MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vieta1 Structured version   Visualization version   GIF version

Theorem vieta1 25578
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 6825 . . . . . . 7 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
32eqeq2d 2747 . . . . . 6 (𝑓 = 𝐹 → (𝑁 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝐹)))
4 cnveq 5815 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
54imaeq1d 5998 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
6 vieta1.3 . . . . . . . . 9 𝑅 = (𝐹 “ {0})
75, 6eqtr4di 2794 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
87fveq2d 6829 . . . . . . 7 (𝑓 = 𝐹 → (♯‘(𝑓 “ {0})) = (♯‘𝑅))
9 vieta1.2 . . . . . . . 8 𝑁 = (deg‘𝐹)
102, 9eqtr4di 2794 . . . . . . 7 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑁)
118, 10eqeq12d 2752 . . . . . 6 (𝑓 = 𝐹 → ((♯‘(𝑓 “ {0})) = (deg‘𝑓) ↔ (♯‘𝑅) = 𝑁))
123, 11anbi12d 631 . . . . 5 (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝐹) ∧ (♯‘𝑅) = 𝑁)))
139biantrur 531 . . . . 5 ((♯‘𝑅) = 𝑁 ↔ (𝑁 = (deg‘𝐹) ∧ (♯‘𝑅) = 𝑁))
1412, 13bitr4di 288 . . . 4 (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (♯‘𝑅) = 𝑁))
157sumeq1d 15512 . . . . 5 (𝑓 = 𝐹 → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = Σ𝑥𝑅 𝑥)
16 fveq2 6825 . . . . . . . . 9 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
17 vieta1.1 . . . . . . . . 9 𝐴 = (coeff‘𝐹)
1816, 17eqtr4di 2794 . . . . . . . 8 (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴)
1910oveq1d 7352 . . . . . . . 8 (𝑓 = 𝐹 → ((deg‘𝑓) − 1) = (𝑁 − 1))
2018, 19fveq12d 6832 . . . . . . 7 (𝑓 = 𝐹 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = (𝐴‘(𝑁 − 1)))
2118, 10fveq12d 6832 . . . . . . 7 (𝑓 = 𝐹 → ((coeff‘𝑓)‘(deg‘𝑓)) = (𝐴𝑁))
2220, 21oveq12d 7355 . . . . . 6 (𝑓 = 𝐹 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
2322negeqd 11316 . . . . 5 (𝑓 = 𝐹 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
2415, 23eqeq12d 2752 . . . 4 (𝑓 = 𝐹 → (Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁))))
2514, 24imbi12d 344 . . 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 2740 . . . . . . . 8 (𝑦 = 1 → (𝑦 = (deg‘𝑓) ↔ 1 = (deg‘𝑓)))
2827anbi1d 630 . . . . . . 7 (𝑦 = 1 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
2928imbi1d 341 . . . . . 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 3170 . . . . 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 2740 . . . . . . . 8 (𝑦 = 𝑑 → (𝑦 = (deg‘𝑓) ↔ 𝑑 = (deg‘𝑓)))
3231anbi1d 630 . . . . . . 7 (𝑦 = 𝑑 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑑 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
3332imbi1d 341 . . . . . 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 3170 . . . . 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 2740 . . . . . . . 8 (𝑦 = (𝑑 + 1) → (𝑦 = (deg‘𝑓) ↔ (𝑑 + 1) = (deg‘𝑓)))
3635anbi1d 630 . . . . . . 7 (𝑦 = (𝑑 + 1) → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
3736imbi1d 341 . . . . . 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 3170 . . . . 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 2740 . . . . . . . 8 (𝑦 = 𝑁 → (𝑦 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝑓)))
4039anbi1d 630 . . . . . . 7 (𝑦 = 𝑁 → ((𝑦 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
4140imbi1d 341 . . . . . 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 3170 . . . . 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 2736 . . . . . . . . . . . . . . 15 (coeff‘𝑓) = (coeff‘𝑓)
4443coef3 25499 . . . . . . . . . . . . . 14 (𝑓 ∈ (Poly‘ℂ) → (coeff‘𝑓):ℕ0⟶ℂ)
4544adantr 481 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (coeff‘𝑓):ℕ0⟶ℂ)
46 0nn0 12349 . . . . . . . . . . . . 13 0 ∈ ℕ0
47 ffvelcdm 7015 . . . . . . . . . . . . 13 (((coeff‘𝑓):ℕ0⟶ℂ ∧ 0 ∈ ℕ0) → ((coeff‘𝑓)‘0) ∈ ℂ)
4845, 46, 47sylancl 586 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) ∈ ℂ)
49 1nn0 12350 . . . . . . . . . . . . 13 1 ∈ ℕ0
50 ffvelcdm 7015 . . . . . . . . . . . . 13 (((coeff‘𝑓):ℕ0⟶ℂ ∧ 1 ∈ ℕ0) → ((coeff‘𝑓)‘1) ∈ ℂ)
5145, 49, 50sylancl 586 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ∈ ℂ)
