Theorem plymulx0 29950

Description: Coefficients of a polynomial multiplyed by X_p. (Contributed by Thierry Arnoux, 25-Sep-2018.)

Assertion

Ref	Expression
plymulx0	⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘𝑓 · 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.

Step	Hyp	Ref	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))
5	2, 3, 4	mp2an 704	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
6	5	a1i 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)
9	7, 8	readdcld 9948	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
10	7, 8	remulcld 9949	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
11	1, 6, 9, 10	plymul 23778	. . . 4 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘𝑓 · Xp) ∈ (Poly‘ℝ))
12		0re 9919	. . . 4 ⊢ 0 ∈ ℝ
13		eqid 2610	. . . . 5 ⊢ (coeff‘(𝐹 ∘𝑓 · Xp)) = (coeff‘(𝐹 ∘𝑓 · Xp))
14	13	coef2 23791	. . . 4 ⊢ (((𝐹 ∘𝑓 · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹 ∘𝑓 · Xp)):ℕ0⟶ℝ)
15	11, 12, 14	sylancl 693	. . 3 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘𝑓 · Xp)):ℕ0⟶ℝ)
16	15	feqmptd 6159	. 2 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹 ∘𝑓 · Xp))‘𝑛)))
17		cnex 9896	. . . . . . . . 9 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19		plyf 23758	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
20	1, 19	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21		plyf 23758	. . . . . . . . . 10 ⊢ (Xp ∈ (Poly‘ℝ) → Xp:ℂ⟶ℂ)
22	5, 21	ax-mp 5	. . . . . . . . 9 ⊢ Xp:ℂ⟶ℂ
23	22	a1i 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)
26	24, 25	mulcomd 9940	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
27	18, 20, 23, 26	caofcom 6827	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘𝑓 · Xp) = (Xp ∘𝑓 · 𝐹))
28	27	fveq2d 6107	. . . . . 6 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘𝑓 · Xp)) = (coeff‘(Xp ∘𝑓 · 𝐹)))
29	28	fveq1d 6105	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹 ∘𝑓 · Xp))‘𝑛) = ((coeff‘(Xp ∘𝑓 · 𝐹))‘𝑛))
30	29	adantr 480	. . . 4 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹 ∘𝑓 · Xp))‘𝑛) = ((coeff‘(Xp ∘𝑓 · 𝐹))‘𝑛))
31	5	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Xp ∈ (Poly‘ℝ))
32	1	adantr 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)
36	34, 35	coemul 23812	. . . . . 6 ⊢ ((Xp ∈ (Poly‘ℝ) ∧ 𝐹 ∈ (Poly‘ℝ) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘𝑓 · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
37	31, 32, 33, 36	syl3anc 1318	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘𝑓 · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
38		elfznn0 12302	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39		coeidp 23823	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ0 → ((coeff‘Xp)‘𝑖) = if(𝑖 = 1, 1, 0))
40	38, 39	syl 17	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝑛) → ((coeff‘Xp)‘𝑖) = if(𝑖 = 1, 1, 0))
41	40	oveq1d 6564	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
42		ovif 6635	. . . . . . . 8 ⊢ (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
43	41, 42	syl6eq 2660	. . . . . . 7 ⊢ (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖)))))
44	43	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖)))))
45	44	sumeq2dv 14281	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖)))))
46		velsn 4141	. . . . . . . . . 10 ⊢ (𝑖 ∈ {1} ↔ 𝑖 = 1)
47	46	bicomi 213	. . . . . . . . 9 ⊢ (𝑖 = 1 ↔ 𝑖 ∈ {1})
48	47	a1i 11	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑖 = 1 ↔ 𝑖 ∈ {1}))
49	35	coef2 23791	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
50	1, 12, 49	sylancl 693	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
51	50	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52		fznn0sub 12244	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑛) → (𝑛 − 𝑖) ∈ ℕ0)
53	52	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛 − 𝑖) ∈ ℕ0)
54	51, 53	ffvelrnd 6268	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
55	54	recnd 9947	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
56	55	mulid2d 9937	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = ((coeff‘𝐹)‘(𝑛 − 𝑖)))
57	55	mul02d 10113	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = 0)
58	48, 56, 57	ifbieq12d 4063	. . . . . . 7 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖)))) = if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0))
59	58	sumeq2dv 14281	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖)))) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0))
60		eqeq2 2621	. . . . . . 7 ⊢ (0 = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))) → (Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0 ↔ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 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)‘(𝑛 − 𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)) ↔ Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 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})
65	63, 64	ax-mp 5	. . . . . . . . . . 11 ⊢ (0...0) = {0}
66	62, 65	syl6eq 2660	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (0...𝑛) = {0})
67		elsni 4142	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ {0} → 𝑖 = 0)
