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Theorem vieta1 24112
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 plyssc 24001 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 vieta1.4 . . 3 (𝜑𝐹 ∈ (Poly‘𝑆))
31, 2sseldi 3634 . 2 (𝜑𝐹 ∈ (Poly‘ℂ))
4 vieta1.6 . . 3 (𝜑𝑁 ∈ ℕ)
5 eqeq1 2655 . . . . . . 7 (𝑦 = 1 → (𝑦 = (deg‘𝑓) ↔ 1 = (deg‘𝑓)))
65anbi1d 741 . . . . . 6 (𝑦 = 1 → ((𝑦 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))))
76imbi1d 330 . . . . 5 (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))))))
87ralbidv 3015 . . . 4 (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))))))
9 eqeq1 2655 . . . . . . 7 (𝑦 = 𝑑 → (𝑦 = (deg‘𝑓) ↔ 𝑑 = (deg‘𝑓)))
109anbi1d 741 . . . . . 6 (𝑦 = 𝑑 → ((𝑦 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑑 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))))
1110imbi1d 330 . . . . 5 (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))))))
1211ralbidv 3015 . . . 4 (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))))))
13 eqeq1 2655 . . . . . . 7 (𝑦 = (𝑑 + 1) → (𝑦 = (deg‘𝑓) ↔ (𝑑 + 1) = (deg‘𝑓)))
1413anbi1d 741 . . . . . 6 (𝑦 = (𝑑 + 1) → ((𝑦 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))))
1514imbi1d 330 . . . . 5 (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))))))
1615ralbidv 3015 . . . 4 (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))))))
17 eqeq1 2655 . . . . . . 7 (𝑦 = 𝑁 → (𝑦 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝑓)))
1817anbi1d 741 . . . . . 6 (𝑦 = 𝑁 → ((𝑦 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))))
1918imbi1d 330 . . . . 5 (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))))))
2019ralbidv 3015 . . . 4 (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))))))
21 eqid 2651 . . . . . . . . . . . . . 14 (coeff‘𝑓) = (coeff‘𝑓)
2221coef3 24033 . . . . . . . . . . . . 13 (𝑓 ∈ (Poly‘ℂ) → (coeff‘𝑓):ℕ0⟶ℂ)
2322adantr 480 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (coeff‘𝑓):ℕ0⟶ℂ)
24 0nn0 11345 . . . . . . . . . . . 12 0 ∈ ℕ0
25 ffvelrn 6397 . . . . . . . . . . . 12 (((coeff‘𝑓):ℕ0⟶ℂ ∧ 0 ∈ ℕ0) → ((coeff‘𝑓)‘0) ∈ ℂ)
2623, 24, 25sylancl 695 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) ∈ ℂ)
27 1nn0 11346 . . . . . . . . . . . 12 1 ∈ ℕ0
28 ffvelrn 6397 . . . . . . . . . . . 12 (((coeff‘𝑓):ℕ0⟶ℂ ∧ 1 ∈ ℕ0) → ((coeff‘𝑓)‘1) ∈ ℂ)
2923, 27, 28sylancl 695 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ∈ ℂ)
30 simpr 476 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 = (deg‘𝑓))
3130fveq2d 6233 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) = ((coeff‘𝑓)‘(deg‘𝑓)))
32 ax-1ne0 10043 . . . . . . . . . . . . . . . 16 1 ≠ 0
3332a1i 11 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 1 ≠ 0)
3430, 33eqnetrrd 2891 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (deg‘𝑓) ≠ 0)
35 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑓 = 0𝑝 → (deg‘𝑓) = (deg‘0𝑝))
36 dgr0 24063 . . . . . . . . . . . . . . . 16 (deg‘0𝑝) = 0
3735, 36syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑓 = 0𝑝 → (deg‘𝑓) = 0)
3837necon3i 2855 . . . . . . . . . . . . . 14 ((deg‘𝑓) ≠ 0 → 𝑓 ≠ 0𝑝)
3934, 38syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 ≠ 0𝑝)
40 eqid 2651 . . . . . . . . . . . . . . . 16 (deg‘𝑓) = (deg‘𝑓)
4140, 21dgreq0 24066 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℂ) → (𝑓 = 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) = 0))
4241necon3bid 2867 . . . . . . . . . . . . . 14 (𝑓 ∈ (Poly‘ℂ) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
4342adantr 480 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 ≠ 0𝑝 ↔ ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0))
4439, 43mpbid 222 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘(deg‘𝑓)) ≠ 0)
4531, 44eqnetrd 2890 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘1) ≠ 0)
4626, 29, 45divcld 10839 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
