bpoly2 - Metamath Proof Explorer

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Theorem bpoly2 16073

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 12518	. . 3 ⊢ 2 ∈ ℕ₀
2		bpolyval 16065	. . 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 690	. 2 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))))
4		2m1e1 12366	. . . . . . 7 ⊢ (2 − 1) = 1
5		0p1e1 12362	. . . . . . 7 ⊢ (0 + 1) = 1
6	4, 5	eqtr4i 2761	. . . . . 6 ⊢ (2 − 1) = (0 + 1)
7	6	oveq2i 7416	. . . . 5 ⊢ (0...(2 − 1)) = (0...(0 + 1))
8	7	sumeq1i 15713	. . . 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 12516	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
10		nn0uz 12894	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtri 2832	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
12	11	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
13		0z 12599	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
14		fzpr 13596	. . . . . . . . . . 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 2826	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, (0 + 1)})
17		vex 3463	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
18	17	elpr 4626	. . . . . . . . 9 ⊢ (𝑘 ∈ {0, (0 + 1)} ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
19	16, 18	bitri 275	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
20		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (2C𝑘) = (2C0))
21		bcn0 14328	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℕ₀ → (2C0) = 1)
22	1, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2C0) = 1
23	20, 22	eqtrdi 2786	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (2C𝑘) = 1)
24		oveq1 7412	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
25		oveq2 7413	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (2 − 𝑘) = (2 − 0))
26	25	oveq1d 7420	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = ((2 − 0) + 1))
27		2cn 12315	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
28	27	subid1i 11555	. . . . . . . . . . . . . . . 16 ⊢ (2 − 0) = 2
29	28	oveq1i 7415	. . . . . . . . . . . . . . 15 ⊢ ((2 − 0) + 1) = (2 + 1)
30		df-3 12304	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
31	29, 30	eqtr4i 2761	. . . . . . . . . . . . . 14 ⊢ ((2 − 0) + 1) = 3
32	26, 31	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = 3)
33	24, 32	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 3))
34	23, 33	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
35		bpoly0 16066	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
36	35	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 3) = (1 / 3))
37	36	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 · (1 / 3)))
38		3cn 12321	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
39		3ne0 12346	. . . . . . . . . . . . . 14 ⊢ 3 ≠ 0
40	38, 39	reccli 11971	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℂ
41	40	mullidi 11240	. . . . . . . . . . . 12 ⊢ (1 · (1 / 3)) = (1 / 3)
42	37, 41	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 / 3))
43	34, 42	sylan9eqr 2792	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
44	43, 40	eqeltrdi 2842	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
45	5	eqeq2i 2748	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
46		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (2C𝑘) = (2C1))
47		bcn1 14331	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ₀ → (2C1) = 2)
48	1, 47	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2C1) = 2
49	46, 48	eqtrdi 2786	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (2C𝑘) = 2)
50		oveq1 7412	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
51		oveq2 7413	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2 − 𝑘) = (2 − 1))
52	51	oveq1d 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = ((2 − 1) + 1))
53		ax-1cn 11187	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
54		npcan 11491	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ) → ((2 − 1) + 1) = 2)
55	27, 53, 54	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ ((2 − 1) + 1) = 2
56	52, 55	eqtrdi 2786	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = 2)
57	50, 56	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 2))
58	49, 57	oveq12d 7423	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
59	45, 58	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
60		bpoly1 16067	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
61	60	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 2) = ((𝑋 − (1 / 2)) / 2))
62	61	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (2 · ((𝑋 − (1 / 2)) / 2)))
63		halfcn 12455	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℂ
64		subcl 11481	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
65	63, 64	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
66		2ne0 12344	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
67		divcan2 11904	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
68	27, 66, 67	mp3an23 1455	. . . . . . . . . . . . 13 ⊢ ((𝑋 − (1 / 2)) ∈ ℂ → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
69	65, 68	syl 17	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
70	62, 69	eqtrd 2770	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (𝑋 − (1 / 2)))
71	59, 70	sylan9eqr 2792	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
72	65	adantr 480	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → (𝑋 − (1 / 2)) ∈ ℂ)
73	71, 72	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
74	44, 73	jaodan 959	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = (0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
75	19, 74	sylan2b 594	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
76	12, 75, 59	fsump1 15772	. . . . . 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 2842	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) ∈ ℂ)
78	34	fsum1 15763	. . . . . . . . 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 587	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
80	79, 42	eqtrd 2770	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
81	80, 70	oveq12d 7423	. . . . . 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 2770	. . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = ((1 / 3) + (𝑋 − (1 / 2))))
83		addsub12 11495	. . . . . . 7 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
84	40, 63, 83	mp3an13 1454	. . . . . 6 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
85	63, 40	negsubdi2i 11569	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = ((1 / 3) − (1 / 2))
86		halfthird 12462	. . . . . . . . 9 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
87	86	negeqi 11475	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = -(1 / 6)
88	85, 87	eqtr3i 2760	. . . . . . 7 ⊢ ((1 / 3) − (1 / 2)) = -(1 / 6)
89	88	oveq2i 7416	. . . . . 6 ⊢ (𝑋 + ((1 / 3) − (1 / 2))) = (𝑋 + -(1 / 6))
90	84, 89	eqtrdi 2786	. . . . 5 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + -(1 / 6)))
91		6cn 12331	. . . . . . 7 ⊢ 6 ∈ ℂ
92		6re 12330	. . . . . . . 8 ⊢ 6 ∈ ℝ
93		6pos 12350	. . . . . . . 8 ⊢ 0 < 6
94	92, 93	gt0ne0ii 11773	. . . . . . 7 ⊢ 6 ≠ 0
95	91, 94	reccli 11971	. . . . . 6 ⊢ (1 / 6) ∈ ℂ
96		negsub 11531	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
97	95, 96	mpan2 691	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
98	82, 90, 97	3eqtrd 2774	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
99	8, 98	eqtrid 2782	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
100	99	oveq2d 7421	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))) = ((𝑋↑2) − (𝑋 − (1 / 6))))
101		sqcl 14136	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
102		subsub 11513	. . . 4 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
103	95, 102	mp3an3 1452	. . 3 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
104	101, 103	mpancom 688	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
105	3, 100, 104	3eqtrd 2774	1 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 {cpr 4603 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 − cmin 11466 -cneg 11467 / cdiv 11894 2c2 12295 3c3 12296 6c6 12299 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13524 ↑cexp 14079 Ccbc 14320 Σcsu 15702 BernPoly cbp 16062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703 df-bpoly 16063

This theorem is referenced by: bpoly3 16074 bpoly4 16075

W3C validator