Theorem plyn0mulidp 26332

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 4082	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2		ax-resscn 11123	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3		1re 11174	. . . . . . 7 ⊢ 1 ∈ ℝ
4		plyid 26256	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 1 ∈ ℝ) → Xp ∈ (Poly‘ℝ))
5	2, 3, 4	mp2an 702	. . . . . 6 ⊢ Xp ∈ (Poly‘ℝ)
6	5	a1i 11	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → Xp ∈ (Poly‘ℝ))
7		simprl 780	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑥 ∈ ℝ)
8		simprr 782	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → 𝑦 ∈ ℝ)
9	7, 8	readdcld 11204	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
10	7, 8	remulcld 11205	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
11	1, 6, 9, 10	plymul 26265	. . . 4 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘f · Xp) ∈ (Poly‘ℝ))
12		0re 11176	. . . 4 ⊢ 0 ∈ ℝ
13		eqid 2761	. . . . 5 ⊢ (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(𝐹 ∘f · Xp))
14	13	coef2 26278	. . . 4 ⊢ (((𝐹 ∘f · Xp) ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘(𝐹 ∘f · Xp)):ℕ0⟶ℝ)
15	11, 12, 14	sylancl 595	. . 3 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)):ℕ0⟶ℝ)
16	15	feqmptd 6929	. 2 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹 ∘f · Xp))‘𝑛)))
17		cnex 11147	. . . . . . . . 9 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19		plyf 26245	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
20	1, 19	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21		plyf 26245	. . . . . . . . . 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 780	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑥 ∈ ℂ)
25		simprr 782	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → 𝑦 ∈ ℂ)
26	24, 25	mulcomd 11196	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
27	18, 20, 23, 26	caofcom 7691	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘f · Xp) = (Xp ∘f · 𝐹))
28	27	fveq2d 6865	. . . . . 6 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(Xp ∘f · 𝐹)))
29	28	fveq1d 6863	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ((coeff‘(𝐹 ∘f · Xp))‘𝑛) = ((coeff‘(Xp ∘f · 𝐹))‘𝑛))
30	29	adantr 484	. . . 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 484	. . . . . 6 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → 𝐹 ∈ (Poly‘ℝ))
33		simpr 488	. . . . . 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)
36	34, 35	coemul 26299	. . . . . 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 1389	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘f · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
38		elfznn0 13618	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39		coeidp 26310	. . . . . . . . . 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 7405	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
42		ovif 7488	. . . . . . . 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 2812	. . . . . . 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 485	. . . . . 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 15719	. . . . 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 4595	. . . . . . . . . 10 ⊢ (𝑖 ∈ {1} ↔ 𝑖 = 1)
47	46	bicomi 226	. . . . . . . . 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 26278	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ)
50	1, 12, 49	sylancl 595	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘𝐹):ℕ0⟶ℝ)
51	50	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (coeff‘𝐹):ℕ0⟶ℝ)
52		fznn0sub 13554	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑛) → (𝑛 − 𝑖) ∈ ℕ0)
53	52	adantl 485	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (𝑛 − 𝑖) ∈ ℕ0)
54	51, 53	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
55	54	recnd 11203	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
56	55	mullidd 11193	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = ((coeff‘𝐹)‘(𝑛 − 𝑖)))
57	55	mul02d 11374	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = 0)
58	48, 56, 57	ifbieq12d 4506	. . . . . . 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 15719	. . . . . 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 2773	. . . . . . 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 2773	. . . . . . 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 7398	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
63		0z 12572	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
64		fzsn 13564	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → (0...0) = {0})
65	63, 64	ax-mp 5	. . . . . . . . . . 11 ⊢ (0...0) = {0}
66	62, 65	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (0...𝑛) = {0})
67		elsni 4596	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ {0} → 𝑖 = 0)
68	67	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69		ax-1ne0 11135	. . . . . . . . . . . . 13 ⊢ 1 ≠ 0
70	69	nesymi 3013	. . . . . . . . . . . 12 ⊢ ¬ 0 = 1
71		eqeq1 2765	. . . . . . . . . . . 12 ⊢ (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
72	70, 71	mtbiri 329	. . . . . . . . . . 11 ⊢ (𝑖 = 0 → ¬ 𝑖 = 1)
73	47	notbii 322	. . . . . . . . . . . 12 ⊢ (¬ 𝑖 = 1 ↔ ¬ 𝑖 ∈ {1})
74	73	biimpi 218	. . . . . . . . . . 11 ⊢ (¬ 𝑖 = 1 → ¬ 𝑖 ∈ {1})
75		iffalse 4486	. . . . . . . . . . 11 ⊢ (¬ 𝑖 ∈ {1} → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
76	68, 72, 74, 75	4syl 19	. . . . . . . . . 10 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
