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Theorem plymulx0 29950
 Description: Coefficients of a polynomial multiplyed 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 3694 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2 ax-resscn 9872 . . . . . . 7 ℝ ⊆ ℂ
3 1re 9918 . . . . . . 7 1 ∈ ℝ
4 plyid 23769 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
52, 3, 4mp2an 704 . . . . . 6 Xp ∈ (Poly‘ℝ)
65a1i 11 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp ∈ (Poly‘ℝ))
7 simprl 790 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8 simprr 792 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
97, 8readdcld 9948 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
107, 8remulcld 9949 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 23778 . . . 4 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹𝑓 · Xp) ∈ (Poly‘ℝ))
12 0re 9919 . . . 4 0 ∈ ℝ
13 eqid 2610 . . . . 5 (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(𝐹𝑓 · Xp))
1413coef2 23791 . . . 4 (((𝐹𝑓 · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹𝑓 · Xp)):ℕ0⟶ℝ)
1511, 12, 14sylancl 693 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)):ℕ0⟶ℝ)
1615feqmptd 6159 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹𝑓 · Xp))‘𝑛)))
17 cnex 9896 . . . . . . . . 9 ℂ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19 plyf 23758 . . . . . . . . 9 (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21 plyf 23758 . . . . . . . . . 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 790 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25 simprr 792 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
2624, 25mulcomd 9940 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2718, 20, 23, 26caofcom 6827 . . . . . . 7 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹𝑓 · Xp) = (Xp𝑓 · 𝐹))
2827fveq2d 6107 . . . . . 6 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(Xp𝑓 · 𝐹)))
2928fveq1d 6105 . . . . 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 476 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
34 eqid 2610 . . . . . . 7 (coeff‘Xp) = (coeff‘Xp)
35 eqid 2610 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
3634, 35coemul 23812 . . . . . 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 1318 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp𝑓 · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))))
38 elfznn0 12302 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39 coeidp 23823 . . . . . . . . . 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 6564 . . . . . . . 8 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))))
42 ovif 6635 . . . . . . . 8 (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖))))
4341, 42syl6eq 2660 . . . . . . 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 14281 . . . . 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 4141 . . . . . . . . . 10 (𝑖 ∈ {1} ↔ 𝑖 = 1)
4746bicomi 213 . . . . . . . . 9 (𝑖 = 1 ↔ 𝑖 ∈ {1})
4847a1i 11 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑖 = 1 ↔ 𝑖 ∈ {1}))
4935coef2 23791 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
501, 12, 49sylancl 693 . . . . . . . . . . . 12 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
5150ad2antrr 758 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52 fznn0sub 12244 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) → (𝑛𝑖) ∈ ℕ0)
5352adantl 481 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛𝑖) ∈ ℕ0)
5451, 53ffvelrnd 6268 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
5554recnd 9947 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
5655mulid2d 9937 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛𝑖))) = ((coeff‘𝐹)‘(𝑛𝑖)))
5755mul02d 10113 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛𝑖))) = 0)
5848, 56, 57ifbieq12d 4063 . . . . . . 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 14281 . . . . . 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 2621 . . . . . . 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 2621 . . . . . . 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 6557 . . . . . . . . . . 11 (𝑛 = 0 → (0...𝑛) = (0...0))
63 0z 11265 . . . . . . . . . . . 12 0 ∈ ℤ
64 fzsn 12254 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65syl6eq 2660 . . . . . . . . . 10 (𝑛 = 0 → (0...𝑛) = {0})
67 elsni 4142 . . . . . . . . . . . . 13 (𝑖 ∈ {0} → 𝑖 = 0)
6867adantl 481 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69 ax-1ne0 9884 . . . . . . . . . . . . . 14 1 ≠ 0
7069nesymi 2839 . . . . . . . . . . . . 13 ¬ 0 = 1
71 eqeq1 2614 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
7270, 71mtbiri 316 . . . . . . . . . . . 12 (𝑖 = 0 → ¬ 𝑖 = 1)
7368, 72syl 17 . . . . . . . . . . 11 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → ¬ 𝑖 = 1)
7447notbii 309 . . . . . . . . . . . 12 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
7574biimpi 205 . . . . . . . . . . 11 𝑖 = 1 → ¬ 𝑖 ∈ {1})
76 iffalse 4045 . . . . . . . . . . 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 14285 . . . . . . . . 9 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 7923 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 405 . . . . . . . . . 10 ({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin)
81 sumz 14300 . . . . . . . . . 10 (({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
8280, 81ax-mp 5 . . . . . . . . 9 Σ𝑖 ∈ {0}0 = 0
8378, 82syl6eq 2660 . . . . . . . 8 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
8483adantl 481 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
85 simpll 786 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
8633adantr 480 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
87 simpr 476 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
8887neqned 2789 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
89 elnnne0 11183 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
9086, 88, 89sylanbrc 695 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
91 1nn0 11185 . . . . . . . . . . . . 13 1 ∈ ℕ0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
93 simpr 476 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9493nnnn0d 11228 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
9593nnge1d 10940 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
96 elfz2nn0 12300 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
9792, 94, 95, 96syl3anbrc 1239 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
9897snssd 4281 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
9950ad2antrr 758 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
100 oveq2 6557 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝑛𝑖) = (𝑛 − 1))
10146, 100sylbi 206 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} → (𝑛𝑖) = (𝑛 − 1))
102101adantl 481 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) = (𝑛 − 1))
103 nnm1nn0 11211 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
104103ad2antlr 759 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
105102, 104eqeltrd 2688 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) ∈ ℕ0)
10699, 105ffvelrnd 6268 . . . . . . . . . . . 12 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
107106recnd 9947 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
108107ralrimiva 2949 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
109 fzfi 12633 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 405 . . . . . . . . . . 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 14304 . . . . . . . . . 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 1317 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
11450adantr 480 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
115103adantl 481 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
116114, 115ffvelrnd 6268 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
117116recnd 9947 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
118100fveq2d 6107 . . . . . . . . . . 11 (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119118sumsn 14319 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
1203, 117, 119sylancr 694 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
121113, 120eqtr3d 2646 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12285, 90, 121syl2anc 691 . . . . . . 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 4073 . . . . . 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 2644 . . . . 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 2648 . . . 4 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp𝑓 · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12630, 125eqtrd 2644 . . 3 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹𝑓 · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
127126mpteq2dva 4672 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹𝑓 · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
12816, 127eqtrd 2644 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  Σcsu 14264  0𝑝c0p 23242  Polycply 23744  Xpcidp 23745  coeffccoe 23746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-idp 23749  df-coe 23750  df-dgr 23751 This theorem is referenced by:  plymulx  29951
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