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Theorem plymulx0 30955
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 3876 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2 ax-resscn 10206 . . . . . . 7 ℝ ⊆ ℂ
3 1re 10252 . . . . . . 7 1 ∈ ℝ
4 plyid 24185 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
52, 3, 4mp2an 710 . . . . . 6 Xp ∈ (Poly‘ℝ)
65a1i 11 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp ∈ (Poly‘ℝ))
7 simprl 811 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8 simprr 813 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
97, 8readdcld 10282 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
107, 8remulcld 10283 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 24194 . . . 4 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹𝑓 · Xp) ∈ (Poly‘ℝ))
12 0re 10253 . . . 4 0 ∈ ℝ
13 eqid 2761 . . . . 5 (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(𝐹𝑓 · Xp))
1413coef2 24207 . . . 4 (((𝐹𝑓 · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹𝑓 · Xp)):ℕ0⟶ℝ)
1511, 12, 14sylancl 697 . . 3 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)):ℕ0⟶ℝ)
1615feqmptd 6413 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹𝑓 · Xp))‘𝑛)))
17 cnex 10230 . . . . . . . . 9 ℂ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19 plyf 24174 . . . . . . . . 9 (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21 plyf 24174 . . . . . . . . . 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 811 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25 simprr 813 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
2624, 25mulcomd 10274 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2718, 20, 23, 26caofcom 7096 . . . . . . 7 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹𝑓 · Xp) = (Xp𝑓 · 𝐹))
2827fveq2d 6358 . . . . . 6 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(Xp𝑓 · 𝐹)))
2928fveq1d 6356 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹𝑓 · Xp))‘𝑛) = ((coeff‘(Xp𝑓 · 𝐹))‘𝑛))
3029adantr 472 . . . 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 472 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℝ))
33 simpr 479 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
34 eqid 2761 . . . . . . 7 (coeff‘Xp) = (coeff‘Xp)
35 eqid 2761 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
3634, 35coemul 24228 . . . . . 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 1477 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp𝑓 · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))))
38 elfznn0 12647 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39 coeidp 24239 . . . . . . . . . 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 6830 . . . . . . . 8 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))))
42 ovif 6904 . . . . . . . 8 (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖))))
4341, 42syl6eq 2811 . . . . . . 7 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))))
4443adantl 473 . . . . . 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 14653 . . . . 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 4338 . . . . . . . . . 10 (𝑖 ∈ {1} ↔ 𝑖 = 1)
4746bicomi 214 . . . . . . . . 9 (𝑖 = 1 ↔ 𝑖 ∈ {1})
4847a1i 11 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑖 = 1 ↔ 𝑖 ∈ {1}))
4935coef2 24207 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
501, 12, 49sylancl 697 . . . . . . . . . . . 12 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
5150ad2antrr 764 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52 fznn0sub 12587 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) → (𝑛𝑖) ∈ ℕ0)
5352adantl 473 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛𝑖) ∈ ℕ0)
5451, 53ffvelrnd 6525 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
5554recnd 10281 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
5655mulid2d 10271 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛𝑖))) = ((coeff‘𝐹)‘(𝑛𝑖)))
5755mul02d 10447 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛𝑖))) = 0)
5848, 56, 57ifbieq12d 4258 . . . . . . 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 14653 . . . . . 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 2772 . . . . . . 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 2772 . . . . . . 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 6823 . . . . . . . . . . 11 (𝑛 = 0 → (0...𝑛) = (0...0))
63 0z 11601 . . . . . . . . . . . 12 0 ∈ ℤ
64 fzsn 12597 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65syl6eq 2811 . . . . . . . . . 10 (𝑛 = 0 → (0...𝑛) = {0})
67 elsni 4339 . . . . . . . . . . . . 13 (𝑖 ∈ {0} → 𝑖 = 0)
6867adantl 473 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69 ax-1ne0 10218 . . . . . . . . . . . . . 14 1 ≠ 0
7069nesymi 2990 . . . . . . . . . . . . 13 ¬ 0 = 1
71 eqeq1 2765 . . . . . . . . . . . . 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 206 . . . . . . . . . . 11 𝑖 = 1 → ¬ 𝑖 ∈ {1})
76 iffalse 4240 . . . . . . . . . . 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 14658 . . . . . . . . 9 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 8206 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 405 . . . . . . . . . 10 ({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin)
