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Theorem plyco 23715
Description: The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
plyco.1 (𝜑𝐹 ∈ (Poly‘𝑆))
plyco.2 (𝜑𝐺 ∈ (Poly‘𝑆))
plyco.3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
plyco.4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
Assertion
Ref Expression
plyco (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦

Proof of Theorem plyco
Dummy variables 𝑘 𝑑 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyco.2 . . . . 5 (𝜑𝐺 ∈ (Poly‘𝑆))
2 plyf 23672 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
31, 2syl 17 . . . 4 (𝜑𝐺:ℂ⟶ℂ)
43ffvelrnda 6249 . . 3 ((𝜑𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
53feqmptd 6141 . . 3 (𝜑𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
6 plyco.1 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
7 eqid 2606 . . . . 5 (coeff‘𝐹) = (coeff‘𝐹)
8 eqid 2606 . . . . 5 (deg‘𝐹) = (deg‘𝐹)
97, 8coeid 23712 . . . 4 (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥𝑘))))
106, 9syl 17 . . 3 (𝜑𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥𝑘))))
11 oveq1 6531 . . . . 5 (𝑥 = (𝐺𝑧) → (𝑥𝑘) = ((𝐺𝑧)↑𝑘))
1211oveq2d 6540 . . . 4 (𝑥 = (𝐺𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1312sumeq2sdv 14225 . . 3 (𝑥 = (𝐺𝑧) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
144, 5, 10, 13fmptco 6285 . 2 (𝜑 → (𝐹𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
15 dgrcl 23707 . . . 4 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
166, 15syl 17 . . 3 (𝜑 → (deg‘𝐹) ∈ ℕ0)
17 oveq2 6532 . . . . . . . 8 (𝑥 = 0 → (0...𝑥) = (0...0))
1817sumeq1d 14222 . . . . . . 7 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1918mpteq2dv 4664 . . . . . 6 (𝑥 = 0 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2019eleq1d 2668 . . . . 5 (𝑥 = 0 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
2120imbi2d 328 . . . 4 (𝑥 = 0 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
22 oveq2 6532 . . . . . . . 8 (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
2322sumeq1d 14222 . . . . . . 7 (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2423mpteq2dv 4664 . . . . . 6 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2524eleq1d 2668 . . . . 5 (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
2625imbi2d 328 . . . 4 (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
27 oveq2 6532 . . . . . . . 8 (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
2827sumeq1d 14222 . . . . . . 7 (𝑥 = (𝑑 + 1) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2928mpteq2dv 4664 . . . . . 6 (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
3029eleq1d 2668 . . . . 5 (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
3130imbi2d 328 . . . 4 (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
32 oveq2 6532 . . . . . . . 8 (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
3332sumeq1d 14222 . . . . . . 7 (𝑥 = (deg‘𝐹) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
3433mpteq2dv 4664 . . . . . 6 (𝑥 = (deg‘𝐹) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
3534eleq1d 2668 . . . . 5 (𝑥 = (deg‘𝐹) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
3635imbi2d 328 . . . 4 (𝑥 = (deg‘𝐹) → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
37 0z 11218 . . . . . . . . 9 0 ∈ ℤ
384exp0d 12816 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑0) = 1)
3938oveq2d 6540 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40 plybss 23668 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
416, 40syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 ⊆ ℂ)
42 0cnd 9886 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ∈ ℂ)
4342snssd 4277 . . . . . . . . . . . . . . 15 (𝜑 → {0} ⊆ ℂ)
4441, 43unssd 3747 . . . . . . . . . . . . . 14 (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
457coef 23704 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
466, 45syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
47 0nn0 11151 . . . . . . . . . . . . . . 15 0 ∈ ℕ0
48 ffvelrn 6247 . . . . . . . . . . . . . . 15 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ0) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
4946, 47, 48sylancl 692 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
5044, 49sseldd 3565 . . . . . . . . . . . . 13 (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
5150adantr 479 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
5251mulid1d 9910 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
5339, 52eqtrd 2640 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = ((coeff‘𝐹)‘0))
5453, 51eqeltrd 2684 . . . . . . . . 9 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) ∈ ℂ)
55 fveq2 6085 . . . . . . . . . . 11 (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56 oveq2 6532 . . . . . . . . . . 11 (𝑘 = 0 → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑0))
5755, 56oveq12d 6542 . . . . . . . . . 10 (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
5857fsum1 14263 . . . . . . . . 9 ((0 ∈ ℤ ∧ (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
5937, 54, 58sylancr 693 . . . . . . . 8 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
6059, 53eqtrd 2640 . . . . . . 7 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