4746negcld 10417 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ)
48 id 22 . . . . . . . . . 10 (𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) → 𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
4948sumsn 14519 . . . . . . . . 9 ((-(((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)))
5047, 47, 49syl2anc 694 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
5150adantrr 753 . . . . . . 7 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
52 eqid 2651 . . . . . . . . . . . . 13 (𝑓 “ {0}) = (𝑓 “ {0})
5352fta1 24108 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 𝑓 ≠ 0𝑝) → ((𝑓 “ {0}) ∈ Fin ∧ (#‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
5439, 53syldan 486 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((𝑓 “ {0}) ∈ Fin ∧ (#‘(𝑓 “ {0})) ≤ (deg‘𝑓)))
5554simpld 474 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓 “ {0}) ∈ Fin)
5655adantrr 753 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → (𝑓 “ {0}) ∈ Fin)
5721, 40coeid2 24040 . . . . . . . . . . . . . 14 ((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)))
5847, 57syldan 486 . . . . . . . . . . . . 13 ((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)))
5930oveq2d 6706 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0...1) = (0...(deg‘𝑓)))
6059sumeq1d 14475 . . . . . . . . . . . . 13 ((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)))
61 nn0uz 11760 . . . . . . . . . . . . . . 15 0 = (ℤ‘0)
62 1e0p1 11590 . . . . . . . . . . . . . . 15 1 = (0 + 1)
63 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘1))
64 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑘 = 1 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))
6563, 64oveq12d 6708 . . . . . . . . . . . . . . 15 (𝑘 = 1 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)))
6623ffvelrnda 6399 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝑓)‘𝑘) ∈ ℂ)
67 expcl 12918 . . . . . . . . . . . . . . . . 17 ((-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
6847, 67sylan 487 . . . . . . . . . . . . . . . 16 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) ∈ ℂ)
6966, 68mulcld 10098 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) ∧ 𝑘 ∈ ℕ0) → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) ∈ ℂ)
70 0z 11426 . . . . . . . . . . . . . . . . . 18 0 ∈ ℤ
7147exp0d 13042 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0) = 1)
7271oveq2d 6706 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = (((coeff‘𝑓)‘0) · 1))
7326mulid1d 10095 . . . . . . . . . . . . . . . . . . . 20 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · 1) = ((coeff‘𝑓)‘0))
7472, 73eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) = ((coeff‘𝑓)‘0))
7574, 26eqeltrd 2730 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)) ∈ ℂ)
76 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((coeff‘𝑓)‘𝑘) = ((coeff‘𝑓)‘0))
77 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘) = (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0))
7876, 77oveq12d 6708 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 0 → (((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = (((coeff‘𝑓)‘0) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑0)))
7978fsum1 14520 . . . . . . . . . . . . . . . . . 18 ((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)))
8070, 75, 79sylancr 696 . . . . . . . . . . . . . . . . 17 ((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)))
8180, 74eqtrd 2685 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0))
8281, 24jctil 559 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (0 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...0)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = ((coeff‘𝑓)‘0)))
8347exp1d 13043 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1) = -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)))
