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Theorem plymulx0 30598
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 3724 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2 ax-resscn 9978 . . . . . . 7 ℝ ⊆ ℂ
3 1re 10024 . . . . . . 7 1 ∈ ℝ
4 plyid 23946 . . . . . . 7 ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
52, 3, 4mp2an 707 . . . . . 6 Xp ∈ (Poly‘ℝ)
65a1i 11 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp ∈ (Poly‘ℝ))
7 simprl 793 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8 simprr 795 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
97, 8readdcld 10054 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
107, 8remulcld 10055 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
111, 6, 9, 10plymul 23955 . . . 4 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹𝑓 · Xp) ∈ (Poly‘ℝ))
12 0re 10025 . . . 4 0 ∈ ℝ
13 eqid 2620 . . . . 5 (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(𝐹𝑓 · Xp))
1413coef2 23968 . . . 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 6236 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹𝑓 · Xp))‘𝑛)))
17 cnex 10002 . . . . . . . . 9 ℂ ∈ V
1817a1i 11 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19 plyf 23935 . . . . . . . . 9 (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
201, 19syl 17 . . . . . . . 8 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21 plyf 23935 . . . . . . . . . 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 793 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25 simprr 795 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
2624, 25mulcomd 10046 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2718, 20, 23, 26caofcom 6914 . . . . . . 7 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹𝑓 · Xp) = (Xp𝑓 · 𝐹))
2827fveq2d 6182 . . . . . 6 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹𝑓 · Xp)) = (coeff‘(Xp𝑓 · 𝐹)))
2928fveq1d 6180 . . . . 5 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹𝑓 · Xp))‘𝑛) = ((coeff‘(Xp𝑓 · 𝐹))‘𝑛))
3029adantr 481 . . . 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 481 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℝ))
33 simpr 477 . . . . . 6 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
34 eqid 2620 . . . . . . 7 (coeff‘Xp) = (coeff‘Xp)
35 eqid 2620 . . . . . . 7 (coeff‘𝐹) = (coeff‘𝐹)
3634, 35coemul 23989 . . . . . 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 1324 . . . . 5 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp𝑓 · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))))
38 elfznn0 12417 . . . . . . . . . 10 (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39 coeidp 24000 . . . . . . . . . 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 6650 . . . . . . . 8 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))))
42 ovif 6722 . . . . . . . 8 (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖))))
4341, 42syl6eq 2670 . . . . . . 7 (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛𝑖))), (0 · ((coeff‘𝐹)‘(𝑛𝑖)))))
4443adantl 482 . . . . . 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 14414 . . . . 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 4184 . . . . . . . . . 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 23968 . . . . . . . . . . . . 13 ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
501, 12, 49sylancl 693 . . . . . . . . . . . 12 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
5150ad2antrr 761 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52 fznn0sub 12358 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑛) → (𝑛𝑖) ∈ ℕ0)
5352adantl 482 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛𝑖) ∈ ℕ0)
5451, 53ffvelrnd 6346 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
5554recnd 10053 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
5655mulid2d 10043 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛𝑖))) = ((coeff‘𝐹)‘(𝑛𝑖)))
5755mul02d 10219 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛𝑖))) = 0)
5848, 56, 57ifbieq12d 4104 . . . . . . 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 14414 . . . . . 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 2631 . . . . . . 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 2631 . . . . . . 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 6643 . . . . . . . . . . 11 (𝑛 = 0 → (0...𝑛) = (0...0))
63 0z 11373 . . . . . . . . . . . 12 0 ∈ ℤ
64 fzsn 12368 . . . . . . . . . . . 12 (0 ∈ ℤ → (0...0) = {0})
6563, 64ax-mp 5 . . . . . . . . . . 11 (0...0) = {0}
6662, 65syl6eq 2670 . . . . . . . . . 10 (𝑛 = 0 → (0...𝑛) = {0})
67 elsni 4185 . . . . . . . . . . . . 13 (𝑖 ∈ {0} → 𝑖 = 0)
6867adantl 482 . . . . . . . . . . . 12 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69 ax-1ne0 9990 . . . . . . . . . . . . . 14 1 ≠ 0
7069nesymi 2848 . . . . . . . . . . . . 13 ¬ 0 = 1
71 eqeq1 2624 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
7270, 71mtbiri 317 . . . . . . . . . . . 12 (𝑖 = 0 → ¬ 𝑖 = 1)
7368, 72syl 17 . . . . . . . . . . 11 ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → ¬ 𝑖 = 1)
7447notbii 310 . . . . . . . . . . . 12 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
