bpoly2 - Metamath Proof Explorer

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Theorem bpoly2 16111

Description: The Bernoulli polynomials at two. (Contributed by Scott Fenton, 8-Jul-2015.)

**Assertion**
Ref	Expression
bpoly2	⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Proof of Theorem bpoly2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 12521	. . 3 ⊢ 2 ∈ ℕ₀
2		bpolyval 16103	. . 3 ⊢ ((2 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (2 BernPoly 𝑋) = ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))))
3	1, 2	mpan 702	. 2 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))))
4		2m1e1 12365	. . . . . . 7 ⊢ (2 − 1) = 1
5		0p1e1 12361	. . . . . . 7 ⊢ (0 + 1) = 1
6	4, 5	eqtr4i 2795	. . . . . 6 ⊢ (2 − 1) = (0 + 1)
7	6	oveq2i 7422	. . . . 5 ⊢ (0...(2 − 1)) = (0...(0 + 1))
8	7	sumeq1i 15748	. . . 4 ⊢ Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))
9		0nn0 12519	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
10		nn0uz 12900	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtri 2867	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
12	11	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
13		0z 12602	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
14		fzpr 13607	. . . . . . . . . . 11 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
15	13, 14	ax-mp 5	. . . . . . . . . 10 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
16	15	eleq2i 2861	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, (0 + 1)})
17		vex 3467	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
18	17	elpr 4619	. . . . . . . . 9 ⊢ (𝑘 ∈ {0, (0 + 1)} ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
19	16, 18	bitri 278	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
20		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (2C𝑘) = (2C0))
21		bcn0 14346	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℕ₀ → (2C0) = 1)
22	1, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2C0) = 1
23	20, 22	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (2C𝑘) = 1)
24		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
25		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (2 − 𝑘) = (2 − 0))
26	25	oveq1d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = ((2 − 0) + 1))
27		2cn 12316	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
28	27	subid1i 11530	. . . . . . . . . . . . . . . 16 ⊢ (2 − 0) = 2
29	28	oveq1i 7421	. . . . . . . . . . . . . . 15 ⊢ ((2 − 0) + 1) = (2 + 1)
30		df-3 12304	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
31	29, 30	eqtr4i 2795	. . . . . . . . . . . . . 14 ⊢ ((2 − 0) + 1) = 3
32	26, 31	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = 3)
33	24, 32	oveq12d 7429	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 3))
34	23, 33	oveq12d 7429	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
35		bpoly0 16104	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
36	35	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 3) = (1 / 3))
37	36	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 · (1 / 3)))
38		3cn 12322	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
39		3ne0 12350	. . . . . . . . . . . . . 14 ⊢ 3 ≠ 0
40	38, 39	reccli 11945	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℂ
41	40	mullidi 11214	. . . . . . . . . . . 12 ⊢ (1 · (1 / 3)) = (1 / 3)
42	37, 41	eqtrdi 2820	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 / 3))
43	34, 42	sylan9eqr 2826	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
44	43, 40	eqeltrdi 2877	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
45	5	eqeq2i 2782	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
46		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (2C𝑘) = (2C1))
47		bcn1 14349	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ₀ → (2C1) = 2)
48	1, 47	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2C1) = 2
49	46, 48	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (2C𝑘) = 2)
50		oveq1 7418	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
51		oveq2 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2 − 𝑘) = (2 − 1))
52	51	oveq1d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = ((2 − 1) + 1))
53		ax-1cn 11158	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
54		npcan 11466	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ) → ((2 − 1) + 1) = 2)
55	27, 53, 54	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ ((2 − 1) + 1) = 2
56	52, 55	eqtrdi 2820	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = 2)
57	50, 56	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 2))
58	49, 57	oveq12d 7429	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
59	45, 58	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
60		bpoly1 16105	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
61	60	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 2) = ((𝑋 − (1 / 2)) / 2))
62	61	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (2 · ((𝑋 − (1 / 2)) / 2)))
63		halfcn 12458	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℂ
64		subcl 11456	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
65	63, 64	mpan2 703	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
66		2ne0 12347	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
67		divcan2 11880	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
68	27, 66, 67	mp3an23 1479	. . . . . . . . . . . . 13 ⊢ ((𝑋 − (1 / 2)) ∈ ℂ → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
69	65, 68	syl 18	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
70	62, 69	eqtrd 2804	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (𝑋 − (1 / 2)))
71	59, 70	sylan9eqr 2826	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
72	65	adantr 485	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → (𝑋 − (1 / 2)) ∈ ℂ)
73	71, 72	eqeltrd 2869	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
74	44, 73	jaodan 972	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = (0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
75	19, 74	sylan2b 605	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
76	12, 75, 59	fsump1 15807	. . . . . 6 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) + (2 · ((1 BernPoly 𝑋) / 2))))
77	42, 40	eqeltrdi 2877	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) ∈ ℂ)
78	34	fsum1 15798	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (1 · ((0 BernPoly 𝑋) / 3)) ∈ ℂ) → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
79	13, 77, 78	sylancr 598	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
80	79, 42	eqtrd 2804	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
81	80, 70	oveq12d 7429	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) + (2 · ((1 BernPoly 𝑋) / 2))) = ((1 / 3) + (𝑋 − (1 / 2))))
82	76, 81	eqtrd 2804	. . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = ((1 / 3) + (𝑋 − (1 / 2))))
83		addsub12 11470	. . . . . . 7 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
84	40, 63, 83	mp3an13 1478	. . . . . 6 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
85	63, 40	negsubdi2i 11544	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = ((1 / 3) − (1 / 2))
86		halfthird 12465	. . . . . . . . 9 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
87	86	negeqi 11450	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = -(1 / 6)
88	85, 87	eqtr3i 2794	. . . . . . 7 ⊢ ((1 / 3) − (1 / 2)) = -(1 / 6)
89	88	oveq2i 7422	. . . . . 6 ⊢ (𝑋 + ((1 / 3) − (1 / 2))) = (𝑋 + -(1 / 6))
90	84, 89	eqtrdi 2820	. . . . 5 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + -(1 / 6)))
91		6cn 12332	. . . . . . 7 ⊢ 6 ∈ ℂ
92		6re 12331	. . . . . . . 8 ⊢ 6 ∈ ℝ
93		6pos 12354	. . . . . . . 8 ⊢ 0 < 6
94	92, 93	gt0ne0ii 11750	. . . . . . 7 ⊢ 6 ≠ 0
95	91, 94	reccli 11945	. . . . . 6 ⊢ (1 / 6) ∈ ℂ
96		negsub 11506	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
97	95, 96	mpan2 703	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
98	82, 90, 97	3eqtrd 2808	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
99	8, 98	eqtrid 2816	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
100	99	oveq2d 7427	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))) = ((𝑋↑2) − (𝑋 − (1 / 6))))
101		sqcl 14154	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
102		subsub 11488	. . . 4 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
103	95, 102	mp3an3 1476	. . 3 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
104	101, 103	mpancom 700	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
105	3, 100, 104	3eqtrd 2808	1 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 {cpr 4596 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 − cmin 11441 -cneg 11442 / cdiv 11871 2c2 12295 3c3 12296 6c6 12299 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ...cfz 13535 ↑cexp 14097 Ccbc 14338 Σcsu 15737 BernPoly cbp 16100

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-bpoly 16101

This theorem is referenced by: bpoly3 16112 bpoly4 16113

W3C validator