68	67	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69		ax-1ne0 9884	. . . . . . . . . . . . . 14 ⊢ 1 ≠ 0
70	69	nesymi 2839	. . . . . . . . . . . . 13 ⊢ ¬ 0 = 1
71		eqeq1 2614	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
72	70, 71	mtbiri 316	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → ¬ 𝑖 = 1)
73	68, 72	syl 17	. . . . . . . . . . 11 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → ¬ 𝑖 = 1)
74	47	notbii 309	. . . . . . . . . . . 12 ⊢ (¬ 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
75	74	biimpi 205	. . . . . . . . . . 11 ⊢ (¬ 𝑖 = 1 → ¬ 𝑖 ∈ {1})
76		iffalse 4045	. . . . . . . . . . 11 ⊢ (¬ 𝑖 ∈ {1} → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
77	73, 75, 76	3syl 18	. . . . . . . . . 10 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
78	66, 77	sumeq12rdv 14285	. . . . . . . . 9 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = Σ𝑖 ∈ {0}0)
79		snfi 7923	. . . . . . . . . . 11 ⊢ {0} ∈ Fin
80	79	olci 405	. . . . . . . . . 10 ⊢ ({0} ⊆ (ℤ≥‘0) ∨ {0} ∈ Fin)
81		sumz 14300	. . . . . . . . . 10 ⊢ (({0} ⊆ (ℤ≥‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
82	80, 81	ax-mp 5	. . . . . . . . 9 ⊢ Σ𝑖 ∈ {0}0 = 0
83	78, 82	syl6eq 2660	. . . . . . . 8 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
84	83	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
85		simpll 786	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
86	33	adantr 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)
88	87	neqned 2789	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
89		elnnne0 11183	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
90	86, 88, 89	sylanbrc 695	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
91		1nn0 11185	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ0
92	91	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
93		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
94	93	nnnn0d 11228	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
95	93	nnge1d 10940	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
96		elfz2nn0 12300	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
97	92, 94, 95, 96	syl3anbrc 1239	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
98	97	snssd 4281	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
99	50	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
100		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → (𝑛 − 𝑖) = (𝑛 − 1))
101	46, 100	sylbi 206	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {1} → (𝑛 − 𝑖) = (𝑛 − 1))
102	101	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) = (𝑛 − 1))
103		nnm1nn0 11211	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
104	103	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
105	102, 104	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) ∈ ℕ0)
106	99, 105	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
107	106	recnd 9947	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
108	107	ralrimiva 2949	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
109		fzfi 12633	. . . . . . . . . . . 12 ⊢ (0...𝑛) ∈ Fin
110	109	olci 405	. . . . . . . . . . 11 ⊢ ((0...𝑛) ⊆ (ℤ≥‘0) ∨ (0...𝑛) ∈ Fin)
111	110	a1i 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)‘(𝑛 − 𝑖)) ∈ ℂ) ∧ ((0...𝑛) ⊆ (ℤ≥‘0) ∨ (0...𝑛) ∈ Fin)) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0))
113	98, 108, 111, 112	syl21anc 1317	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0))
114	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
115	103	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
116	114, 115	ffvelrnd 6268	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
117	116	recnd 9947	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
118	100	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119	118	sumsn 14319	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
120	3, 117, 119	sylancr 694	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
121	113, 120	eqtr3d 2646	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
122	85, 90, 121	syl2anc 691	. . . . . . 7 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
123	60, 61, 84, 122	ifbothda 4073	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
124	59, 123	eqtrd 2644	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 = 1, (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))), (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖)))) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
125	37, 45, 124	3eqtrd 2648	. . . 4 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘𝑓 · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
126	30, 125	eqtrd 2644	. . 3 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹 ∘𝑓 · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
127	126	mpteq2dva 4672	. 2 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹 ∘𝑓 · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
128	16, 127	eqtrd 2644	1 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘𝑓 · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))

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  0_𝑝c0p 23242  Polycply 23744  X_pcidp 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