7574biimpi 206 . . . . . . . . . . 11 𝑖 = 1 → ¬ 𝑖 ∈ {1})
76 iffalse 4086 . . . . . . . . . . 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 14419 . . . . . . . . 9 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = Σ𝑖 ∈ {0}0)
79 snfi 8023 . . . . . . . . . . 11 {0} ∈ Fin
8079olci 406 . . . . . . . . . 10 ({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin)
81 sumz 14434 . . . . . . . . . 10 (({0} ⊆ (ℤ‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
8280, 81ax-mp 5 . . . . . . . . 9 Σ𝑖 ∈ {0}0 = 0
8378, 82syl6eq 2670 . . . . . . . 8 (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
8483adantl 482 . . . . . . 7 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = 0)
85 simpll 789 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
8633adantr 481 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
87 simpr 477 . . . . . . . . . 10 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
8887neqned 2798 . . . . . . . . 9 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
89 elnnne0 11291 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0𝑛 ≠ 0))
9086, 88, 89sylanbrc 697 . . . . . . . 8 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
91 1nn0 11293 . . . . . . . . . . . . 13 1 ∈ ℕ0
9291a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
93 simpr 477 . . . . . . . . . . . . 13 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9493nnnn0d 11336 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
9593nnge1d 11048 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
96 elfz2nn0 12415 . . . . . . . . . . . 12 (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
9792, 94, 95, 96syl3anbrc 1244 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
9897snssd 4331 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
9950ad2antrr 761 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
100 oveq2 6643 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝑛𝑖) = (𝑛 − 1))
10146, 100sylbi 207 . . . . . . . . . . . . . . 15 (𝑖 ∈ {1} → (𝑛𝑖) = (𝑛 − 1))
102101adantl 482 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) = (𝑛 − 1))
103 nnm1nn0 11319 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
104103ad2antlr 762 . . . . . . . . . . . . . 14 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
105102, 104eqeltrd 2699 . . . . . . . . . . . . 13 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛𝑖) ∈ ℕ0)
10699, 105ffvelrnd 6346 . . . . . . . . . . . 12 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℝ)
107106recnd 10053 . . . . . . . . . . 11 (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
108107ralrimiva 2963 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) ∈ ℂ)
109 fzfi 12754 . . . . . . . . . . . 12 (0...𝑛) ∈ Fin
110109olci 406 . . . . . . . . . . 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 14438 . . . . . . . . . 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 1323 . . . . . . . . 9 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0))
11450adantr 481 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
115103adantl 482 . . . . . . . . . . . 12 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
116114, 115ffvelrnd 6346 . . . . . . . . . . 11 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
117116recnd 10053 . . . . . . . . . 10 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
118100fveq2d 6182 . . . . . . . . . . 11 (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119118sumsn 14456 . . . . . . . . . 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 2656 . . . . . . . 8 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
12285, 90, 121syl2anc 692 . . . . . . 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 4114 . . . . . 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 2654 . . . . 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 2658 . . . 4 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp𝑓 · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
12630, 125eqtrd 2654 . . 3 ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹𝑓 · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
127126mpteq2dva 4735 . 2 (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹𝑓 · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
12816, 127eqtrd 2654 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 383  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  Vcvv 3195  cdif 3564  wss 3567  ifcif 4077  {csn 4168   class class class wbr 4644  cmpt 4720  wf 5872  cfv 5876  (class class class)co 6635  𝑓 cof 6880  Fincfn 7940  cc 9919  cr 9920  0cc0 9921  1c1 9922   · cmul 9926  cle 10060  cmin 10251  cn 11005  0cn0 11277  cz 11362  cuz 11672  ...cfz 12311  Σcsu 14397  0𝑝c0p 23417  Polycply 23921  Xpcidp 23922  coeffccoe 23923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-fz 12312  df-fzo 12450  df-fl 12576  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-clim 14200  df-rlim 14201  df-sum 14398  df-0p 23418  df-ply 23925  df-idp 23926  df-coe 23927  df-dgr 23928
This theorem is referenced by:  plymulx  30599
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