plyrecj - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > plyrecj		GIF version

Theorem plyrecj 15279

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 15250	. . . 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 9391	. . . . . . . . 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 9500	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑛 ∈ ℤ)
8	5, 7	fzfigd 10583	. . . . . . . 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 8079	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
11		snssi 3779	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {0} ⊆ ℝ
13		ssequn2 3347	. . . . . . . . . . . . . . . 16 ⊢ ({0} ⊆ ℝ ↔ (ℝ ∪ {0}) = ℝ)
14	12, 13	mpbi 145	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∪ {0}) = ℝ
15		reex 8066	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
16	14, 15	eqeltri 2279	. . . . . . . . . . . . . 14 ⊢ (ℝ ∪ {0}) ∈ V
17		nn0ex 9308	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
18	16, 17	elmap 6771	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀) ↔ 𝑎:ℕ₀⟶(ℝ ∪ {0}))
19		feq3 5416	. . . . . . . . . . . . . 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 10243	. . . . . . . . . . . 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 5723	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
26	25	recnd 8108	. . . . . . . . 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 10825	. . . . . . . . 9 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝐴↑𝑘) ∈ ℂ)
29	26, 28	mulcld 8100	. . . . . . . 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 11829	. . . . . . 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 11321	. . . . . . . . 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 5723	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (𝑎‘𝑘) ∈ ℝ)
36	35	cjred 11326	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝑎‘𝑘)) = (𝑎‘𝑘))
37	27, 24	cjexpd 11313	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → (∗‘(𝐴↑𝑘)) = ((∗‘𝐴)↑𝑘))
38	36, 37	oveq12d 5969	. . . . . . . . 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 2239	. . . . . . . 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 11723	. . . . . . 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 2239	. . . . . 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 5585	. . . . . . 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 2206	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)))
46		oveq1 5958	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
47	46	oveq2d 5967	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · (𝐴↑𝑘)))
48	47	sumeq2sdv 11725	. . . . . . . . 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 11755	. . . . . . . . 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 5680	. . . . . . . 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 2239	. . . . . 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 5587	. . . . 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 5585	. . . . . 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 5958	. . . . . . . . . 10 ⊢ (𝑥 = (∗‘𝐴) → (𝑥↑𝑘) = ((∗‘𝐴)↑𝑘))
57	56	oveq2d 5967	. . . . . . . . 9 ⊢ (𝑥 = (∗‘𝐴) → ((𝑎‘𝑘) · (𝑥↑𝑘)) = ((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
58	57	sumeq2sdv 11725	. . . . . . . 8 ⊢ (𝑥 = (∗‘𝐴) → Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · ((∗‘𝐴)↑𝑘)))
59	49	cjcld 11295	. . . . . . . 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 10825	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ (𝑛 ∈ ℕ₀ ∧ 𝑎 ∈ ((ℝ ∪ {0}) ↑_𝑚 ℕ₀))) ∧ 𝑘 ∈ (0...𝑛)) → ((∗‘𝐴)↑𝑘) ∈ ℂ)
62	26, 61	mulcld 8100	. . . . . . . . 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 11755	. . . . . . . 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 5680	. . . . . . 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 2239	. . . . 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 2249	. . . 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 2632	. 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 1373 ∈ wcel 2177 ∃wrex 2486 Vcvv 2773 ∪ cun 3165 ⊆ wss 3167 {csn 3634 ↦ cmpt 4109 ⟶wf 5272 ‘cfv 5276 (class class class)co 5951 ↑_𝑚 cmap 6742 ℂcc 7930 ℝcr 7931 0cc0 7932 · cmul 7937 ℕ₀cn0 9302 ...cfz 10137 ↑cexp 10690 ∗ccj 11194 Σcsu 11708 Polycply 15244

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051 ax-caucvg 8052

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-isom 5285 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-irdg 6463 df-frec 6484 df-1o 6509 df-oadd 6513 df-er 6627 df-map 6744 df-en 6835 df-dom 6836 df-fin 6837 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-3 9103 df-4 9104 df-n0 9303 df-z 9380 df-uz 9656 df-q 9748 df-rp 9783 df-fz 10138 df-fzo 10272 df-seqfrec 10600 df-exp 10691 df-ihash 10928 df-cj 11197 df-re 11198 df-im 11199 df-rsqrt 11353 df-abs 11354 df-clim 11634 df-sumdc 11709 df-ply 15246

This theorem is referenced by: plyreres 15280

W3C validator