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Theorem plyrecj 14933

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 14905	. . . 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 9332	. . . . . . . . 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 9440	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℤ)
8	5, 7	fzfigd 10505	. . . . . . . 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 8021	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
11		snssi 3763	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {0} ⊆ ℝ
13		ssequn2 3333	. . . . . . . . . . . . . . . 16 ⊢ ({0} ⊆ ℝ ↔ (ℝ ∪ {0}) = ℝ)
14	12, 13	mpbi 145	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∪ {0}) = ℝ
15		reex 8008	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
16	14, 15	eqeltri 2266	. . . . . . . . . . . . . 14 ⊢ (ℝ ∪ {0}) ∈ V
17		nn0ex 9249	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
18	16, 17	elmap 6733	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝑎:ℕ₀⟶(ℝ ∪ {0}))
19		feq3 5389	. . . . . . . . . . . . . 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 10183	. . . . . . . . . . . 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 5695	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
26	25	recnd 8050	. . . . . . . . 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 10747	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴↑𝑘) ∈ ℂ)
29	26, 28	mulcld 8042	. . . . . . . 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 11620	. . . . . . 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 11113	. . . . . . . . 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 5695	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
36	35	cjred 11118	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝑎‘𝑘)) = (𝑎‘𝑘))
37	27, 24	cjexpd 11105	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝐴↑𝑘)) = ((∗‘𝐴)↑𝑘))
38	36, 37	oveq12d 5937	. . . . . . . . 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 2226	. . . . . . . 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 11514	. . . . . . 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 2226	. . . . . 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 5557	. . . . . . 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 2193	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))
46		oveq1 5926	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
47	46	oveq2d 5935	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · (𝐴↑𝑘)))
48	47	sumeq2sdv 11516	. . . . . . . . 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 11546	. . . . . . . . 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 5652	. . . . . . . 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 2226	. . . . . 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 5559	. . . . 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 5557	. . . . . 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 5926	. . . . . . . . . 10 ⊢ (𝑥 = (∗‘𝐴) → (𝑥↑𝑘) = ((∗‘𝐴)↑𝑘))
57	56	oveq2d 5935	. . . . . . . . 9 ⊢ (𝑥 = (∗‘𝐴) → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
58	57	sumeq2sdv 11516	. . . . . . . 8 ⊢ (𝑥 = (∗‘𝐴) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
59	49	cjcld 11087	. . . . . . . 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 10747	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗‘𝐴)↑𝑘) ∈ ℂ)
62	26, 61	mulcld 8042	. . . . . . . . 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 11546	. . . . . . . 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 5652	. . . . . . 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 2226	. . . . 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 2236	. . . 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 2619	. 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 1364 ∈ wcel 2164 ∃wrex 2473 Vcvv 2760 ∪ cun 3152 ⊆ wss 3154 {csn 3619 ↦ cmpt 4091 ⟶wf 5251 ‘cfv 5255 (class class class)co 5919 ↑_𝑚 cmap 6704 ℂcc 7872 ℝcr 7873 0cc0 7874 · cmul 7879 ℕ₀cn0 9243 ...cfz 10077 ↑cexp 10612 ∗ccj 10986 Σcsu 11499 Polycply 14899

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-map 6706 df-en 6797 df-dom 6798 df-fin 6799 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500 df-ply 14901

This theorem is referenced by: plyreres 14934

W3C validator