8483oveq2d 6706 . . . . . . . . . . . . . . . . . 18 ((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))))
8529, 46mulneg2d 10522 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))))
8626, 29, 45divcan2d 10841 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = ((coeff‘𝑓)‘0))
8786negeqd 10313 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘1) · (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = -((coeff‘𝑓)‘0))
8884, 85, 873eqtrd 2689 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1)) = -((coeff‘𝑓)‘0))
8988oveq2d 6706 . . . . . . . . . . . . . . . 16 ((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)))
9026negidd 10420 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + -((coeff‘𝑓)‘0)) = 0)
9189, 90eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) + (((coeff‘𝑓)‘1) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑1))) = 0)
9261, 62, 65, 69, 82, 91fsump1i 14544 . . . . . . . . . . . . . 14 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 ∈ ℕ0 ∧ Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0))
9392simprd 478 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → Σ𝑘 ∈ (0...1)(((coeff‘𝑓)‘𝑘) · (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))↑𝑘)) = 0)
9458, 60, 933eqtr2d 2691 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (𝑓‘-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))) = 0)
95 plyf 23999 . . . . . . . . . . . . . . 15 (𝑓 ∈ (Poly‘ℂ) → 𝑓:ℂ⟶ℂ)
96 ffn 6083 . . . . . . . . . . . . . . 15 (𝑓:ℂ⟶ℂ → 𝑓 Fn ℂ)
9795, 96syl 17 . . . . . . . . . . . . . 14 (𝑓 ∈ (Poly‘ℂ) → 𝑓 Fn ℂ)
9897adantr 480 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 𝑓 Fn ℂ)
99 fniniseg 6378 . . . . . . . . . . . . 13 (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)))
10098, 99syl 17 . . . . . . . . . . . 12 ((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)))
10147, 94, 100mpbir2and 977 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ (𝑓 “ {0}))
102101snssd 4372 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}))
103102adantrr 753 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ⊆ (𝑓 “ {0}))
104 hashsng 13197 . . . . . . . . . . . . . 14 (-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) ∈ ℂ → (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1)
10547, 104syl 17 . . . . . . . . . . . . 13 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = 1)
106105, 30eqtrd 2685 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
107106adantrr 753 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (deg‘𝑓))
108 simprr 811 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → (#‘(𝑓 “ {0})) = (deg‘𝑓))
109107, 108eqtr4d 2688 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → (#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (#‘(𝑓 “ {0})))
110 snfi 8079 . . . . . . . . . . . 12 {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ∈ Fin
111 hashen 13175 . . . . . . . . . . . 12 (({-(((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})))
112110, 55, 111sylancr 696 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((#‘{-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}) = (#‘(𝑓 “ {0})) ↔ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0})))
113112adantrr 753 . . . . . . . . . 10 ((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})))
114109, 113mpbid 222 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} ≈ (𝑓 “ {0}))
115 fisseneq 8212 . . . . . . . . 9 (((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}))
11656, 103, 114, 115syl3anc 1366 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))} = (𝑓 “ {0}))
117116sumeq1d 14475 . . . . . . 7 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ {-(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1))}𝑥 = Σ𝑥 ∈ (𝑓 “ {0})𝑥)
118 1m1e0 11127 . . . . . . . . . . . 12 (1 − 1) = 0
11930oveq1d 6705 . . . . . . . . . . . 12 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (1 − 1) = ((deg‘𝑓) − 1))
