𝑓 ≠ 0𝑝)
6156, 60syl 17 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 ≠ 0𝑝)
62 eqid 2778 . . . . . . . . . . . . . . . . 17 (deg‘𝑓) = (deg‘𝑓)
6362, 43dgreq0 24469 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (Poly‘ℂ) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
6463necon3bid 3013 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℂ) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
6564adantr 474 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
6661, 65mpbid 224 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
6753, 66eqnetrd 3036 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ≠ 0)
6848, 51, 67divcld 11154 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
6968negcld 10723 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
70 id 22 . . . . . . . . . . 11 (𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) → 𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
7170sumsn 14891 . . . . . . . . . 10 ((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
7269, 69, 71syl2anc 579 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
7372adantrr 707 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
74 eqid 2778 . . . . . . . . . . . . . 14 (𝑓 “ {0}) = (𝑓 “ {0})
7574fta1 24511 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 𝑓 ≠ 0𝑝) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
7661, 75syldan 585 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
7776simpld 490 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 “ {0}) ∈ Fin)
7877adantrr 707 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (𝑓 “ {0}) ∈ Fin)
7943, 62coeid2 24443 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)))
8069, 79syldan 585 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)))
8152oveq2d 6940 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0...1) = (0...(deg‘𝑓)))
8281sumeq1d 14848 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝑓))(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)))
83 nn0uz 12033 . . . . . . . . . . . . . . . 16 0 = (ℤ‘0)
84 1e0p1 11893 . . . . . . . . . . . . . . . 16 1 = (0 + 1)
85 fveq2 6448 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘1))
86 oveq2 6932 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))
8785, 86oveq12d 6942 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)))
8845ffvelrnda 6625 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝑓)‘𝑘) ∈ ℂ)
89 expcl 13201 . . . . . . . . . . . . . . . . . 18 ((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
9069, 89sylan 575 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
9188, 90mulcld 10399 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) ∈ ℂ)
92 0z 11744 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
9369exp0d 13326 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0) = 1)
9493oveq2d 6940 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = (((coeff‘𝑓)‘0) · 1))
9548mulid1d 10396 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · 1) = ((coeff‘𝑓)‘0))
9694, 95eqtrd 2814 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = ((coeff‘𝑓)‘0))
9796, 48eqeltrd 2859 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈ ℂ)
98 fveq2 6448 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘0))
99 oveq2 6932 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))
10098, 99oveq12d 6942 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
101100fsum1 14892 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℤ ∧ (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
10292, 97, 101sylancr 581 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
103102, 96eqtrd 2814 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0))
104103, 46jctil 515 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0)))
10569exp1d 13327 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1) = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
106105oveq2d 6940 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) = (((coeff‘𝑓)‘1) · -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))))
10751, 68mulneg2d 10832 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))))
10848, 51, 67divcan2d 11156 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = ((coeff‘𝑓)‘0))
109108negeqd 10618 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -((coeff‘𝑓)‘0))
110106, 107, 1093eqtrd 2818 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) = -((coeff‘𝑓)‘0))
111110oveq2d 6940 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)))
11248negidd 10726 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)) = 0)