52 simpr 485 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 = (deg‘𝑓))
5352fveq2d 6829 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) = ((coeff‘𝑓)‘(deg‘𝑓)))
54 ax-1ne0 11041 . . . . . . . . . . . . . . . . 17 1 ≠ 0
5554a1i 11 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 ≠ 0)
5652, 55eqnetrrd 3009 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (deg‘𝑓) ≠ 0)
57 fveq2 6825 . . . . . . . . . . . . . . . . 17 (𝑓 = 0𝑝 → (deg‘𝑓) = (deg‘0𝑝))
58 dgr0 25529 . . . . . . . . . . . . . . . . 17 (deg‘0𝑝) = 0
5957, 58eqtrdi 2792 . . . . . . . . . . . . . . . 16 (𝑓 = 0𝑝 → (deg‘𝑓) = 0)
6059necon3i 2973 . . . . . . . . . . . . . . 15 ((deg‘𝑓) ≠ 0 → 𝑓 ≠ 0𝑝)
6156, 60syl 17 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 ≠ 0𝑝)
62 eqid 2736 . . . . . . . . . . . . . . . . 17 (deg‘𝑓) = (deg‘𝑓)
6362, 43dgreq0 25532 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (Poly‘ℂ) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
6463necon3bid 2985 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℂ) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
6564adantr 481 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
6661, 65mpbid 231 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
6753, 66eqnetrd 3008 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ≠ 0)
6848, 51, 67divcld 11852 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
6968negcld 11420 . . . . . . . . . 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 15557 . . . . . . . . . 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 584 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
7372adantrr 714 . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (𝑓 “ {0}) = (𝑓 “ {0})
7574fta1 25574 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 𝑓 ≠ 0𝑝) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
7661, 75syldan 591 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((𝑓 “ {0}) ∈ Fin ∧ (♯‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
7776simpld 495 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 “ {0}) ∈ Fin)
7877adantrr 714 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (𝑓 “ {0}) ∈ Fin)
7943, 62coeid2 25506 . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . 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 7353 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0...1) = (0...(deg‘𝑓)))
8281sumeq1d 15512 . . . . . . . . . . . . . 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 12721 . . . . . . . . . . . . . . . 16 0 = (ℤ‘0)
84 1e0p1 12580 . . . . . . . . . . . . . . . 16 1 = (0 + 1)
85 fveq2 6825 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘1))
86 oveq2 7345 . . . . . . . . . . . . . . . . 17 (𝑘 = 1 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))
8785, 86oveq12d 7355 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)))
8845ffvelcdmda 7017 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝑓)‘𝑘) ∈ ℂ)
89 expcl 13901 . . . . . . . . . . . . . . . . . 18 ((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
9069, 89sylan 580 . . . . . . . . . . . . . . . . 17 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
9188, 90mulcld 11096 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) ∈ ℂ)
92 0z 12431 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
9369exp0d 13959 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0) = 1)
9493oveq2d 7353 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = (((coeff‘𝑓)‘0) · 1))
9548mulid1d 11093 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · 1) = ((coeff‘𝑓)‘0))
9694, 95eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = ((coeff‘𝑓)‘0))
9796, 48eqeltrd 2837 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈ ℂ)
98 fveq2 6825 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘0))
99 oveq2 7345 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))
10098, 99oveq12d 7355 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
101100fsum1 15558 . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . 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 13960 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1) = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
