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Theorem plyrecj 15615

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 15586	. . . 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 9585	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 0 ∈ ℤ)
6		simprl 531	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℕ₀)
7	6	nn0zd 9694	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℤ)
8	5, 7	fzfigd 10789	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → (0...𝑛) ∈ Fin)
9		simplrr 538	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))
10		0re 8270	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
11		snssi 3837	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {0} ⊆ ℝ
13		ssequn2 3391	. . . . . . . . . . . . . . . 16 ⊢ ({0} ⊆ ℝ ↔ (ℝ ∪ {0}) = ℝ)
14	12, 13	mpbi 145	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∪ {0}) = ℝ
15		reex 8257	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
16	14, 15	eqeltri 2305	. . . . . . . . . . . . . 14 ⊢ (ℝ ∪ {0}) ∈ V
17		nn0ex 9498	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
18	16, 17	elmap 6910	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝑎:ℕ₀⟶(ℝ ∪ {0}))
19		feq3 5492	. . . . . . . . . . . . . 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 10444	. . . . . . . . . . . 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 5812	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
26	25	recnd 8298	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℂ)
27		simpllr 536	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → 𝐴 ∈ ℂ)
28	27, 24	expcld 11031	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴↑𝑘) ∈ ℂ)
29	26, 28	mulcld 8290	. . . . . . . 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 12153	. . . . . . 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 11644	. . . . . . . . 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 533	. . . . . . . . . . . . . 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 5812	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
36	35	cjred 11649	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝑎‘𝑘)) = (𝑎‘𝑘))
37	27, 24	cjexpd 11636	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝐴↑𝑘)) = ((∗‘𝐴)↑𝑘))
38	36, 37	oveq12d 6067	. . . . . . . . 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 2265	. . . . . . . 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 12046	. . . . . . 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 2265	. . . . . 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 5671	. . . . . . 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 2232	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))
46		oveq1 6056	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
47	46	oveq2d 6065	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · (𝐴↑𝑘)))
48	47	sumeq2sdv 12048	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝐴↑𝑘)))
49		simplr 529	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝐴 ∈ ℂ)
50	8, 29	fsumcl 12079	. . . . . . . . 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 5770	. . . . . . . 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 2265	. . . . . 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 5673	. . . . 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 5671	. . . . . 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 6056	. . . . . . . . . 10 ⊢ (𝑥 = (∗‘𝐴) → (𝑥↑𝑘) = ((∗‘𝐴)↑𝑘))
57	56	oveq2d 6065	. . . . . . . . 9 ⊢ (𝑥 = (∗‘𝐴) → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
58	57	sumeq2sdv 12048	. . . . . . . 8 ⊢ (𝑥 = (∗‘𝐴) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
59	49	cjcld 11618	. . . . . . . 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 11031	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗‘𝐴)↑𝑘) ∈ ℂ)
62	26, 61	mulcld 8290	. . . . . . . . 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 12079	. . . . . . . 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 5770	. . . . . . 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 2265	. . . . 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 2275	. . . 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 2668	. 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 1398 ∈ wcel 2203 ∃wrex 2521 Vcvv 2812 ∪ cun 3208 ⊆ wss 3210 {csn 3688 ↦ cmpt 4170 ⟶wf 5347 ‘cfv 5351 (class class class)co 6049 ↑_𝑚 cmap 6881 ℂcc 8121 ℝcr 8122 0cc0 8123 · cmul 8128 ℕ₀cn0 9492 ...cfz 10338 ↑cexp 10896 ∗ccj 11517 Σcsu 12031 Polycply 15580

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-isom 5360 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-frec 6621 df-1o 6646 df-oadd 6650 df-er 6766 df-map 6883 df-en 6975 df-dom 6976 df-fin 6977 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-fz 10339 df-fzo 10473 df-seqfrec 10806 df-exp 10897 df-ihash 11134 df-cj 11520 df-re 11521 df-im 11522 df-rsqrt 11676 df-abs 11677 df-clim 11957 df-sumdc 12032 df-ply 15582

This theorem is referenced by: plyreres 15616

W3C validator