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Theorem plyco 24749
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 24706 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
31, 2syl 17 . . . 4 (𝜑𝐺:ℂ⟶ℂ)
43ffvelrnda 6847 . . 3 ((𝜑𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
53feqmptd 6730 . . 3 (𝜑𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
6 plyco.1 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
7 eqid 2826 . . . . 5 (coeff‘𝐹) = (coeff‘𝐹)
8 eqid 2826 . . . . 5 (deg‘𝐹) = (deg‘𝐹)
97, 8coeid 24746 . . . 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 7155 . . . . 5 (𝑥 = (𝐺𝑧) → (𝑥𝑘) = ((𝐺𝑧)↑𝑘))
1211oveq2d 7164 . . . 4 (𝑥 = (𝐺𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1312sumeq2sdv 15051 . . 3 (𝑥 = (𝐺𝑧) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
144, 5, 10, 13fmptco 6887 . 2 (𝜑 → (𝐹𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
15 dgrcl 24741 . . . 4 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
166, 15syl 17 . . 3 (𝜑 → (deg‘𝐹) ∈ ℕ0)
17 oveq2 7156 . . . . . . . 8 (𝑥 = 0 → (0...𝑥) = (0...0))
1817sumeq1d 15048 . . . . . . 7 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1918mpteq2dv 5159 . . . . . 6 (𝑥 = 0 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2019eleq1d 2902 . . . . 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 342 . . . 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 7156 . . . . . . . 8 (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
2322sumeq1d 15048 . . . . . . 7 (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2423mpteq2dv 5159 . . . . . 6 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2524eleq1d 2902 . . . . 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 342 . . . 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 7156 . . . . . . . 8 (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
2827sumeq1d 15048 . . . . . . 7 (𝑥 = (𝑑 + 1) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2928mpteq2dv 5159 . . . . . 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 2902 . . . . 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 342 . . . 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 7156 . . . . . . . 8 (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
3332sumeq1d 15048 . . . . . . 7 (𝑥 = (deg‘𝐹) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
3433mpteq2dv 5159 . . . . . 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 2902 . . . . 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 342 . . . 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 11981 . . . . . . . . 9 0 ∈ ℤ
384exp0d 13494 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑0) = 1)
3938oveq2d 7164 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40 plybss 24702 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
416, 40syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 ⊆ ℂ)
42 0cnd 10623 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ∈ ℂ)
4342snssd 4741 . . . . . . . . . . . . . . 15 (𝜑 → {0} ⊆ ℂ)
4441, 43unssd 4166 . . . . . . . . . . . . . 14 (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
457coef 24738 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
466, 45syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
47 0nn0 11901 . . . . . . . . . . . . . . 15 0 ∈ ℕ0
48 ffvelrn 6845 . . . . . . . . . . . . . . 15 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ0) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
4946, 47, 48sylancl 586 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
5044, 49sseldd 3972 . . . . . . . . . . . . 13 (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
5150adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
5251mulid1d 10647 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
5339, 52eqtrd 2861 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = ((coeff‘𝐹)‘0))
5453, 51eqeltrd 2918 . . . . . . . . 9 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) ∈ ℂ)
55 fveq2 6667 . . . . . . . . . . 11 (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56 oveq2 7156 . . . . . . . . . . 11 (𝑘 = 0 → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑0))
5755, 56oveq12d 7166 . . . . . . . . . 10 (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
5857fsum1 15092 . . . . . . . . 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 587 . . . . . . . 8 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
6059, 53eqtrd 2861 . . . . . . 7 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
6160mpteq2dva 5158 . . . . . 6 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62 fconstmpt 5613 . . . . . 6 (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
6361, 62syl6eqr 2879 . . . . 5 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64 plyconst 24714 . . . . . . 7 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
6544, 49, 64syl2anc 584 . . . . . 6 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66 plyun0 24705 . . . . . 6 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
6765, 66syl6eleq 2928 . . . . 5 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
6863, 67eqeltrd 2918 . . . 4 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69 simprr 769 . . . . . . . . 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 481 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → (𝑆 ∪ {0}) ⊆ ℂ)
71 peano2nn0 11926 . . . . . . . . . . . . . 14 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
72 ffvelrn 6845 . . . . . . . . . . . . . 14 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
7346, 71, 72syl2an 595 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74 plyconst 24714 . . . . . . . . . . . . 13 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7570, 73, 74syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7675, 66syl6eleq 2928 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77 nn0p1nn 11925 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
78 oveq2 7156 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑1))
7978mpteq2dv 5159 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)))
8079eleq1d 2902 . . . . . . . . . . . . . . 15 (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆)))
8180imbi2d 342 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))))
82 oveq2 7156 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑑 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑𝑑))
8382mpteq2dv 5159 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
8483eleq1d 2902 . . . . . . . . . . . . . . 15 (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
8584imbi2d 342 . . . . . . . . . . . . . 14 (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86 oveq2 7156 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑑 + 1) → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑(𝑑 + 1)))
