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Theorem plyco 24288
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 24245 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
31, 2syl 17 . . . 4 (𝜑𝐺:ℂ⟶ℂ)
43ffvelrnda 6549 . . 3 ((𝜑𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
53feqmptd 6438 . . 3 (𝜑𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
6 plyco.1 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
7 eqid 2765 . . . . 5 (coeff‘𝐹) = (coeff‘𝐹)
8 eqid 2765 . . . . 5 (deg‘𝐹) = (deg‘𝐹)
97, 8coeid 24285 . . . 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 6849 . . . . 5 (𝑥 = (𝐺𝑧) → (𝑥𝑘) = ((𝐺𝑧)↑𝑘))
1211oveq2d 6858 . . . 4 (𝑥 = (𝐺𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1312sumeq2sdv 14722 . . 3 (𝑥 = (𝐺𝑧) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
144, 5, 10, 13fmptco 6587 . 2 (𝜑 → (𝐹𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
15 dgrcl 24280 . . . 4 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
166, 15syl 17 . . 3 (𝜑 → (deg‘𝐹) ∈ ℕ0)
17 oveq2 6850 . . . . . . . 8 (𝑥 = 0 → (0...𝑥) = (0...0))
1817sumeq1d 14718 . . . . . . 7 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1918mpteq2dv 4904 . . . . . 6 (𝑥 = 0 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2019eleq1d 2829 . . . . 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 331 . . . 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 6850 . . . . . . . 8 (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
2322sumeq1d 14718 . . . . . . 7 (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2423mpteq2dv 4904 . . . . . 6 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2524eleq1d 2829 . . . . 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 331 . . . 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 6850 . . . . . . . 8 (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
2827sumeq1d 14718 . . . . . . 7 (𝑥 = (𝑑 + 1) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2928mpteq2dv 4904 . . . . . 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 2829 . . . . 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 331 . . . 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 6850 . . . . . . . 8 (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
3332sumeq1d 14718 . . . . . . 7 (𝑥 = (deg‘𝐹) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
3433mpteq2dv 4904 . . . . . 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 2829 . . . . 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 331 . . . 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 11635 . . . . . . . . 9 0 ∈ ℤ
384exp0d 13209 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑0) = 1)
3938oveq2d 6858 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40 plybss 24241 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
416, 40syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 ⊆ ℂ)
42 0cnd 10286 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ∈ ℂ)
4342snssd 4494 . . . . . . . . . . . . . . 15 (𝜑 → {0} ⊆ ℂ)
4441, 43unssd 3951 . . . . . . . . . . . . . 14 (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
457coef 24277 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
466, 45syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
47 0nn0 11555 . . . . . . . . . . . . . . 15 0 ∈ ℕ0
48 ffvelrn 6547 . . . . . . . . . . . . . . 15 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ0) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
4946, 47, 48sylancl 580 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
5044, 49sseldd 3762 . . . . . . . . . . . . 13 (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
5150adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
5251mulid1d 10311 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
5339, 52eqtrd 2799 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = ((coeff‘𝐹)‘0))
5453, 51eqeltrd 2844 . . . . . . . . 9 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) ∈ ℂ)
55 fveq2 6375 . . . . . . . . . . 11 (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56 oveq2 6850 . . . . . . . . . . 11 (𝑘 = 0 → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑0))
5755, 56oveq12d 6860 . . . . . . . . . 10 (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
5857fsum1 14763 . . . . . . . . 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 581 . . . . . . . 8 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
6059, 53eqtrd 2799 . . . . . . 7 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
6160mpteq2dva 4903 . . . . . 6 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62 fconstmpt 5333 . . . . . 6 (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
6361, 62syl6eqr 2817 . . . . 5 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64 plyconst 24253 . . . . . . 7 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
6544, 49, 64syl2anc 579 . . . . . 6 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66 plyun0 24244 . . . . . 6 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
6765, 66syl6eleq 2854 . . . . 5 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
6863, 67eqeltrd 2844 . . . 4 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69 simprr 789 . . . . . . . . 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 472 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → (𝑆 ∪ {0}) ⊆ ℂ)
71 peano2nn0 11580 . . . . . . . . . . . . . 14 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
72 ffvelrn 6547 . . . . . . . . . . . . . 14 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
7346, 71, 72syl2an 589 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74 plyconst 24253 . . . . . . . . . . . . 13 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7570, 73, 74syl2anc 579 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7675, 66syl6eleq 2854 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77 nn0p1nn 11579 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
78 oveq2 6850 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑1))
7978mpteq2dv 4904 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)))
8079eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆)))
8180imbi2d 331 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))))
82 oveq2 6850 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑑 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑𝑑))
8382mpteq2dv 4904 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
8483eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
8584imbi2d 331 . . . . . . . . . . . . . 14 (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86 oveq2 6850 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑑 + 1) → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑(𝑑 + 1)))
8786mpteq2dv 4904 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
8887eleq1d 2829 . . . . . . . . . . . . . . 15 (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
8988imbi2d 331 . . . . . . . . . . . . . 14 (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
