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Theorem plyrecj 15847

Description: A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)

**Assertion**
Ref	Expression
plyrecj	⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴)))

Proof of Theorem plyrecj Dummy variables 𝑎 𝑘 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → 𝐹 ∈ (Poly‘ℝ))
2		elply 15818	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) ↔ (ℝ ⊆ ℂ ∧ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀)𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))))
3	1, 2	sylib 122	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (ℝ ⊆ ℂ ∧ ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀)𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))))
4	3	simprd 114	. 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → ∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀)𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))))
5		0zd 9639	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 0 ∈ ℤ)
6		simprl 535	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℕ₀)
7	6	nn0zd 9749	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℤ)
8	5, 7	fzfigd 10851	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (0...𝑛) ∈ Fin)
9		simplrr 542	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))
10		0re 8320	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
11		snssi 3857	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {0} ⊆ ℝ
13		ssequn2 3402	. . . . . . . . . . . . . . . 16 ⊢ ({0} ⊆ ℝ ↔ (ℝ ∪ {0}) = ℝ)
14	12, 13	mpbi 145	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∪ {0}) = ℝ
15		reex 8307	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
16	14, 15	eqeltri 2311	. . . . . . . . . . . . . 14 ⊢ (ℝ ∪ {0}) ∈ V
17		nn0ex 9552	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
18	16, 17	elmap 6952	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝑎:ℕ₀⟶(ℝ ∪ {0}))
19		feq3 5516	. . . . . . . . . . . . . 14 ⊢ ((ℝ ∪ {0}) = ℝ → (𝑎:ℕ₀⟶(ℝ ∪ {0}) ↔ 𝑎:ℕ₀⟶ℝ))
20	14, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑎:ℕ₀⟶(ℝ ∪ {0}) ↔ 𝑎:ℕ₀⟶ℝ)
21	18, 20	bitri 184	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝑎:ℕ₀⟶ℝ)
22	9, 21	sylib 122	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑎:ℕ₀⟶ℝ)
23		elfznn0 10504	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...𝑛) → 𝑘 ∈ ℕ₀)
24	23	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑘 ∈ ℕ₀)
25	22, 24	ffvelcdmd 5838	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
26	25	recnd 8348	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℂ)
27		simpllr 540	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ ℂ)
28	27, 24	expcld 11094	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴↑𝑘) ∈ ℂ)
29	26, 28	mulcld 8340	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎‘𝑘) · (𝐴↑𝑘)) ∈ ℂ)
30	8, 29	fsumcj 12224	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (∗‘Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘))) = Σ𝑘 ∈ (0...𝑛)(∗‘((𝑎‘𝑘) · (𝐴↑𝑘))))
31	26, 28	cjmuld 11715	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘((𝑎‘𝑘) · (𝐴↑𝑘))) = ((∗‘(𝑎‘𝑘)) · (∗‘(𝐴↑𝑘))))
32		simprr 537	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))
33	32, 21	sylib 122	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑎:ℕ₀⟶ℝ)
34	33	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑎:ℕ₀⟶ℝ)
35	34, 24	ffvelcdmd 5838	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
36	35	cjred 11720	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝑎‘𝑘)) = (𝑎‘𝑘))
37	27, 24	cjexpd 11707	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝐴↑𝑘)) = ((∗‘𝐴)↑𝑘))
38	36, 37	oveq12d 6097	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗‘(𝑎‘𝑘)) · (∗‘(𝐴↑𝑘))) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
39	31, 38	eqtrd 2271	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘((𝑎‘𝑘) · (𝐴↑𝑘))) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
40	39	sumeq2dv 12117	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → Σ𝑘 ∈ (0...𝑛)(∗‘((𝑎‘𝑘) · (𝐴↑𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
41	30, 40	eqtrd 2271	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (∗‘Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
42	41	adantr 276	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (∗‘Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘))) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
43		simpr 110	. . . . . . . 8 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))))
44	43	fveq1d 5695	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (𝐹‘𝐴) = ((𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))‘𝐴))
45		eqid 2238	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))
46		oveq1 6086	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
47	46	oveq2d 6095	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · (𝐴↑𝑘)))
48	47	sumeq2sdv 12119	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)))
49		simplr 533	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝐴 ∈ ℂ)
50	8, 29	fsumcl 12150	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)) ∈ ℂ)
51	45, 48, 49, 50	fvmptd3 5796	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → ((𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)))
52	51	adantr 276	. . . . . . 7 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → ((𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)))
53	44, 52	eqtrd 2271	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (𝐹‘𝐴) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)))
54	53	fveq2d 5697	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (∗‘(𝐹‘𝐴)) = (∗‘Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘))))
55	43	fveq1d 5695	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (𝐹‘(∗‘𝐴)) = ((𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))‘(∗‘𝐴)))
56		oveq1 6086	. . . . . . . . . 10 ⊢ (𝑥 = (∗‘𝐴) → (𝑥↑𝑘) = ((∗‘𝐴)↑𝑘))
57	56	oveq2d 6095	. . . . . . . . 9 ⊢ (𝑥 = (∗‘𝐴) → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
58	57	sumeq2sdv 12119	. . . . . . . 8 ⊢ (𝑥 = (∗‘𝐴) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
59	49	cjcld 11689	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (∗‘𝐴) ∈ ℂ)
60	59	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘𝐴) ∈ ℂ)
61	60, 24	expcld 11094	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗‘𝐴)↑𝑘) ∈ ℂ)
62	26, 61	mulcld 8340	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)) ∈ ℂ)
63	8, 62	fsumcl 12150	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)) ∈ ℂ)
64	45, 58, 59, 63	fvmptd3 5796	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → ((𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))‘(∗‘𝐴)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
65	64	adantr 276	. . . . . 6 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → ((𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))‘(∗‘𝐴)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
66	55, 65	eqtrd 2271	. . . . 5 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (𝐹‘(∗‘𝐴)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
67	42, 54, 66	3eqtr4d 2281	. . . 4 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴)))
68	67	ex 115	. . 3 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴))))
69	68	rexlimdvva 2676	. 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∃𝑛 ∈ ℕ₀ ∃𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀)𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴))))
70	4, 69	mpd 13	1 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 Vcvv 2821 ∪ cun 3218 ⊆ wss 3220 {csn 3708 ↦ cmpt 4190 ⟶wf 5371 ‘cfv 5375 (class class class)co 6079 ↑_𝑚 cmap 6916 ℂcc 8171 ℝcr 8172 0cc0 8173 · cmul 8178 ℕ₀cn0 9546 ...cfz 10394 ↑cexp 10958 ∗ccj 11587 Σcsu 12102 Polycply 15812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-isom 5384 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-irdg 6635 df-frec 6656 df-1o 6681 df-oadd 6685 df-er 6801 df-map 6918 df-en 7017 df-dom 7018 df-fin 7019 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-fz 10395 df-fzo 10533 df-seqfrec 10868 df-exp 10959 df-ihash 11198 df-cj 11590 df-re 11591 df-im 11592 df-rsqrt 11747 df-abs 11748 df-clim 12028 df-sumdc 12103 df-ply 15814

This theorem is referenced by: plyreres 15848

W3C validator