8786mpteq2dv 5159 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
8887eleq1d 2902 . . . . . . . . . . . . . . 15 (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
8988imbi2d 342 . . . . . . . . . . . . . 14 (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
904exp1d 13495 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑1) = (𝐺𝑧))
9190mpteq2dva 5158 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
9291, 5eqtr4d 2864 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = 𝐺)
9392, 1eqeltrd 2918 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))
94 simprr 769 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))
951adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → 𝐺 ∈ (Poly‘𝑆))
96 plyco.3 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9796adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
98 plyco.4 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
9998adantlr 711 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
10094, 95, 97, 99plymul 24726 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘f · 𝐺) ∈ (Poly‘𝑆))
101100expr 457 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘f · 𝐺) ∈ (Poly‘𝑆)))
102 cnex 10607 . . . . . . . . . . . . . . . . . . . . 21 ℂ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → ℂ ∈ V)
104 ovexd 7183 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑𝑑) ∈ V)
1054adantlr 711 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
106 eqidd 2827 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
1075adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
108103, 104, 105, 106, 107offval2 7416 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
109 nnnn0 11893 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ℕ → 𝑑 ∈ ℕ0)
110109ad2antlr 723 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
111105, 110expp1d 13501 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) = (((𝐺𝑧)↑𝑑) · (𝐺𝑧)))
112111mpteq2dva 5158 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
113108, 112eqtr4d 2864 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘f · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
114113eleq1d 2902 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → (((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘f · 𝐺) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
115101, 114sylibd 240 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
116115expcom 414 . . . . . . . . . . . . . . 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 11645 . . . . . . . . . . . . 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 408 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
12196adantlr 711 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12298adantlr 711 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
12376, 120, 121, 122plymul 24726 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘f · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124123adantrr 713 . . . . . . . . 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 711 . . . . . . . . 9 (((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12669, 124, 125plyadd 24725 . . . . . . . 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 457 . . . . . . 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 15034 . . . . . . . . . . 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 7183 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))) ∈ V)
132 eqidd 2827 . . . . . . . . . 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 6682 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134 ovexd 7183 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) ∈ V)
135 fconstmpt 5613 . . . . . . . . . . . 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 2827 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
138128, 133, 134, 136, 137offval2 7416 . . . . . . . . . 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 7416 . . . . . . . . 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 765 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
141 nn0uz 12269 . . . . . . . . . . . 12 0 = (ℤ‘0)
142140, 141syl6eleq 2928 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ‘0))
1437coef3 24740 . . . . . . . . . . . . . . 15 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
1446, 143syl 17 . . . . . . . . . . . . . 14 (𝜑 → (coeff‘𝐹):ℕ0⟶ℂ)
145144ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ0⟶ℂ)
146 elfznn0 12990 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ0)
147 ffvelrn 6845 . . . . . . . . . . . . 13 (((coeff‘𝐹):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148145, 146, 147syl2an 595 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
1494adantlr 711 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
150 expcl 13437 . . . . . . . . . . . . 13 (((𝐺𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
151149, 146, 150syl2an 595 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
152148, 151mulcld 10650 . . . . . . . . . . 11 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ ℂ)
153 fveq2 6667 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154 oveq2 7156 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑(𝑑 + 1)))
155153, 154oveq12d 7166 . . . . . . . . . . 11 (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))))
156142, 152, 155fsump1 15101 . . . . . . . . . 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 5158 . . . . . . . . 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 2864 . . . . . . . 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 2902 . . . . . . 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 240 . . . . . 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 414 . . . . 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 12066 . . 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 2918 1 (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cun 3938  wss 3940  {csn 4564  cmpt 5143   × cxp 5552  ccom 5558  wf 6348  cfv 6352  (class class class)co 7148  f cof 7397  cc 10524  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  cn 11627  0cn0 11886  cz 11970  cuz 12232  ...cfz 12882  cexp 13419  Σcsu 15032  Polycply 24692  coeffccoe 24694  degcdgr 24695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-addf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-seq 13360  df-exp 13420  df-hash 13681  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-clim 14835  df-rlim 14836  df-sum 15033  df-0p 24189  df-ply 24696  df-coe 24698  df-dgr 24699
This theorem is referenced by:  dgrcolem1  24781  dgrcolem2  24782  taylply2  24874  ftalem7  25573
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