113111, 112eqtrd 2814 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = 0)
11483, 84, 87, 91, 104, 113fsump1i 14914 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0))
115114simprd 491 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0)
11680, 82, 1153eqtr2d 2820 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)
117 plyf 24402 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
118117ffnd 6294 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℂ) → 𝑓 Fn ℂ)
119118adantr 474 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 Fn ℂ)
120 fniniseg 6604 . . . . . . . . . . . . . 14 (𝑓 Fn ℂ → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (𝑓 “ {0}) ↔ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)))
121119, 120syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (𝑓 “ {0}) ↔ (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)))
12269, 116, 121mpbir2and 703 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (𝑓 “ {0}))
123122snssd 4573 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}))
124123adantrr 707 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}))
125 hashsng 13480 . . . . . . . . . . . . . . 15 (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1)
12669, 125syl 17 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1)
127126, 52eqtrd 2814 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
128127adantrr 707 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
129 simprr 763 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (♯‘(𝑓 “ {0})) = (deg‘𝑓))
130128, 129eqtr4d 2817 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(𝑓 “ {0})))
131 snfi 8328 . . . . . . . . . . . . 13 {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin
132 hashen 13458 . . . . . . . . . . . . 13 (({-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin ∧ (𝑓 “ {0}) ∈ Fin) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0})))
133131, 77, 132sylancr 581 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0})))
134133adantrr 707 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0})))
135130, 134mpbid 224 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0}))
136 fisseneq 8461 . . . . . . . . . 10 (((𝑓 “ {0}) ∈ Fin ∧ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}) ∧ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0})) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (𝑓 “ {0}))
13778, 124, 135, 136syl3anc 1439 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (𝑓 “ {0}))
138137sumeq1d 14848 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = Σ𝑥 ∈ (𝑓 “ {0})𝑥)
139 1m1e0 11452 . . . . . . . . . . . . 13 (1 − 1) = 0
14052oveq1d 6939 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 − 1) = ((deg‘𝑓) − 1))
141139, 140syl5eqr 2828 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 0 = ((deg‘𝑓) − 1))
142141fveq2d 6452 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
143142, 53oveq12d 6942 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
144143negeqd 10618 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
145144adantrr 707 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
14673, 138, 1453eqtr3d 2822 . . . . . . 7 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
147146ex 403 . . . . . 6 (𝑓 ∈ (Poly‘ℂ) → ((1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
148147rgen 3104 . . . . 5 𝑓 ∈ (Poly‘ℂ)((1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
149 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 = 𝑥)
150149cbvsumv 14843 . . . . . . . . . . 11 Σ𝑦 ∈ (𝑓 “ {0})𝑦 = Σ𝑥 ∈ (𝑓 “ {0})𝑥
151150eqeq1i 2783 . . . . . . . . . 10 𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
152151imbi2i 328 . . . . . . . . 9 (((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
153152ralbii 3162 . . . . . . . 8 (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
154 eqid 2778 . . . . . . . . . 10 (coeff‘𝑔) = (coeff‘𝑔)
155 eqid 2778 . . . . . . . . . 10 (deg‘𝑔) = (deg‘𝑔)
156 eqid 2778 . . . . . . . . . 10 (𝑔 “ {0}) = (𝑔 “ {0})
157 simp1r 1212 . . . . . . . . . 10 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → 𝑔 ∈ (Poly‘ℂ))
158 simp3r 1216 . . . . . . . . . 10 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → (♯‘(𝑔 “ {0})) = (deg‘𝑔))
159 simp1l 1211 . . . . . . . . . 10 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → 𝑑 ∈ ℕ)
160 simp3l 1215 . . . . . . . . . 10 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → (𝑑 + 1) = (deg‘𝑔))
161 simp2 1128 . . . . . . . . . . 11 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
