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Theorem plymulx0 34212
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 4127 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2 ax-resscn 11203 . . . . . . 7 ℝ ⊆ ℂ
3 1re 11252 . . . . . . 7 1 ∈ ℝ
4 plyid 26163 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
52, 3, 4mp2an 690 . . . . . 6 Xp ∈ (Poly‘ℝ)
65a1i 11 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp ∈ (Poly‘ℝ))
7 simprl 769 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8 simprr 771 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
97, 8readdcld 11281 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
107, 8remulcld 11282 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 26172 . . . 4 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹f · Xp) ∈ (Poly‘ℝ))
12 0re 11254 . . . 4 0 ∈ ℝ
13 eqid 2728 . . . . 5 (coeff‘(𝐹f · Xp)) = (coeff‘(𝐹f · Xp))
1413coef2 26185 . . . 4 (((𝐹f · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹f · Xp)):ℕ0⟶ℝ)
1511, 12, 14sylancl 584 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)):ℕ0⟶ℝ)
1615feqmptd 6972 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹f · Xp))‘𝑛)))
17 cnex 11227 . . . . . . . . 9 ℂ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19 plyf 26152 . . . . . . . . 9 (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21 plyf 26152 . . . . . . . . . 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 769 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25 simprr 771 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
2624, 25mulcomd 11273 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2718, 20, 23, 26caofcom 7726 . . . . . . 7 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹f · Xp) = (Xpf · 𝐹))
2827fveq2d 6906 . . . . . 6 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹f · Xp)) = (coeff‘(Xpf · 𝐹)))
2928fveq1d 6904 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹f · Xp))‘𝑛) = ((coeff‘(Xpf · 𝐹))‘𝑛))
3029adantr 479 . . . 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 479 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℝ))
33 simpr 483 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
34 eqid 2728 . . . . . . 7 (coeff‘Xp) = (coeff‘Xp)
35 eqid 2728 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
3634, 35coemul 26206 . . . . . 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 1368 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xpf · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))))
38 elfznn0 13634 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39 coeidp 26218 . . . . . . . . . 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 7441 . . . . . . . 8 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))))
42 ovif 7524 . . . . . . . 8 (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖))))
4341, 42eqtrdi 2784 . . . . . . 7 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))))
4443adantl 480 . . . . . 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 15689 . . . . 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 4648 . . . . . . . . . 10 (𝑖 ∈ {1} ↔ 𝑖 = 1)
4746bicomi 223 . . . . . . . . 9 (𝑖 = 1 ↔ 𝑖 ∈ {1})
4847a1i 11 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑖 = 1 ↔ 𝑖 ∈ {1}))
4935coef2 26185 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
501, 12, 49sylancl 584 . . . . . . . . . . . 12 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
5150ad2antrr 724 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52 fznn0sub 13573 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) → (𝑛𝑖) ∈ ℕ0)
5352adantl 480 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛𝑖) ∈ ℕ0)
5451, 53ffvelcdmd 7100 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
5554recnd 11280 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
5655mullidd 11270 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛𝑖))) = ((coeff‘𝐹)‘(𝑛𝑖)))
5755mul02d 11450 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛𝑖))) = 0)
5848, 56, 57ifbieq12d 4560 . . . . . . 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 15689 . . . . . 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 2740 . . . . . . 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 2740 . . . . . . 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 7434 . . . . . . . . . . 11 (𝑛 = 0 → (0...𝑛) = (0...0))
63 0z 12607 . . . . . . . . . . . 12 0 ∈ ℤ
64 fzsn 13583 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65eqtrdi 2784 . . . . . . . . . 10 (𝑛 = 0 → (0...𝑛) = {0})
67 elsni 4649 . . . . . . . . . . . . 13 (𝑖 ∈ {0} → 𝑖 = 0)
6867adantl 480 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69 ax-1ne0 11215 . . . . . . . . . . . . . 14 1 ≠ 0
7069nesymi 2995 . . . . . . . . . . . . 13 ¬ 0 = 1
71 eqeq1 2732 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
7270, 71mtbiri 326 . . . . . . . . . . . 12 (𝑖 = 0 → ¬ 𝑖 = 1)
7368, 72syl 17 . . . . . . . . . . 11 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → ¬ 𝑖 = 1)
7447notbii 319 . . . . . . . . . . . 12 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
7574biimpi 215 . . . . . . . . . . 11 𝑖 = 1 → ¬ 𝑖 ∈ {1})