106105oveq2d 7353 . . . . . . . . . . . . . . . . . . 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 11530 . . . . . . . . . . . . . . . . . . 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 11854 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = ((coeff‘𝑓)‘0))
109108negeqd 11316 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -((coeff‘𝑓)‘0))
110106, 107, 1093eqtrd 2780 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) = -((coeff‘𝑓)‘0))
111110oveq2d 7353 . . . . . . . . . . . . . . . . 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 11423 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)) = 0)
113111, 112eqtrd 2776 . . . . . . . . . . . . . . . 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 15580 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0))
115114simprd 496 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0)
11680, 82, 1153eqtr2d 2782 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)
117 plyf 25465 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
118117ffnd 6652 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℂ) → 𝑓 Fn ℂ)
119118adantr 481 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 Fn ℂ)
120 fniniseg 6993 . . . . . . . . . . . . . 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 710 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (𝑓 “ {0}))
123122snssd 4756 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}))
124123adantrr 714 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}))
125 hashsng 14184 . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
128127adantrr 714 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
129 simprr 770 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (♯‘(𝑓 “ {0})) = (deg‘𝑓))
130128, 129eqtr4d 2779 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → (♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(𝑓 “ {0})))
131 snfi 8909 . . . . . . . . . . . . 13 {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin
132 hashen 14162 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((♯‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (♯‘(𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0})))
134133adantrr 714 . . . . . . . . . . 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 231 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0}))
136 fisseneq 9122 . . . . . . . . . 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 1370 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (𝑓 “ {0}))
138137sumeq1d 15512 . . . . . . . 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 12146 . . . . . . . . . . . . 13 (1 − 1) = 0
14052oveq1d 7352 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 − 1) = ((deg‘𝑓) − 1))
141139, 140eqtr3id 2790 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 0 = ((deg‘𝑓) − 1))
142141fveq2d 6829 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
143142, 53oveq12d 7355 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
144143negeqd 11316 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
145144adantrr 714 . . . . . . . 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 2784 . . . . . . 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 413 . . . . . 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 3063 . . . . 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 15507 . . . . . . . . . . 11 Σ𝑦 ∈ (𝑓 “ {0})𝑦 = Σ𝑥 ∈ (𝑓 “ {0})𝑥
151150eqeq1i 2741 . . . . . . . . . 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 335 . . . . . . . . 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 3092 . . . . . . . 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 2736 . . . . . . . . . 10 (coeff‘𝑔) = (coeff‘𝑔)
155 eqid 2736 . . . . . . . . . 10 (deg‘𝑔) = (deg‘𝑔)
156 eqid 2736 . . . . . . . . . 10 (𝑔 “ {0}) = (𝑔 “ {0})
