Theorem plyn0mulidp 26451

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 4085	. . . . 5 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹 ∈ (Poly‘ℝ))
2		ax-resscn 11161	. . . . . . 7 ⊢ ℝ ⊆ ℂ
3		1re 11212	. . . . . . 7 ⊢ 1 ∈ ℝ
4		plyid 26375	. . . . . . 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 11242	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
10	7, 8	remulcld 11243	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
11	1, 6, 9, 10	plymul 26384	. . . 4 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘f · Xp) ∈ (Poly‘ℝ))
12		0re 11214	. . . 4 ⊢ 0 ∈ ℝ
13		eqid 2763	. . . . 5 ⊢ (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(𝐹 ∘f · Xp))
14	13	coef2 26397	. . . 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 6949	. 2 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ ((coeff‘(𝐹 ∘f · Xp))‘𝑛)))
17		cnex 11185	. . . . . . . . 9 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → ℂ ∈ V)
19		plyf 26364	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
20	1, 19	syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → 𝐹:ℂ⟶ℂ)
21		plyf 26364	. . . . . . . . . 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 11234	. . . . . . . 8 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
27	18, 20, 23, 26	caofcom 7711	. . . . . . 7 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (𝐹 ∘f · Xp) = (Xp ∘f · 𝐹))
28	27	fveq2d 6885	. . . . . 6 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (coeff‘(Xp ∘f · 𝐹)))
29	28	fveq1d 6883	. . . . 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 2763	. . . . . . 7 ⊢ (coeff‘Xp) = (coeff‘Xp)
35		eqid 2763	. . . . . . 7 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
36	34, 35	coemul 26418	. . . . . 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 1398	. . . . 5 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘f · 𝐹))‘𝑛) = Σ𝑖 ∈ (0...𝑛)(((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
38		elfznn0 13653	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...𝑛) → 𝑖 ∈ ℕ0)
39		coeidp 26429	. . . . . . . . . 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 7425	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑛) → (((coeff‘Xp)‘𝑖) · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = (if(𝑖 = 1, 1, 0) · ((coeff‘𝐹)‘(𝑛 − 𝑖))))
42		ovif 7508	. . . . . . . 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 2814	. . . . . . 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 15758	. . . . 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 4605	. . . . . . . . . 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 26397	. . . . . . . . . . . . 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 13589	. . . . . . . . . . . 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 7080	. . . . . . . . . 10 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
55	54	recnd 11241	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℂ)
56	55	mullidd 11231	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (1 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = ((coeff‘𝐹)‘(𝑛 − 𝑖)))
57	55	mul02d 11412	. . . . . . . 8 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ (0...𝑛)) → (0 · ((coeff‘𝐹)‘(𝑛 − 𝑖))) = 0)
58	48, 56, 57	ifbieq12d 4516	. . . . . . 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 15758	. . . . . 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 2775	. . . . . . 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 2775	. . . . . . 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 7418	. . . . . . . . . . 11 ⊢ (𝑛 = 0 → (0...𝑛) = (0...0))
63		0z 12606	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
64		fzsn 13599	. . . . . . . . . . . 12 ⊢ (0 ∈ ℤ → (0...0) = {0})
65	63, 64	ax-mp 5	. . . . . . . . . . 11 ⊢ (0...0) = {0}
66	62, 65	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑛 = 0 → (0...𝑛) = {0})
67		elsni 4606	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ {0} → 𝑖 = 0)
68	67	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑛 = 0 ∧ 𝑖 ∈ {0}) → 𝑖 = 0)
69		ax-1ne0 11173	. . . . . . . . . . . . 13 ⊢ 1 ≠ 0
70	69	nesymi 3015	. . . . . . . . . . . 12 ⊢ ¬ 0 = 1
71		eqeq1 2767	. . . . . . . . . . . 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 4496	. . . . . . . . . . 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 15763	. . . . . . . . 9 ⊢ (𝑛 = 0 → Σ𝑖 ∈ (0...𝑛)if(𝑖 ∈ {1}, ((coeff‘𝐹)‘(𝑛 − 𝑖)), 0) = Σ𝑖 ∈ {0}0)
78		snfi 9036	. . . . . . . . . . 11 ⊢ {0} ∈ Fin
79	78	olci 879	. . . . . . . . . 10 ⊢ ({0} ⊆ (ℤ≥‘0) ∨ {0} ∈ Fin)
80		sumz 15778	. . . . . . . . . 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 2814	. . . . . . . 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 2965	. . . . . . . . 9 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ 0)
88		elnnne0 12522	. . . . . . . . 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 12524	. . . . . . . . . . . . 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 12569	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
94	92	nnge1d 12288	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ≤ 𝑛)
95		elfz2nn0 13651	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...𝑛) ↔ (1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 1 ≤ 𝑛))
96	91, 93, 94, 95	syl3anbrc 1362	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → 1 ∈ (0...𝑛))
97	96	snssd 4752	. . . . . . . . . 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 7418	. . . . . . . . . . . . . . . 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 12549	. . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → (𝑛 − 𝑖) ∈ ℕ0)
105	98, 104	ffvelcdmd 7080	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) ∧ 𝑖 ∈ {1}) → ((coeff‘𝐹)‘(𝑛 − 𝑖)) ∈ ℝ)
106	105	recnd 11241	. . . . . . . . . . 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 14013	. . . . . . . . . . . 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 15782	. . . . . . . . . 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 7080	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℝ)
116	115	recnd 11241	. . . . . . . . . 10 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ) → ((coeff‘𝐹)‘(𝑛 − 1)) ∈ ℂ)
117	99	fveq2d 6885	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((coeff‘𝐹)‘(𝑛 − 𝑖)) = ((coeff‘𝐹)‘(𝑛 − 1)))
118	117	sumsn 15802	. . . . . . . . . 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 2800	. . . . . . . 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 4526	. . . . . 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 2798	. . . . 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 2802	. . . 4 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(Xp ∘f · 𝐹))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
125	30, 124	eqtrd 2798	. . 3 ⊢ ((𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) ∧ 𝑛 ∈ ℕ0) → ((coeff‘(𝐹 ∘f · Xp))‘𝑛) = if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1))))
126	125	mpteq2dva 5204	. 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 2798	1 ⊢ (𝐹 ∈ ((Poly‘ℝ) ∖ {0𝑝}) → (coeff‘(𝐹 ∘f · Xp)) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 0, ((coeff‘𝐹)‘(𝑛 − 1)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ifcif 4487  {csn 4589   class class class wbr 5109   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672  Fincfn 8939  ℂcc 11102  ℝcr 11103  0cc0 11104  1c1 11105   · cmul 11109   ≤ cle 11248   − cmin 11445  ℕcn 12237  ℕ₀cn0 12508  ℤcz 12595  ℤ_≥cuz 12866  ...cfz 13539  Σcsu 15742  0_𝑝c0p 25837  Polycply 26350  X_pcidp 26351  coeffccoe 26352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181  ax-pre-sup 11182

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9930  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-div 11876  df-nn 12238  df-2 12307  df-3 12308  df-n0 12509  df-z 12596  df-uz 12867  df-rp 13021  df-fz 13540  df-fzo 13688  df-fl 13830  df-seq 14043  df-exp 14103  df-hash 14372  df-cj 15155  df-re 15156  df-im 15157  df-sqrt 15291  df-abs 15292  df-clim 15544  df-rlim 15545  df-sum 15743  df-0p 25838  df-ply 26354  df-idp 26355  df-coe 26356  df-dgr 26357

This theorem is used by:  plymulidp  26452