120118, 119syl5eqr 2699 . . . . . . . . . . 11 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → 0 = ((deg‘𝑓) − 1))
121120fveq2d 6233 . . . . . . . . . 10 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → ((coeff‘𝑓)‘0) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
122121, 31oveq12d 6708 . . . . . . . . 9 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → (((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
123122negeqd 10313 . . . . . . . 8 ((𝑓 ∈ (Poly‘ℂ) ∧ 1 = (deg‘𝑓)) → -(((coeff‘𝑓)‘0) / ((coeff‘𝑓)‘1)) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
124123adantrr 753 . . . . . . 7 ((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))))
12551, 117, 1243eqtr3d 2693 . . . . . 6 ((𝑓 ∈ (Poly‘ℂ) ∧ (1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
126125ex 449 . . . . 5 (𝑓 ∈ (Poly‘ℂ) → ((1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
127126rgen 2951 . . . 4 𝑓 ∈ (Poly‘ℂ)((1 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
128 id 22 . . . . . . . . . . 11 (𝑦 = 𝑥𝑦 = 𝑥)
129128cbvsumv 14470 . . . . . . . . . 10 Σ𝑦 ∈ (𝑓 “ {0})𝑦 = Σ𝑥 ∈ (𝑓 “ {0})𝑥
130129eqeq1i 2656 . . . . . . . . 9 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))))
131130imbi2i 325 . . . . . . . 8 (((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)))))
132131ralbii 3009 . . . . . . 7 (∀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)))))
133 eqid 2651 . . . . . . . . 9 (coeff‘𝑔) = (coeff‘𝑔)
134 eqid 2651 . . . . . . . . 9 (deg‘𝑔) = (deg‘𝑔)
135 eqid 2651 . . . . . . . . 9 (𝑔 “ {0}) = (𝑔 “ {0})
136 simp1r 1106 . . . . . . . . 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))) → 𝑔 ∈ (Poly‘ℂ))
137 simp3r 1110 . . . . . . . . 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))) → (#‘(𝑔 “ {0})) = (deg‘𝑔))
138 simp1l 1105 . . . . . . . . 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))) → 𝑑 ∈ ℕ)
139 simp3l 1109 . . . . . . . . 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))) → (𝑑 + 1) = (deg‘𝑔))
140 simp2 1082 . . . . . . . . . 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)) → Σ𝑦 ∈ (𝑓 “ {0})𝑦 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
141140, 132sylib 208 . . . . . . . . 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))) → ∀𝑓 ∈ (Poly‘ℂ)((𝑑 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
142 eqid 2651 . . . . . . . . 9 (𝑔 quot (Xp𝑓 − (ℂ × {𝑧}))) = (𝑔 quot (Xp𝑓 − (ℂ × {𝑧})))
143133, 134, 135, 136, 137, 138, 139, 141, 142vieta1lem2 24111 . . . . . . . 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))))
1441433exp 1283 . . . . . . 7 ((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))))))
145132, 144syl5bir 233 . . . . . 6 ((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))))))
146145ralrimdva 2998 . . . . 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)))) → ∀𝑔 ∈ (Poly‘ℂ)(((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(𝑔 “ {0})) = (deg‘𝑔)) → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))))))
147 fveq2 6229 . . . . . . . . 9 (𝑔 = 𝑓 → (deg‘𝑔) = (deg‘𝑓))
148147eqeq2d 2661 . . . . . . . 8 (𝑔 = 𝑓 → ((𝑑 + 1) = (deg‘𝑔) ↔ (𝑑 + 1) = (deg‘𝑓)))
149 cnveq 5328 . . . . . . . . . . 11 (𝑔 = 𝑓𝑔 = 𝑓)
150149imaeq1d 5500 . . . . . . . . . 10 (𝑔 = 𝑓 → (𝑔 “ {0}) = (𝑓 “ {0}))
151150fveq2d 6233 . . . . . . . . 9 (𝑔 = 𝑓 → (#‘(𝑔 “ {0})) = (#‘(𝑓 “ {0})))
152151, 147eqeq12d 2666 . . . . . . . 8 (𝑔 = 𝑓 → ((#‘(𝑔 “ {0})) = (deg‘𝑔) ↔ (#‘(𝑓 “ {0})) = (deg‘𝑓)))
153148, 152anbi12d 747 . . . . . . 7 (𝑔 = 𝑓 → (((𝑑 + 1) = (deg‘𝑔) ∧ (#‘(𝑔 “ {0})) = (deg‘𝑔)) ↔ ((𝑑 + 1) = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓))))
154150sumeq1d 14475 . . . . . . . 8 (𝑔 = 𝑓 → Σ𝑥 ∈ (𝑔 “ {0})𝑥 = Σ𝑥 ∈ (𝑓 “ {0})𝑥)
155 fveq2 6229 . . . . . . . . . . 11 (𝑔 = 𝑓 → (coeff‘𝑔) = (coeff‘𝑓))
156147oveq1d 6705 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((deg‘𝑔) − 1) = ((deg‘𝑓) − 1))