162161, 153sylib 210 . . . . . . . . . 10 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
163 eqid 2778 . . . . . . . . . 10 (𝑔 quot (Xp𝑓 − (ℂ × {𝑧}))) = (𝑔 quot (Xp𝑓 − (ℂ × {𝑧})))
164154, 155, 156, 157, 158, 159, 160, 162, 163vieta1lem2 24514 . . . . . . . . 9 (((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) ∧ ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) ∧ ((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔))) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))
1651643exp 1109 . . . . . . . 8 ((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
166153, 165syl5bir 235 . . . . . . 7 ((𝑑 ∈ ℕ ∧ 𝑔 ∈ (Poly‘ℂ)) → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
167166ralrimdva 3151 . . . . . 6 (𝑑 ∈ ℕ → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ∀𝑔 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
168 fveq2 6448 . . . . . . . . . 10 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
169168eqeq2d 2788 . . . . . . . . 9 (𝑔 = 𝑓 → ((𝑑 + 1) = (deg‘𝑔) ↔ (𝑑 + 1) = (deg‘𝑓)))
170 cnveq 5543 . . . . . . . . . . . 12 (𝑔 = 𝑓𝑔 = 𝑓)
171170imaeq1d 5721 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑔 “ {0}) = (𝑓 “ {0}))
172171fveq2d 6452 . . . . . . . . . 10 (𝑔 = 𝑓 → (♯‘(𝑔 “ {0})) = (♯‘(𝑓 “ {0})))
173172, 168eqeq12d 2793 . . . . . . . . 9 (𝑔 = 𝑓 → ((♯‘(𝑔 “ {0})) = (deg‘𝑔) ↔ (♯‘(𝑓 “ {0})) = (deg‘𝑓)))
174169, 173anbi12d 624 . . . . . . . 8 (𝑔 = 𝑓 → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
175171sumeq1d 14848 . . . . . . . . 9 (𝑔 = 𝑓 → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = Σ𝑥 ∈ (𝑓 “ {0})𝑥)
176 fveq2 6448 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (coeff‘𝑔) = (coeff‘𝑓))
177168oveq1d 6939 . . . . . . . . . . . 12 (𝑔 = 𝑓 → ((deg‘𝑔) − 1) = ((deg‘𝑓) − 1))
178176, 177fveq12d 6455 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((coeff‘𝑔)‘((deg‘𝑔) − 1)) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
179176, 168fveq12d 6455 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((coeff‘𝑔)‘(deg‘𝑔)) = ((coeff‘𝑓)‘(deg‘𝑓)))
180178, 179oveq12d 6942 . . . . . . . . . 10 (𝑔 = 𝑓 → (((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
181180negeqd 10618 . . . . . . . . 9 (𝑔 = 𝑓 → -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
182175, 181eqeq12d 2793 . . . . . . . 8 (𝑔 = 𝑓 → (Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) ↔ Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
183174, 182imbi12d 336 . . . . . . 7 (𝑔 = 𝑓 → ((((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) ↔ (((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
184183cbvralv 3367 . . . . . 6 (∀𝑔 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔)))) ↔ ∀𝑓 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
185167, 184syl6ib 243 . . . . 5 (𝑑 ∈ ℕ → (∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))) → ∀𝑓 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))))
18630, 34, 38, 42, 148, 185nnind 11399 . . . 4 (𝑁 ∈ ℕ → ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
18726, 186syl 17 . . 3 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
188 plyssc 24404 . . . 4 (Poly‘𝑆) ⊆ (Poly‘ℂ)
189 vieta1.4 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
190188, 189sseldi 3819 . . 3 (𝜑𝐹 ∈ (Poly‘ℂ))
19125, 187, 190rspcdva 3517 . 2 (𝜑 → ((♯‘𝑅) = 𝑁 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁))))
1921, 191mpd 15 1 (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  wral 3090  wss 3792  {csn 4398   class class class wbr 4888   × cxp 5355  ccnv 5356  cima 5360   Fn wfn 6132  wf 6133  cfv 6137  (class class class)co 6924  𝑓 cof 7174  cen 8240  Fincfn 8243  cc 10272  0cc0 10274  1c1 10275   + caddc 10277   · cmul 10279  cle 10414  cmin 10608  -cneg 10609   / cdiv 11035  cn 11379  0cn0 11647  cz 11733  ...cfz 12648  cexp 13183  chash 13441  Σcsu 14833  0𝑝c0p 23884  Polycply 24388  Xpcidp 24389  coeffccoe 24390  degcdgr 24391   quot cquot 24493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352  ax-addf 10353
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-inf 8639  df-oi 8706  df-card 9100  df-cda 9327  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-n0 11648  df-xnn0 11720  df-z 11734  df-uz 11998  df-rp 12143  df-fz 12649  df-fzo 12790  df-fl 12917  df-seq 13125  df-exp 13184  df-hash 13442  df-cj 14252  df-re 14253  df-im 14254  df-sqrt 14388  df-abs 14389  df-clim 14636  df-rlim 14637  df-sum 14834  df-0p 23885  df-ply 24392  df-idp 24393  df-coe 24394  df-dgr 24395  df-quot 24494
This theorem is referenced by:  basellem5  25274
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