6160mpteq2dva 4663 . . . . . 6 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62 fconstmpt 5072 . . . . . 6 (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
6361, 62syl6eqr 2658 . . . . 5 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64 plyconst 23680 . . . . . . 7 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
6544, 49, 64syl2anc 690 . . . . . 6 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66 plyun0 23671 . . . . . 6 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
6765, 66syl6eleq 2694 . . . . 5 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
6863, 67eqeltrd 2684 . . . 4 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69 simprr 791 . . . . . . . . 9 ((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
7044adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → (𝑆 ∪ {0}) ⊆ ℂ)
71 peano2nn0 11177 . . . . . . . . . . . . . 14 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
72 ffvelrn 6247 . . . . . . . . . . . . . 14 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
7346, 71, 72syl2an 492 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74 plyconst 23680 . . . . . . . . . . . . 13 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7570, 73, 74syl2anc 690 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7675, 66syl6eleq 2694 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77 nn0p1nn 11176 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
78 oveq2 6532 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑1))
7978mpteq2dv 4664 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)))
8079eleq1d 2668 . . . . . . . . . . . . . . 15 (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆)))
8180imbi2d 328 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))))
82 oveq2 6532 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑑 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑𝑑))
8382mpteq2dv 4664 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
8483eleq1d 2668 . . . . . . . . . . . . . . 15 (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
8584imbi2d 328 . . . . . . . . . . . . . 14 (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86 oveq2 6532 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑑 + 1) → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑(𝑑 + 1)))
8786mpteq2dv 4664 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
8887eleq1d 2668 . . . . . . . . . . . . . . 15 (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
8988imbi2d 328 . . . . . . . . . . . . . 14 (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
904exp1d 12817 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑1) = (𝐺𝑧))
9190mpteq2dva 4663 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
9291, 5eqtr4d 2643 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = 𝐺)
9392, 1eqeltrd 2684 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))
94 simprr 791 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))
951adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → 𝐺 ∈ (Poly‘𝑆))
96 plyco.3 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9796adantlr 746 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
98 plyco.4 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
9998adantlr 746 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
10094, 95, 97, 99plymul 23692 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆))
101100expr 640 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆)))
102 cnex 9870 . . . . . . . . . . . . . . . . . . . . 21 ℂ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → ℂ ∈ V)
104 ovex 6552 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺𝑧)↑𝑑) ∈ V
105104a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑𝑑) ∈ V)
1064adantlr 746 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
107 eqidd 2607 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
1085adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
109103, 105, 106, 107, 108offval2 6786 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
110 nnnn0 11143 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ℕ → 𝑑 ∈ ℕ0)
111110ad2antlr 758 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
112106, 111expp1d 12823 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) = (((𝐺𝑧)↑𝑑) · (𝐺𝑧)))
113112mpteq2dva 4663 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
114109, 113eqtr4d 2643 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
115114eleq1d 2668 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → (((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
116101, 115sylibd 227 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
117116expcom 449 . . . . . . . . . . . . . . 15 (𝑑 ∈ ℕ → (𝜑 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
118117a2d 29 . . . . . . . . . . . . . 14 (𝑑 ∈ ℕ → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
11981, 85, 89, 89, 93, 118nnind 10882 . . . . . . . . . . . . 13 ((𝑑 + 1) ∈ ℕ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
12077, 119syl 17 . . . . . . . . . . . 12 (𝑑 ∈ ℕ0 → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
121120impcom 444 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
12296adantlr 746 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12398adantlr 746 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
