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Theorem plyrecj 15458

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 15429	. . . 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 9474	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 0 ∈ ℤ)
6		simprl 529	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℕ₀)
7	6	nn0zd 9583	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℤ)
8	5, 7	fzfigd 10670	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (0...𝑛) ∈ Fin)
9		simplrr 536	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))
10		0re 8162	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
11		snssi 3812	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {0} ⊆ ℝ
13		ssequn2 3377	. . . . . . . . . . . . . . . 16 ⊢ ({0} ⊆ ℝ ↔ (ℝ ∪ {0}) = ℝ)
14	12, 13	mpbi 145	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∪ {0}) = ℝ
15		reex 8149	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
16	14, 15	eqeltri 2302	. . . . . . . . . . . . . 14 ⊢ (ℝ ∪ {0}) ∈ V
17		nn0ex 9391	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
18	16, 17	elmap 6837	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝑎:ℕ₀⟶(ℝ ∪ {0}))
19		feq3 5461	. . . . . . . . . . . . . 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 10327	. . . . . . . . . . . 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 5776	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
26	25	recnd 8191	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℂ)
27		simpllr 534	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ ℂ)
28	27, 24	expcld 10912	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴↑𝑘) ∈ ℂ)
29	26, 28	mulcld 8183	. . . . . . . 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 12006	. . . . . . 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 11498	. . . . . . . . 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 531	. . . . . . . . . . . . . 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 5776	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
36	35	cjred 11503	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝑎‘𝑘)) = (𝑎‘𝑘))
37	27, 24	cjexpd 11490	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝐴↑𝑘)) = ((∗‘𝐴)↑𝑘))
38	36, 37	oveq12d 6028	. . . . . . . . 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 2262	. . . . . . . 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 11900	. . . . . . 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 2262	. . . . . 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 5634	. . . . . . 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 2229	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))
46		oveq1 6017	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
47	46	oveq2d 6026	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · (𝐴↑𝑘)))
48	47	sumeq2sdv 11902	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)))
49		simplr 528	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝐴 ∈ ℂ)
50	8, 29	fsumcl 11932	. . . . . . . . 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 5733	. . . . . . . 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 2262	. . . . . 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 5636	. . . . 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 5634	. . . . . 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 6017	. . . . . . . . . 10 ⊢ (𝑥 = (∗‘𝐴) → (𝑥↑𝑘) = ((∗‘𝐴)↑𝑘))
57	56	oveq2d 6026	. . . . . . . . 9 ⊢ (𝑥 = (∗‘𝐴) → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
58	57	sumeq2sdv 11902	. . . . . . . 8 ⊢ (𝑥 = (∗‘𝐴) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
59	49	cjcld 11472	. . . . . . . 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 10912	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗‘𝐴)↑𝑘) ∈ ℂ)
62	26, 61	mulcld 8183	. . . . . . . . 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 11932	. . . . . . . 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 5733	. . . . . . 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 2262	. . . . 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 2272	. . . 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 2656	. 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 1395 ∈ wcel 2200 ∃wrex 2509 Vcvv 2799 ∪ cun 3195 ⊆ wss 3197 {csn 3666 ↦ cmpt 4145 ⟶wf 5317 ‘cfv 5321 (class class class)co 6010 ↑_𝑚 cmap 6808 ℂcc 8013 ℝcr 8014 0cc0 8015 · cmul 8020 ℕ₀cn0 9385 ...cfz 10221 ↑cexp 10777 ∗ccj 11371 Σcsu 11885 Polycply 15423

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-isom 5330 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-irdg 6527 df-frec 6548 df-1o 6573 df-oadd 6577 df-er 6693 df-map 6810 df-en 6901 df-dom 6902 df-fin 6903 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-fz 10222 df-fzo 10356 df-seqfrec 10687 df-exp 10778 df-ihash 11015 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-clim 11811 df-sumdc 11886 df-ply 15425

This theorem is referenced by: plyreres 15459

W3C validator