Theorem plyn0mulidp 26399

Description: Coefficients of a non-zero polynomial multiplied by the identity polynomial. (Contributed by Thierry Arnoux, 25-Sep-2018.)

Assertion

Ref	Expression
plyn0mulidp	⊢ (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 plyn0mulidp

Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4087	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2		ax-resscn 11145	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3		1re 11196	. . . . . . 7 ⊢ 1 ∈ ℝ
4		plyid 26323	. . . . . . 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 782	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8		simprr 784	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
9	7, 8	readdcld 11226	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
10	7, 8	remulcld 11227	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
11	1, 6, 9, 10	plymul 26332	. . . 4 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘f · Xp) ∈ (Poly‘ℝ))
12		0re 11198	. . . 4 ⊢ 0 ∈ ℝ
13		eqid 2765	. . . . 5 ⊢ (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(𝐹 ∘f · Xp))
14	13	coef2 26345	. . . 4 ⊢ (((𝐹 ∘f · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹 ∘f · Xp)):ℕ0⟶ℝ)
15	11, 12, 14	sylancl 597	. . 3 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)):ℕ0⟶ℝ)
16	15	feqmptd 6939	. 2 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹 ∘f · Xp))‘𝑛)))
17		cnex 11169	. . . . . . . . 9 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19		plyf 26312	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
20	1, 19	syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21		plyf 26312	. . . . . . . . . 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 782	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25		simprr 784	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
26	24, 25	mulcomd 11218	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
27	18, 20, 23, 26	caofcom 7701	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘f · Xp) = (Xp ∘f · 𝐹))
28	27	fveq2d 6875	. . . . . 6 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(Xp ∘f · 𝐹)))
29	28	fveq1d 6873	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹 ∘f · Xp))‘𝑛) = ((coeff‘(Xp ∘f · 𝐹))‘𝑛))
30	29	adantr 485	. . . 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 485	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℝ))
33		simpr 489	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
34		eqid 2765	. . . . . . 7 ⊢ (coeff‘Xp) = (coeff‘Xp)
35		eqid 2765	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
36	34, 35	coemul 26366	. . . . . 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 1394	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘f · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
38		elfznn0 13636	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39		coeidp 26377	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ0 → ((coeff‘Xp)‘𝑖) = if(𝑖 = 1, 1, 0))
40	38, 39	syl 18	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...𝑛) → ((coeff‘Xp)‘𝑖) = if(𝑖 = 1, 1, 0))
41	40	oveq1d 7415	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
42		ovif 7498	. . . . . . . 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	eqtrdi 2816	. . . . . . 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 486	. . . . . 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 15741	. . . . 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 4601	. . . . . . . . . 10 ⊢ (𝑖 ∈ {1} ↔ 𝑖 = 1)
47	46	bicomi 227	. . . . . . . . 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 26345	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
50	1, 12, 49	sylancl 597	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
51	50	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52		fznn0sub 13572	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑛) → (𝑛 − 𝑖) ∈ ℕ0)
53	52	adantl 486	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛 − 𝑖) ∈ ℕ0)
54	51, 53	ffvelcdmd 7070	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
55	54	recnd 11225	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
56	55	mullidd 11215	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = ((coeff‘𝐹)‘(𝑛 − 𝑖)))
57	55	mul02d 11396	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = 0)
58	48, 56, 57	ifbieq12d 4512	. . . . . . 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 15741	. . . . . 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 2777	. . . . . . 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 2777	. . . . . . 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 7408	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
63		0z 12590	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
64		fzsn 13582	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → (0...0) = {0})
65	63, 64	ax-mp 5	. . . . . . . . . . 11 ⊢ (0...0) = {0}
66	62, 65	eqtrdi 2816	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (0...𝑛) = {0})
67		elsni 4602	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ {0} → 𝑖 = 0)
68	67	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69		ax-1ne0 11157	. . . . . . . . . . . . 13 ⊢ 1 ≠ 0
70	69	nesymi 3017	. . . . . . . . . . . 12 ⊢ ¬ 0 = 1
71		eqeq1 2769	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
72	70, 71	mtbiri 330	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → ¬ 𝑖 = 1)