81 sumz 14673 . . . . . . . . . 10 (({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
8280, 81ax-mp 5 . . . . . . . . 9 Σ𝑖 ∈ {0}0 = 0
8378, 82syl6eq 2811 . . . . . . . 8 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
8483adantl 473 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
85 simpll 807 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
8633adantr 472 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
87 simpr 479 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
8887neqned 2940 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
89 elnnne0 11519 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
9086, 88, 89sylanbrc 701 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
91 1nn0 11521 . . . . . . . . . . . . 13 1 ∈ ℕ0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
93 simpr 479 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9493nnnn0d 11564 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
9593nnge1d 11276 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
96 elfz2nn0 12645 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
9792, 94, 95, 96syl3anbrc 1429 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
9897snssd 4486 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
9950ad2antrr 764 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
100 oveq2 6823 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝑛𝑖) = (𝑛 − 1))
10146, 100sylbi 207 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} → (𝑛𝑖) = (𝑛 − 1))
102101adantl 473 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) = (𝑛 − 1))
103 nnm1nn0 11547 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
104103ad2antlr 765 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
105102, 104eqeltrd 2840 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) ∈ ℕ0)
10699, 105ffvelrnd 6525 . . . . . . . . . . . 12 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
107106recnd 10281 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
108107ralrimiva 3105 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
109 fzfi 12986 . . . . . . . . . . . 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 14677 . . . . . . . . . 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 1476 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
11450adantr 472 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
115103adantl 473 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
116114, 115ffvelrnd 6525 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
117116recnd 10281 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
118100fveq2d 6358 . . . . . . . . . . 11 (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119118sumsn 14695 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
1203, 117, 119sylancr 698 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
121113, 120eqtr3d 2797 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12285, 90, 121syl2anc 696 . . . . . . 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 4268 . . . . . 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 2795 . . . . 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 2799 . . . 4 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp𝑓 · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12630, 125eqtrd 2795 . . 3 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹𝑓 · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
127126mpteq2dva 4897 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹𝑓 · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
12816, 127eqtrd 2795 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 196  wo 382  wa 383   = wceq 1632  wcel 2140  wne 2933  wral 3051  Vcvv 3341  cdif 3713  wss 3716  ifcif 4231  {csn 4322   class class class wbr 4805  cmpt 4882  wf 6046  cfv 6050  (class class class)co 6815  𝑓 cof 7062  Fincfn 8124  cc 10147  cr 10148  0cc0 10149  1c1 10150   · cmul 10154  cle 10288  cmin 10479  cn 11233  0cn0 11505  cz 11590  cuz 11900  ...cfz 12540  Σcsu 14636  0𝑝c0p 23656  Polycply 24160  Xpcidp 24161  coeffccoe 24162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-inf2 8714  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-pre-sup 10227  ax-addf 10228
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-of 7064  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-oadd 7735  df-er 7914  df-map 8028  df-pm 8029  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-sup 8516  df-inf 8517  df-oi 8583  df-card 8976  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-nn 11234  df-2 11292  df-3 11293  df-n0 11506  df-z 11591  df-uz 11901  df-rp 12047  df-fz 12541  df-fzo 12681  df-fl 12808  df-seq 13017  df-exp 13076  df-hash 13333  df-cj 14059  df-re 14060  df-im 14061  df-sqrt 14195  df-abs 14196  df-clim 14439  df-rlim 14440  df-sum 14637  df-0p 23657  df-ply 24164  df-idp 24165  df-coe 24166  df-dgr 24167
This theorem is referenced by:  plymulx  30956
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