12476, 121, 122, 123plymul 23692 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
125124adantrr 748 . . . . . . . . 9 ((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
12696adantlr 746 . . . . . . . . 9 (((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12769, 125, 126plyadd 23691 . . . . . . . 8 ((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆))
128127expr 640 . . . . . . 7 ((𝜑𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆)))
129102a1i 11 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ℂ ∈ V)
130 sumex 14209 . . . . . . . . . . 11 Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ V
131130a1i 11 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ V)
132 ovex 6552 . . . . . . . . . . 11 (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))) ∈ V
133132a1i 11 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))) ∈ V)
134 eqidd 2607 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
135 fvex 6095 . . . . . . . . . . . 12 ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V
136135a1i 11 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
137 ovex 6552 . . . . . . . . . . . 12 ((𝐺𝑧)↑(𝑑 + 1)) ∈ V
138137a1i 11 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) ∈ V)
139 fconstmpt 5072 . . . . . . . . . . . 12 (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘(𝑑 + 1)))
140139a1i 11 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘(𝑑 + 1))))
141 eqidd 2607 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
142129, 136, 138, 140, 141offval2 6786 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) = (𝑧 ∈ ℂ ↦ (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1)))))
143129, 131, 133, 134, 142offval2 6786 . . . . . . . . 9 ((𝜑𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) + (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))))))
144 simplr 787 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
145 nn0uz 11551 . . . . . . . . . . . 12 0 = (ℤ‘0)
146144, 145syl6eleq 2694 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ‘0))
1477coef3 23706 . . . . . . . . . . . . . . 15 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
1486, 147syl 17 . . . . . . . . . . . . . 14 (𝜑 → (coeff‘𝐹):ℕ0⟶ℂ)
149148ad2antrr 757 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ0⟶ℂ)
150 elfznn0 12254 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ0)
151 ffvelrn 6247 . . . . . . . . . . . . 13 (((coeff‘𝐹):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
152149, 150, 151syl2an 492 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
1534adantlr 746 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
154 expcl 12692 . . . . . . . . . . . . 13 (((𝐺𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
155153, 150, 154syl2an 492 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
156152, 155mulcld 9913 . . . . . . . . . . 11 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ ℂ)
157 fveq2 6085 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
158 oveq2 6532 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑(𝑑 + 1)))
159157, 158oveq12d 6542 . . . . . . . . . . 11 (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))))
160146, 156, 159fsump1 14272 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) + (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1)))))
161160mpteq2dva 4663 . . . . . . . . 9 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) + (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))))))
162143, 161eqtr4d 2643 . . . . . . . 8 ((𝜑𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
163162eleq1d 2668 . . . . . . 7 ((𝜑𝑑 ∈ ℕ0) → (((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
164128, 163sylibd 227 . . . . . 6 ((𝜑𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
165164expcom 449 . . . . 5 (𝑑 ∈ ℕ0 → (𝜑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
166165a2d 29 . . . 4 (𝑑 ∈ ℕ0 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
16721, 26, 31, 36, 68, 166nn0ind 11301 . . 3 ((deg‘𝐹) ∈ ℕ0 → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
16816, 167mpcom 37 . 2 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
16914, 168eqeltrd 2684 1 (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3169  cun 3534  wss 3536  {csn 4121  cmpt 4634   × cxp 5023  ccom 5029  wf 5783  cfv 5787  (class class class)co 6524  𝑓 cof 6767  cc 9787  0cc0 9789  1c1 9790   + caddc 9792   · cmul 9794  cn 10864  0cn0 11136  cz 11207  cuz 11516  ...cfz 12149  cexp 12674  Σcsu 14207  Polycply 23658  coeffccoe 23660  degcdgr 23661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-inf2 8395  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867  ax-addf 9868
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-of 6769  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-oi 8272  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-fz 12150  df-fzo 12287  df-fl 12407  df-seq 12616  df-exp 12675  df-hash 12932  df-cj 13630  df-re 13631  df-im 13632  df-sqrt 13766  df-abs 13767  df-clim 14010  df-rlim 14011  df-sum 14208  df-0p 23157  df-ply 23662  df-coe 23664  df-dgr 23665
This theorem is referenced by:  dgrcolem1  23747  dgrcolem2  23748  taylply2  23840  ftalem7  24519
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