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Theorem plyco 24042
 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 23999 . . . . 5 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
31, 2syl 17 . . . 4 (𝜑𝐺:ℂ⟶ℂ)
43ffvelrnda 6399 . . 3 ((𝜑𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
53feqmptd 6288 . . 3 (𝜑𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
6 plyco.1 . . . 4 (𝜑𝐹 ∈ (Poly‘𝑆))
7 eqid 2651 . . . . 5 (coeff‘𝐹) = (coeff‘𝐹)
8 eqid 2651 . . . . 5 (deg‘𝐹) = (deg‘𝐹)
97, 8coeid 24039 . . . 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 6697 . . . . 5 (𝑥 = (𝐺𝑧) → (𝑥𝑘) = ((𝐺𝑧)↑𝑘))
1211oveq2d 6706 . . . 4 (𝑥 = (𝐺𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1312sumeq2sdv 14479 . . 3 (𝑥 = (𝐺𝑧) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
144, 5, 10, 13fmptco 6436 . 2 (𝜑 → (𝐹𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
15 dgrcl 24034 . . . 4 (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0)
166, 15syl 17 . . 3 (𝜑 → (deg‘𝐹) ∈ ℕ0)
17 oveq2 6698 . . . . . . . 8 (𝑥 = 0 → (0...𝑥) = (0...0))
1817sumeq1d 14475 . . . . . . 7 (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
1918mpteq2dv 4778 . . . . . 6 (𝑥 = 0 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2019eleq1d 2715 . . . . 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 329 . . . 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 6698 . . . . . . . 8 (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
2322sumeq1d 14475 . . . . . . 7 (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2423mpteq2dv 4778 . . . . . 6 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))))
2524eleq1d 2715 . . . . 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 329 . . . 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 6698 . . . . . . . 8 (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
2827sumeq1d 14475 . . . . . . 7 (𝑥 = (𝑑 + 1) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
2928mpteq2dv 4778 . . . . . 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 2715 . . . . 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 329 . . . 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 6698 . . . . . . . 8 (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
3332sumeq1d 14475 . . . . . . 7 (𝑥 = (deg‘𝐹) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)))
3433mpteq2dv 4778 . . . . . 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 2715 . . . . 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 329 . . . 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 11426 . . . . . . . . 9 0 ∈ ℤ
384exp0d 13042 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑0) = 1)
3938oveq2d 6706 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40 plybss 23995 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
416, 40syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑆 ⊆ ℂ)
42 0cnd 10071 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ∈ ℂ)
4342snssd 4372 . . . . . . . . . . . . . . 15 (𝜑 → {0} ⊆ ℂ)
4441, 43unssd 3822 . . . . . . . . . . . . . 14 (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
457coef 24031 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
466, 45syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}))
47 0nn0 11345 . . . . . . . . . . . . . . 15 0 ∈ ℕ0
48 ffvelrn 6397 . . . . . . . . . . . . . . 15 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ0) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
4946, 47, 48sylancl 695 . . . . . . . . . . . . . 14 (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
5044, 49sseldd 3637 . . . . . . . . . . . . 13 (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
5150adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
5251mulid1d 10095 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
5339, 52eqtrd 2685 . . . . . . . . . 10 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) = ((coeff‘𝐹)‘0))
5453, 51eqeltrd 2730 . . . . . . . . 9 ((𝜑𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)) ∈ ℂ)
55 fveq2 6229 . . . . . . . . . . 11 (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56 oveq2 6698 . . . . . . . . . . 11 (𝑘 = 0 → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑0))
5755, 56oveq12d 6708 . . . . . . . . . 10 (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
5857fsum1 14520 . . . . . . . . 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 696 . . . . . . . 8 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺𝑧)↑0)))
6059, 53eqtrd 2685 . . . . . . 7 ((𝜑𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
6160mpteq2dva 4777 . . . . . 6 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62 fconstmpt 5197 . . . . . 6 (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
6361, 62syl6eqr 2703 . . . . 5 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64 plyconst 24007 . . . . . . 7 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
6544, 49, 64syl2anc 694 . . . . . 6 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66 plyun0 23998 . . . . . 6 (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
6765, 66syl6eleq 2740 . . . . 5 (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
6863, 67eqeltrd 2730 . . . 4 (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69 simprr 811 . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → (𝑆 ∪ {0}) ⊆ ℂ)
71 peano2nn0 11371 . . . . . . . . . . . . . 14 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
72 ffvelrn 6397 . . . . . . . . . . . . . 14 (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
7346, 71, 72syl2an 493 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ ℕ0) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74 plyconst 24007 . . . . . . . . . . . . 13 (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7570, 73, 74syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
7675, 66syl6eleq 2740 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77 nn0p1nn 11370 . . . . . . . . . . . . 13 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
78 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑥 = 1 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑1))
7978mpteq2dv 4778 . . . . . . . . . . . . . . . 16 (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)))
8079eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆)))
8180imbi2d 329 . . . . . . . . . . . . . 14 (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))))
82 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑑 → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑𝑑))
8382mpteq2dv 4778 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
8483eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
