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Theorem plymulx0 34540
Description: Coefficients of a polynomial multiplied by Xp. (Contributed by Thierry Arnoux, 25-Sep-2018.)
Assertion
Ref Expression
plymulx0 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
Distinct variable group:   𝑛,𝐹

Proof of Theorem plymulx0
Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 4140 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2 ax-resscn 11209 . . . . . . 7 ℝ ⊆ ℂ
3 1re 11258 . . . . . . 7 1 ∈ ℝ
4 plyid 26262 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
52, 3, 4mp2an 692 . . . . . 6 Xp ∈ (Poly‘ℝ)
65a1i 11 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp ∈ (Poly‘ℝ))
7 simprl 771 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8 simprr 773 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
97, 8readdcld 11287 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
107, 8remulcld 11288 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 26271 . . . 4 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹f · Xp) ∈ (Poly‘ℝ))
12 0re 11260 . . . 4 0 ∈ ℝ
13 eqid 2734 . . . . 5 (coeff‘(𝐹f · Xp)) = (coeff‘(𝐹f · Xp))
1413coef2 26284 . . . 4 (((𝐹f · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹f · Xp)):ℕ0⟶ℝ)
1511, 12, 14sylancl 586 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)):ℕ0⟶ℝ)
1615feqmptd 6976 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹f · Xp))‘𝑛)))
17 cnex 11233 . . . . . . . . 9 ℂ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19 plyf 26251 . . . . . . . . 9 (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21 plyf 26251 . . . . . . . . . 10 (Xp ∈ (Poly‘ℝ) → Xp:ℂ⟶ℂ)
225, 21ax-mp 5 . . . . . . . . 9 Xp:ℂ⟶ℂ
2322a1i 11 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp:ℂ⟶ℂ)
24 simprl 771 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25 simprr 773 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
2624, 25mulcomd 11279 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2718, 20, 23, 26caofcom 7733 . . . . . . 7 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹f · Xp) = (Xpf · 𝐹))
2827fveq2d 6910 . . . . . 6 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (coeff‘(Xpf · 𝐹)))
2928fveq1d 6908 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹f · Xp))‘𝑛) = ((coeff‘(Xpf · 𝐹))‘𝑛))
3029adantr 480 . . . 4 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹f · Xp))‘𝑛) = ((coeff‘(Xpf · 𝐹))‘𝑛))
315a1i 11 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Xp ∈ (Poly‘ℝ))
321adantr 480 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℝ))
33 simpr 484 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
34 eqid 2734 . . . . . . 7 (coeff‘Xp) = (coeff‘Xp)
35 eqid 2734 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
3634, 35coemul 26305 . . . . . 6 ((Xp ∈ (Poly‘ℝ) ∧ 𝐹 ∈ (Poly‘ℝ) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xpf · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))))
3731, 32, 33, 36syl3anc 1370 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xpf · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))))
38 elfznn0 13656 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39 coeidp 26317 . . . . . . . . . 10 (𝑖 ∈ ℕ0 → ((coeff‘Xp)‘𝑖) = if(𝑖 = 1, 1, 0))
4038, 39syl 17 . . . . . . . . 9 (𝑖 ∈ (0...𝑛) → ((coeff‘Xp)‘𝑖) = if(𝑖 = 1, 1, 0))
4140oveq1d 7445 . . . . . . . 8 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))))
42 ovif 7530 . . . . . . . 8 (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖))))
4341, 42eqtrdi 2790 . . . . . . 7 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))))
4443adantl 481 . . . . . 6 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))))
4544sumeq2dv 15734 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))))
46 velsn 4646 . . . . . . . . . 10 (𝑖 ∈ {1} ↔ 𝑖 = 1)
4746bicomi 224 . . . . . . . . 9 (𝑖 = 1 ↔ 𝑖 ∈ {1})
4847a1i 11 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑖 = 1 ↔ 𝑖 ∈ {1}))
4935coef2 26284 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
501, 12, 49sylancl 586 . . . . . . . . . . . 12 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
5150ad2antrr 726 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52 fznn0sub 13592 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) → (𝑛𝑖) ∈ ℕ0)
5352adantl 481 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛𝑖) ∈ ℕ0)
5451, 53ffvelcdmd 7104 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
5554recnd 11286 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
5655mullidd 11276 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛𝑖))) = ((coeff‘𝐹)‘(𝑛𝑖)))
5755mul02d 11456 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛𝑖))) = 0)
5848, 56, 57ifbieq12d 4558 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))) = if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
5958sumeq2dv 15734 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
60 eqeq2 2746 . . . . . . 7 (0 = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) → (Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0 ↔ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
61 eqeq2 2746 . . . . . . 7 (((coeff‘𝐹)‘(𝑛 − 1)) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) → (Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)) ↔ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
62 oveq2 7438 . . . . . . . . . . 11 (𝑛 = 0 → (0...𝑛) = (0...0))
63 0z 12621 . . . . . . . . . . . 12 0 ∈ ℤ
64 fzsn 13602 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65eqtrdi 2790 . . . . . . . . . 10 (𝑛 = 0 → (0...𝑛) = {0})