73	47	notbii 323	. . . . . . . . . . . 12 ⊢ (¬ 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
74	73	biimpi 219	. . . . . . . . . . 11 ⊢ (¬ 𝑖 = 1 → ¬ 𝑖 ∈ {1})
75		iffalse 4492	. . . . . . . . . . 11 ⊢ (¬ 𝑖 ∈ {1} → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
76	68, 72, 74, 75	4syl 20	. . . . . . . . . 10 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
77	66, 76	sumeq12rdv 15746	. . . . . . . . 9 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = Σ𝑖 ∈ {0}0)
78		snfi 9028	. . . . . . . . . . 11 ⊢ {0} ∈ Fin
79	78	olci 879	. . . . . . . . . 10 ⊢ ({0} ⊆ (ℤ≥‘0) ∨ {0} ∈ Fin)
80		sumz 15761	. . . . . . . . . 10 ⊢ (({0} ⊆ (ℤ≥‘0) ∨ {0} ∈ Fin) → Σ𝑖 ∈ {0}0 = 0)
81	79, 80	ax-mp 5	. . . . . . . . 9 ⊢ Σ𝑖 ∈ {0}0 = 0
82	77, 81	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
83	82	adantl 486	. . . . . . 7 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
84		simpll 778	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
85	33	adantr 485	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
86		simpr 489	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
87	86	neqned 2967	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
88		elnnne0 12506	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
89	85, 87, 88	sylanbrc 594	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
90		1nn0 12508	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ0
91	90	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
92		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
93	92	nnnn0d 12553	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
94	92	nnge1d 12272	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
95		elfz2nn0 13634	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
96	91, 93, 94, 95	syl3anbrc 1360	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
97	96	snssd 4748	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
98	50	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
99		oveq2 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → (𝑛 − 𝑖) = (𝑛 − 1))
100	46, 99	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {1} → (𝑛 − 𝑖) = (𝑛 − 1))
101	100	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) = (𝑛 − 1))
102		nnm1nn0 12533	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
103	102	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
104	101, 103	eqeltrd 2865	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) ∈ ℕ0)
105	98, 104	ffvelcdmd 7070	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
106	105	recnd 11225	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
107	106	ralrimiva 3157	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
108		fzfi 13996	. . . . . . . . . . . 12 ⊢ (0...𝑛) ∈ Fin
109	108	olci 879	. . . . . . . . . . 11 ⊢ ((0...𝑛) ⊆ (ℤ≥‘0) ∨ (0...𝑛) ∈ Fin)
110	109	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((0...𝑛) ⊆ (ℤ≥‘0) ∨ (0...𝑛) ∈ Fin))
111		sumss2 15765	. . . . . . . . . 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))
112	97, 107, 110, 111	syl21anc 850	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0))
113	50	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
114	102	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
115	113, 114	ffvelcdmd 7070	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
116	115	recnd 11225	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
117	99	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
118	117	sumsn 15785	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119	3, 116, 118	sylancr 598	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
120	112, 119	eqtr3d 2802	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
121	84, 89, 120	syl2anc 595	. . . . . . 7 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = ((coeff‘𝐹)‘(𝑛 − 1)))
122	60, 61, 83, 121	ifbothda 4522	. . . . . 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))))
123	59, 122	eqtrd 2800	. . . . 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))))
124	37, 45, 123	3eqtrd 2804	. . . 4 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘f · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
125	30, 124	eqtrd 2800	. . 3 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹 ∘f · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
126	125	mpteq2dva 5197	. 2 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹 ∘f · Xp))‘𝑛)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))
127	16, 126	eqtrd 2800	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 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ifcif 4483  {csn 4585   class class class wbr 5104   ↦ cmpt 5185  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∘_f cof 7662  Fincfn 8931  ℂcc 11086  ℝcr 11087  0cc0 11088  1c1 11089   · cmul 11093   ≤ cle 11232   − cmin 11429  ℕcn 12221  ℕ₀cn0 12492  ℤcz 12579  ℤ_≥cuz 12850  ...cfz 13523  Σcsu 15725  0_𝑝c0p 25785  Polycply 26298  X_pcidp 26299  coeffccoe 26300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12222  df-2 12291  df-3 12292  df-n0 12493  df-z 12580  df-uz 12851  df-rp 13005  df-fz 13524  df-fzo 13671  df-fl 13813  df-seq 14026  df-exp 14086  df-hash 14355  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15527  df-rlim 15528  df-sum 15726  df-0p 25786  df-ply 26302  df-idp 26303  df-coe 26304  df-dgr 26305

This theorem is referenced by:  plymulidp  26400