157155, 156fveq12d 6235 . . . . . . . . . 10 (𝑔 = 𝑓 → ((coeff‘𝑔)‘((deg‘𝑔) − 1)) = ((coeff‘𝑓)‘((deg‘𝑓) − 1)))
158155, 147fveq12d 6235 . . . . . . . . . 10 (𝑔 = 𝑓 → ((coeff‘𝑔)‘(deg‘𝑔)) = ((coeff‘𝑓)‘(deg‘𝑓)))
159157, 158oveq12d 6708 . . . . . . . . 9 (𝑔 = 𝑓 → (((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
160159negeqd 10313 . . . . . . . 8 (𝑔 = 𝑓 → -(((coeff‘𝑔)‘((deg‘𝑔) − 1)) / ((coeff‘𝑔)‘(deg‘𝑔))) = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))))
161154, 160eqeq12d 2666 . . . . . . 7 (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)))))
162153, 161imbi12d 333 . . . . . 6 (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))))))
163162cbvralv 3201 . . . . 5 (∀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)))))
164146, 163syl6ib 241 . . . 4 (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))))))
1658, 12, 16, 20, 127, 164nnind 11076 . . 3 (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)))))
1664, 165syl 17 . 2 (𝜑 → ∀𝑓 ∈ (Poly‘ℂ)((𝑁 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓)))))
167 vieta1.5 . 2 (𝜑 → (#‘𝑅) = 𝑁)
168 fveq2 6229 . . . . . . 7 (𝑓 = 𝐹 → (deg‘𝑓) = (deg‘𝐹))
169168eqeq2d 2661 . . . . . 6 (𝑓 = 𝐹 → (𝑁 = (deg‘𝑓) ↔ 𝑁 = (deg‘𝐹)))
170 cnveq 5328 . . . . . . . . . 10 (𝑓 = 𝐹𝑓 = 𝐹)
171170imaeq1d 5500 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ {0}) = (𝐹 “ {0}))
172 vieta1.3 . . . . . . . . 9 𝑅 = (𝐹 “ {0})
173171, 172syl6eqr 2703 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓 “ {0}) = 𝑅)
174173fveq2d 6233 . . . . . . 7 (𝑓 = 𝐹 → (#‘(𝑓 “ {0})) = (#‘𝑅))
175 vieta1.2 . . . . . . . 8 𝑁 = (deg‘𝐹)
176168, 175syl6eqr 2703 . . . . . . 7 (𝑓 = 𝐹 → (deg‘𝑓) = 𝑁)
177174, 176eqeq12d 2666 . . . . . 6 (𝑓 = 𝐹 → ((#‘(𝑓 “ {0})) = (deg‘𝑓) ↔ (#‘𝑅) = 𝑁))
178169, 177anbi12d 747 . . . . 5 (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (𝑁 = (deg‘𝐹) ∧ (#‘𝑅) = 𝑁)))
179175biantrur 526 . . . . 5 ((#‘𝑅) = 𝑁 ↔ (𝑁 = (deg‘𝐹) ∧ (#‘𝑅) = 𝑁))
180178, 179syl6bbr 278 . . . 4 (𝑓 = 𝐹 → ((𝑁 = (deg‘𝑓) ∧ (#‘(𝑓 “ {0})) = (deg‘𝑓)) ↔ (#‘𝑅) = 𝑁))
181173sumeq1d 14475 . . . . 5 (𝑓 = 𝐹 → Σ𝑥 ∈ (𝑓 “ {0})𝑥 = Σ𝑥𝑅 𝑥)
182 fveq2 6229 . . . . . . . . 9 (𝑓 = 𝐹 → (coeff‘𝑓) = (coeff‘𝐹))
183 vieta1.1 . . . . . . . . 9 𝐴 = (coeff‘𝐹)
184182, 183syl6eqr 2703 . . . . . . . 8 (𝑓 = 𝐹 → (coeff‘𝑓) = 𝐴)
185176oveq1d 6705 . . . . . . . 8 (𝑓 = 𝐹 → ((deg‘𝑓) − 1) = (𝑁 − 1))
186184, 185fveq12d 6235 . . . . . . 7 (𝑓 = 𝐹 → ((coeff‘𝑓)‘((deg‘𝑓) − 1)) = (𝐴‘(𝑁 − 1)))
187184, 176fveq12d 6235 . . . . . . 7 (𝑓 = 𝐹 → ((coeff‘𝑓)‘(deg‘𝑓)) = (𝐴𝑁))
188186, 187oveq12d 6708 . . . . . 6 (𝑓 = 𝐹 → (((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = ((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
189188negeqd 10313 . . . . 5 (𝑓 = 𝐹 → -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
190181, 189eqeq12d 2666 . . . 4 (𝑓 = 𝐹 → (Σ𝑥 ∈ (𝑓 “ {0})𝑥 = -(((coeff‘𝑓)‘((deg‘𝑓) − 1)) / ((coeff‘𝑓)‘(deg‘𝑓))) ↔ Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁))))
191180, 190imbi12d 333 . . 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)))))
192191rspcv 3336 . 2 (𝐹 ∈ (Poly‘ℂ) → (∀𝑓 ∈ (Poly‘ℂ)((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)))))
1933, 166, 167, 192syl3c 66 1 (𝜑 → Σ𝑥𝑅 𝑥 = -((𝐴‘(𝑁 − 1)) / (𝐴𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wss 3607  {csn 4210   class class class wbr 4685   × cxp 5141  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  cen 7994  Fincfn 7997  cc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  cmin 10304  -cneg 10305   / cdiv 10722  cn 11058  0cn0 11330  cz 11415  ...cfz 12364  cexp 12900  #chash 13157  Σcsu 14460  0𝑝c0p 23481  Polycply 23985  Xpcidp 23986  coeffccoe 23987  degcdgr 23988   quot cquot 24090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-0p 23482  df-ply 23989  df-idp 23990  df-coe 23991  df-dgr 23992  df-quot 24091
This theorem is referenced by:  basellem5  24856
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