76 iffalse 4541 . . . . . . . . . . 11 𝑖 ∈ {1} → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
7773, 75, 763syl 18 . . . . . . . . . 10 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
7866, 77sumeq12rdv 15693 . . . . . . . . 9 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 9075 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 864 . . . . . . . . . 10 ({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin)
81 sumz 15708 . . . . . . . . . 10 (({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
8280, 81ax-mp 5 . . . . . . . . 9 Σ𝑖 ∈ {0}0 = 0
8378, 82eqtrdi 2784 . . . . . . . 8 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
8483adantl 480 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
85 simpll 765 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
8633adantr 479 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
87 simpr 483 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
8887neqned 2944 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
89 elnnne0 12524 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
9086, 88, 89sylanbrc 581 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
91 1nn0 12526 . . . . . . . . . . . . 13 1 ∈ ℕ0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
93 simpr 483 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9493nnnn0d 12570 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
9593nnge1d 12298 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
96 elfz2nn0 13632 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
9792, 94, 95, 96syl3anbrc 1340 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
9897snssd 4817 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
9950ad2antrr 724 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
100 oveq2 7434 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝑛𝑖) = (𝑛 − 1))
10146, 100sylbi 216 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} → (𝑛𝑖) = (𝑛 − 1))
102101adantl 480 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) = (𝑛 − 1))
103 nnm1nn0 12551 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
104103ad2antlr 725 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
105102, 104eqeltrd 2829 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) ∈ ℕ0)
10699, 105ffvelcdmd 7100 . . . . . . . . . . . 12 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
107106recnd 11280 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
108107ralrimiva 3143 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
109 fzfi 13977 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 864 . . . . . . . . . . 11 ((0...𝑛) ⊆ (ℤ‘0) ∨ (0...𝑛) ∈ Fin)
111110a1i 11 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((0...𝑛) ⊆ (ℤ‘0) ∨ (0...𝑛) ∈ Fin))
112 sumss2 15712 . . . . . . . . . 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))
11398, 108, 111, 112syl21anc 836 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
11450adantr 479 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
115103adantl 480 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
116114, 115ffvelcdmd 7100 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
117116recnd 11280 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
118100fveq2d 6906 . . . . . . . . . . 11 (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119118sumsn 15732 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
1203, 117, 119sylancr 585 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
121113, 120eqtr3d 2770 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12285, 90, 121syl2anc 582 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12360, 61, 84, 122ifbothda 4570 . . . . . 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))))
12459, 123eqtrd 2768 . . . . 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))))
12537, 45, 1243eqtrd 2772 . . . 4 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xpf · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12630, 125eqtrd 2768 . . 3 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹f · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
127126mpteq2dva 5252 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹f · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
12816, 127eqtrd 2768 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 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2937  wral 3058  Vcvv 3473  cdif 3946  wss 3949  ifcif 4532  {csn 4632   class class class wbr 5152  cmpt 5235  wf 6549  cfv 6553  (class class class)co 7426  f cof 7689  Fincfn 8970  cc 11144  cr 11145  0cc0 11146  1c1 11147   · cmul 11151  cle 11287  cmin 11482  cn 12250  0cn0 12510  cz 12596  cuz 12860  ...cfz 13524  Σcsu 15672  0𝑝c0p 25618  Polycply 26138  Xpcidp 26139  coeffccoe 26140
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-inf 9474  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-fl 13797  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-rlim 15473  df-sum 15673  df-0p 25619  df-ply 26142  df-idp 26143  df-coe 26144  df-dgr 26145
This theorem is referenced by:  plymulx  34213
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