77	66, 76	sumeq12rdv 15724	. . . . . . . . 9 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = Σ𝑖 ∈ {0}0)
78		snfi 9017	. . . . . . . . . . 11 ⊢ {0} ∈ Fin
79	78	olci 877	. . . . . . . . . 10 ⊢ ({0} ⊆ (ℤ≥‘0) ∨ {0} ∈ Fin)
80		sumz 15739	. . . . . . . . . 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 2812	. . . . . . . 8 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
83	82	adantl 485	. . . . . . 7 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = 0) → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = 0)
84		simpll 776	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}))
85	33	adantr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ0)
86		simpr 488	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = 0)
87	86	neqned 2963	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
88		elnnne0 12488	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ0 ∧ 𝑛 ≠ 0))
89	85, 87, 88	sylanbrc 592	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ∈ ℕ)
90		1nn0 12490	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ0
91	90	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ ℕ0)
92		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
93	92	nnnn0d 12535	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
94	92	nnge1d 12254	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
95		elfz2nn0 13616	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
96	91, 93, 94, 95	syl3anbrc 1356	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
97	96	snssd 4742	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → {1} ⊆ (0...𝑛))
98	50	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (coeff‘𝐹):ℕ0⟶ℝ)
99		oveq2 7398	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → (𝑛 − 𝑖) = (𝑛 − 1))
100	46, 99	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ {1} → (𝑛 − 𝑖) = (𝑛 − 1))
101	100	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) = (𝑛 − 1))
102		nnm1nn0 12515	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
103	102	ad2antlr 737	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 1) ∈ ℕ0)
104	101, 103	eqeltrd 2861	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) ∈ ℕ0)
105	98, 104	ffvelcdmd 7060	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
106	105	recnd 11203	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
107	106	ralrimiva 3153	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ∀𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
108		fzfi 13978	. . . . . . . . . . . 12 ⊢ (0...𝑛) ∈ Fin
109	108	olci 877	. . . . . . . . . . 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 15743	. . . . . . . . . 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 848	. . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (coeff‘𝐹):ℕ0⟶ℝ)
114	102	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → (𝑛 − 1) ∈ ℕ0)
115	113, 114	ffvelcdmd 7060	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
116	115	recnd 11203	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
117	99	fveq2d 6865	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
118	117	sumsn 15763	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
119	3, 116, 118	sylancr 596	. . . . . . . . 9 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → Σ𝑖 ∈ {1} ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
120	112, 119	eqtr3d 2798	. . . . . . . 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 593	. . . . . . 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 4516	. . . . . 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 2796	. . . . 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 2800	. . . 4 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘f · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
125	30, 124	eqtrd 2796	. . 3 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹 ∘f · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
126	125	mpteq2dva 5190	. 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 2796	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 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ifcif 4477  {csn 4579   class class class wbr 5097   ↦ cmpt 5178  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390   ∘_f cof 7652  Fincfn 8920  ℂcc 11064  ℝcr 11065  0cc0 11066  1c1 11067   · cmul 11071   ≤ cle 11210   − cmin 11407  ℕcn 12203  ℕ₀cn0 12474  ℤcz 12561  ℤ_≥cuz 12832  ...cfz 13505  Σcsu 15703  0_𝑝c0p 25718  Polycply 26231  X_pcidp 26232  coeffccoe 26233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-inf2 9589  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143  ax-pre-sup 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-isom 6524  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7654  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-pm 8804  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-sup 9381  df-inf 9382  df-oi 9451  df-card 9890  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-2 12273  df-3 12274  df-n0 12475  df-z 12562  df-uz 12833  df-rp 12987  df-fz 13506  df-fzo 13653  df-fl 13795  df-seq 14008  df-exp 14068  df-hash 14337  df-cj 15116  df-re 15117  df-im 15118  df-sqrt 15252  df-abs 15253  df-clim 15505  df-rlim 15506  df-sum 15704  df-0p 25719  df-ply 26235  df-idp 26236  df-coe 26237  df-dgr 26238

This theorem is referenced by:  plymulidp  26333