8584imbi2d 329 . . . . . . . . . . . . . 14 (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑑 + 1) → ((𝐺𝑧)↑𝑥) = ((𝐺𝑧)↑(𝑑 + 1)))
8786mpteq2dv 4778 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
8887eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
8988imbi2d 329 . . . . . . . . . . . . . 14 (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
904exp1d 13043 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧 ∈ ℂ) → ((𝐺𝑧)↑1) = (𝐺𝑧))
9190mpteq2dva 4777 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
9291, 5eqtr4d 2688 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) = 𝐺)
9392, 1eqeltrd 2730 . . . . . . . . . . . . . 14 (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑1)) ∈ (Poly‘𝑆))
94 simprr 811 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))
951adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → 𝐺 ∈ (Poly‘𝑆))
96 plyco.3 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
9796adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
98 plyco.4 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
9998adantlr 751 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
10094, 95, 97, 99plymul 24019 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆))
101100expr 642 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆)))
102 cnex 10055 . . . . . . . . . . . . . . . . . . . . 21 ℂ ∈ V
103102a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → ℂ ∈ V)
104 ovexd 6720 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑𝑑) ∈ V)
1054adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
106 eqidd 2652 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)))
1075adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺𝑧)))
108103, 104, 105, 106, 107offval2 6956 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
109 nnnn0 11337 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ ℕ → 𝑑 ∈ ℕ0)
110109ad2antlr 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
111105, 110expp1d 13049 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) = (((𝐺𝑧)↑𝑑) · (𝐺𝑧)))
112111mpteq2dva 4777 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺𝑧)↑𝑑) · (𝐺𝑧))))
113108, 112eqtr4d 2688 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
114113eleq1d 2715 . . . . . . . . . . . . . . . . 17 ((𝜑𝑑 ∈ ℕ) → (((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
115101, 114sylibd 229 . . . . . . . . . . . . . . . 16 ((𝜑𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
116115expcom 450 . . . . . . . . . . . . . . 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 11076 . . . . . . . . . . . . 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 445 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
12196adantlr 751 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12298adantlr 751 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
12376, 120, 121, 122plymul 24019 . . . . . . . . . 10 ((𝜑𝑑 ∈ ℕ0) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124123adantrr 753 . . . . . . . . 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 751 . . . . . . . . 9 (((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘))) ∈ (Poly‘𝑆))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
12669, 124, 125plyadd 24018 . . . . . . . 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 642 . . . . . . 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 14462 . . . . . . . . . . 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 6720 . . . . . . . . . 10 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))) ∈ V)
132 eqidd 2652 . . . . . . . . . 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 6241 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134 ovexd 6720 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐺𝑧)↑(𝑑 + 1)) ∈ V)
135 fconstmpt 5197 . . . . . . . . . . . 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 2652 . . . . . . . . . . 11 ((𝜑𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺𝑧)↑(𝑑 + 1))))
138128, 133, 134, 136, 137offval2 6956 . . . . . . . . . 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 6956 . . . . . . . . 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 807 . . . . . . . . . . . 12 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
141 nn0uz 11760 . . . . . . . . . . . 12 0 = (ℤ‘0)
142140, 141syl6eleq 2740 . . . . . . . . . . 11 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ‘0))
1437coef3 24033 . . . . . . . . . . . . . . 15 (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶ℂ)
1446, 143syl 17 . . . . . . . . . . . . . 14 (𝜑 → (coeff‘𝐹):ℕ0⟶ℂ)
145144ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ0⟶ℂ)
146 elfznn0 12471 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ0)
147 ffvelrn 6397 . . . . . . . . . . . . 13 (((coeff‘𝐹):ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148145, 146, 147syl2an 493 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
1494adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐺𝑧) ∈ ℂ)
150 expcl 12918 . . . . . . . . . . . . 13 (((𝐺𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
151149, 146, 150syl2an 493 . . . . . . . . . . . 12 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺𝑧)↑𝑘) ∈ ℂ)
152148, 151mulcld 10098 . . . . . . . . . . 11 ((((𝜑𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) ∈ ℂ)
153 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154 oveq2 6698 . . . . . . . . . . . 12 (𝑘 = (𝑑 + 1) → ((𝐺𝑧)↑𝑘) = ((𝐺𝑧)↑(𝑑 + 1)))
155153, 154oveq12d 6708 . . . . . . . . . . 11 (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺𝑧)↑(𝑑 + 1))))
156142, 152, 155fsump1 14531 . . . . . . . . . 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 4777 . . . . . . . . 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 2688 . . . . . . . 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 2715 . . . . . . 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 229 . . . . . 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 450 . . . . 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 11510 . . 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 2730 1 (𝜑 → (𝐹𝐺) ∈ (Poly‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  {csn 4210   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937  ℂcc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ℤ≥cuz 11725  ...cfz 12364  ↑cexp 12900  Σcsu 14460  Polycply 23985  coeffccoe 23987  degcdgr 23988 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-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-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-coe 23991  df-dgr 23992 This theorem is referenced by:  dgrcolem1  24074  dgrcolem2  24075  taylply2  24167  ftalem7  24850
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