157 simp1r 1197 . . . . . . . . . 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 1201 . . . . . . . . . 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 1196 . . . . . . . . . 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 1200 . . . . . . . . . 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 1136 . . . . . . . . . . 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 217 . . . . . . . . . 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 2736 . . . . . . . . . 10 (𝑔 quot (Xpf − (ℂ × {𝑧}))) = (𝑔 quot (Xpf − (ℂ × {𝑧})))
164154, 155, 156, 157, 158, 159, 160, 162, 163vieta1lem2 25577 . . . . . . . . 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 1118 . . . . . . . 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 242 . . . . . . 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 3147 . . . . . 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 6825 . . . . . . . . . 10 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
169168eqeq2d 2747 . . . . . . . . 9 (𝑔 = 𝑓 → ((𝑑 + 1) = (deg‘𝑔) ↔ (𝑑 + 1) = (deg‘𝑓)))
170 cnveq 5815 . . . . . . . . . . . 12 (𝑔 = 𝑓𝑔 = 𝑓)
171170imaeq1d 5998 . . . . . . . . . . 11 (𝑔 = 𝑓 → (𝑔 “ {0}) = (𝑓 “ {0}))
172171fveq2d 6829 . . . . . . . . . 10 (𝑔 = 𝑓 → (♯‘(𝑔 “ {0})) = (♯‘(𝑓 “ {0})))
173172, 168eqeq12d 2752 . . . . . . . . 9 (𝑔 = 𝑓 → ((♯‘(𝑔 “ {0})) = (deg‘𝑔) ↔ (♯‘(𝑓 “ {0})) = (deg‘𝑓)))
174169, 173anbi12d 631 . . . . . . . 8 (𝑔 = 𝑓 → (((𝑑 + 1) = (deg‘𝑔) ∧ (♯‘(𝑔 “ {0})) = (deg‘𝑔)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (♯‘(𝑓 “ {0})) = (deg‘𝑓))))
175171sumeq1d 15512 . . . . . . . . 9 (𝑔 = 𝑓 → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = Σ𝑥 ∈ (𝑓 “ {0})𝑥)
176 fveq2 6825 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (coeff‘𝑔) = (coeff‘𝑓))
177168oveq1d 7352 . . . . . . . . . . . 12 (𝑔 = 𝑓 → ((deg‘𝑔) − 1) = ((deg‘𝑓) − 1))
178176, 177fveq12d 6832 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((coeff‘𝑔)‘((deg‘𝑔) − 1)) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
179176, 168fveq12d 6832 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((coeff‘𝑔)‘(deg‘𝑔)) = ((coeff‘𝑓)‘(deg‘𝑓)))
180178, 179oveq12d 7355 . . . . . . . . . 10 (𝑔 = 𝑓 → (((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
181180negeqd 11316 . . . . . . . . 9 (𝑔 = 𝑓 → -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
182175, 181eqeq12d 2752 . . . . . . . 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 344 . . . . . . 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))))))
184183cbvralvw 3221 . . . . . 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 250 . . . . 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 12092 . . . 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 25467 . . . 4 (Poly‘𝑆) ⊆ (Poly‘ℂ)
189 vieta1.4 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
190188, 189sselid 3930 . . 3 (𝜑𝐹 ∈ (Poly‘ℂ))
19125, 187, 190rspcdva 3571 . 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 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940  wral 3061  wss 3898  {csn 4573   class class class wbr 5092   × cxp 5618  ccnv 5619  cima 5623   Fn wfn 6474  wf 6475  cfv 6479  (class class class)co 7337  f cof 7593  cen 8801  Fincfn 8804  cc 10970  0cc0 10972  1c1 10973   + caddc 10975   · cmul 10977  cle 11111  cmin 11306  -cneg 11307   / cdiv 11733  cn 12074  0cn0 12334  cz 12420  ...cfz 13340  cexp 13883  chash 14145  Σcsu 15496  0𝑝c0p 24939  Polycply 25451  Xpcidp 25452  coeffccoe 25453  degcdgr 25454   quot cquot 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-inf2 9498  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-oadd 8371  df-er 8569  df-map 8688  df-pm 8689  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-sup 9299  df-inf 9300  df-oi 9367  df-dju 9758  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-xnn0 12407  df-z 12421  df-uz 12684  df-rp 12832  df-fz 13341  df-fzo 13484  df-fl 13613  df-seq 13823  df-exp 13884  df-hash 14146  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-rlim 15297  df-sum 15497  df-0p 24940  df-ply 25455  df-idp 25456  df-coe 25457  df-dgr 25458  df-quot 25557
This theorem is referenced by:  basellem5  26340
  Copyright terms: Public domain W3C validator