904exp1d 13210 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑1) = (𝐺𝑧))
9190mpteq2dva 4903 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
9291, 5eqtr4d 2802 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = 𝐺)
9392, 1eqeltrd 2844 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))
94 simprr 789 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))
951adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → 𝐺 ∈ (Poly‘𝑆))
96 plyco.3 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9796adantlr 706 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
98 plyco.4 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
9998adantlr 706 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
10094, 95, 97, 99plymul 24265 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆))
101100expr 448 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆)))
102 cnex 10270 . . . . . . . . . . . . . . . . . . . . 21 ℂ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → ℂ ∈ V)
104 ovexd 6876 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑𝑑) ∈ V)
1054adantlr 706 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
106 eqidd 2766 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
1075adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
108103, 104, 105, 106, 107offval2 7112 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
109 nnnn0 11546 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ℕ → 𝑑 ∈ ℕ0)
110109ad2antlr 718 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
111105, 110expp1d 13216 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) = (((𝐺𝑧)↑𝑑) · (𝐺𝑧)))
112111mpteq2dva 4903 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
113108, 112eqtr4d 2802 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
114113eleq1d 2829 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → (((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
115101, 114sylibd 230 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
116115expcom 402 . . . . . . . . . . . . . . 15 (𝑑 ∈ ℕ → (𝜑 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
117116a2d 29 . . . . . . . . . . . . . 14 (𝑑 ∈ ℕ → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
11881, 85, 89, 89, 93, 117nnind 11294 . . . . . . . . . . . . 13 ((𝑑 + 1) ∈ ℕ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
11977, 118syl 17 . . . . . . . . . . . 12 (𝑑 ∈ ℕ0 → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
120119impcom 396 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
12196adantlr 706 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12298adantlr 706 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
12376, 120, 121, 122plymul 24265 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124123adantrr 708 . . . . . . . . 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))
12596adantlr 706 . . . . . . . . 9 (((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12669, 124, 125plyadd 24264 . . . . . . . 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))
127126expr 448 . . . . . . 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)))
128102a1i 11 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ℂ ∈ V)
129 sumex 14705 . . . . . . . . . . 11 Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ V
130129a1i 11 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ V)
131 ovexd 6876 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))) ∈ V)
132 eqidd 2766 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
133 fvexd 6390 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134 ovexd 6876 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) ∈ V)
135 fconstmpt 5333 . . . . . . . . . . . 12 (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘(𝑑 + 1)))
136135a1i 11 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘(𝑑 + 1))))
137 eqidd 2766 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
138128, 133, 134, 136, 137offval2 7112 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) = (𝑧 ∈ ℂ ↦ (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1)))))
139128, 130, 131, 132, 138offval2 7112 . . . . . . . . 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))))))
140 simplr 785 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
141 nn0uz 11922 . . . . . . . . . . . 12 0 = (ℤ‘0)
142140, 141syl6eleq 2854 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ‘0))
1437coef3 24279 . . . . . . . . . . . . . . 15 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
1446, 143syl 17 . . . . . . . . . . . . . 14 (𝜑 → (coeff‘𝐹):ℕ0⟶ℂ)
145144ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ0⟶ℂ)
146 elfznn0 12640 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ0)
147 ffvelrn 6547 . . . . . . . . . . . . 13 (((coeff‘𝐹):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148145, 146, 147syl2an 589 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
1494adantlr 706 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
150 expcl 13085 . . . . . . . . . . . . 13 (((𝐺𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
151149, 146, 150syl2an 589 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
152148, 151mulcld 10314 . . . . . . . . . . 11 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ ℂ)
153 fveq2 6375 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154 oveq2 6850 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑(𝑑 + 1)))
155153, 154oveq12d 6860 . . . . . . . . . . 11 (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))))
156142, 152, 155fsump1 14774 . . . . . . . . . 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)))))
157156mpteq2dva 4903 . . . . . . . . 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))))))
158139, 157eqtr4d 2802 . . . . . . . 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))))
159158eleq1d 2829 . . . . . . 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)))
160127, 159sylibd 230 . . . . . 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)))
161160expcom 402 . . . . 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))))
162161a2d 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))))
16321, 26, 31, 36, 68, 162nn0ind 11719 . . 3 ((deg‘𝐹) ∈ ℕ0 → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
16416, 163mpcom 38 . 2 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
16514, 164eqeltrd 2844 1 (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  Vcvv 3350  cun 3730  wss 3732  {csn 4334  cmpt 4888   × cxp 5275  ccom 5281  wf 6064  cfv 6068  (class class class)co 6842  𝑓 cof 7093  cc 10187  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194  cn 11274  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  cexp 13067  Σcsu 14703  Polycply 24231  coeffccoe 24233  degcdgr 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-rlim 14507  df-sum 14704  df-0p 23728  df-ply 24235  df-coe 24237  df-dgr 24238
This theorem is referenced by:  dgrcolem1  24320  dgrcolem2  24321  taylply2  24413  ftalem7  25096
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