67 elsni 4647 . . . . . . . . . . . 12 (𝑖 ∈ {0} → 𝑖 = 0)
6867adantl 481 . . . . . . . . . . 11 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69 ax-1ne0 11221 . . . . . . . . . . . . 13 1 ≠ 0
7069nesymi 2995 . . . . . . . . . . . 12 ¬ 0 = 1
71 eqeq1 2738 . . . . . . . . . . . 12 (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
7270, 71mtbiri 327 . . . . . . . . . . 11 (𝑖 = 0 → ¬ 𝑖 = 1)
7347notbii 320 . . . . . . . . . . . 12 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
7473biimpi 216 . . . . . . . . . . 11 𝑖 = 1 → ¬ 𝑖 ∈ {1})
75 iffalse 4539 . . . . . . . . . . 11 𝑖 ∈ {1} → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
7668, 72, 74, 754syl 19 . . . . . . . . . 10 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
7766, 76sumeq12rdv 15739 . . . . . . . . 9 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = Σ𝑖 ∈ {0}0)
78 snfi 9081 . . . . . . . . . . 11 {0} ∈ Fin
7978olci 866 . . . . . . . . . 10 ({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin)
80 sumz 15754 . . . . . . . . . 10 (({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
8179, 80ax-mp 5 . . . . . . . . 9 Σ𝑖 ∈ {0}0 = 0
8277, 81eqtrdi 2790 . . . . . . . 8 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
8382adantl 481 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
84 simpll 767 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
8533adantr 480 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
86 simpr 484 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
8786neqned 2944 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
88 elnnne0 12537 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
8985, 87, 88sylanbrc 583 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
90 1nn0 12539 . . . . . . . . . . . . 13 1 ∈ ℕ0
9190a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
92 simpr 484 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9392nnnn0d 12584 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
9492nnge1d 12311 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
95 elfz2nn0 13654 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
9691, 93, 94, 95syl3anbrc 1342 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
9796snssd 4813 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
9850ad2antrr 726 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
99 oveq2 7438 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝑛𝑖) = (𝑛 − 1))
10046, 99sylbi 217 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} → (𝑛𝑖) = (𝑛 − 1))
101100adantl 481 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) = (𝑛 − 1))
102 nnm1nn0 12564 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
103102ad2antlr 727 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
104101, 103eqeltrd 2838 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) ∈ ℕ0)
10598, 104ffvelcdmd 7104 . . . . . . . . . . . 12 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
106105recnd 11286 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
107106ralrimiva 3143 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
108 fzfi 14009 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
109108olci 866 . . . . . . . . . . 11 ((0...𝑛) ⊆ (ℤ‘0) ∨ (0...𝑛) ∈ Fin)
110109a1i 11 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((0...𝑛) ⊆ (ℤ‘0) ∨ (0...𝑛) ∈ Fin))
111 sumss2 15758 . . . . . . . . . 10 ((({1} ⊆ (0...𝑛) ∧ ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ) ∧ ((0...𝑛) ⊆ (ℤ‘0) ∨ (0...𝑛) ∈ Fin)) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
11297, 107, 110, 111syl21anc 838 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
11350adantr 480 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
114102adantl 481 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
115113, 114ffvelcdmd 7104 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
116115recnd 11286 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
11799fveq2d 6910 . . . . . . . . . . 11 (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
118117sumsn 15778 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
1193, 116, 118sylancr 587 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
120112, 119eqtr3d 2776 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12184, 89, 120syl2anc 584 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12260, 61, 83, 121ifbothda 4568 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12359, 122eqtrd 2774 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12437, 45, 1233eqtrd 2778 . . . 4 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xpf · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12530, 124eqtrd 2774 . . 3 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹f · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
126125mpteq2dva 5247 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹f · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
12716, 126eqtrd 2774 1 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cdif 3959  wss 3962  ifcif 4530  {csn 4630   class class class wbr 5147  cmpt 5230  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   · cmul 11157  cle 11293  cmin 11489  cn 12263  0cn0 12523  cz 12610  cuz 12875  ...cfz 13543  Σcsu 15718  0𝑝c0p 25717  Polycply 26237  Xpcidp 26238  coeffccoe 26239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-0p 25718  df-ply 26241  df-idp 26242  df-coe 26243  df-dgr 26244
This theorem is referenced by:  plymulx  34541
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