bpoly2 - Metamath Proof Explorer

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Theorem bpoly2 14994

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 11511	. . 3 ⊢ 2 ∈ ℕ₀
2		bpolyval 14986	. . 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 670	. 2 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))))
4		2m1e1 11337	. . . . . . 7 ⊢ (2 − 1) = 1
5		0p1e1 11334	. . . . . . 7 ⊢ (0 + 1) = 1
6	4, 5	eqtr4i 2796	. . . . . 6 ⊢ (2 − 1) = (0 + 1)
7	6	oveq2i 6804	. . . . 5 ⊢ (0...(2 − 1)) = (0...(0 + 1))
8	7	sumeq1i 14636	. . . 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 11509	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
10		nn0uz 11924	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtri 2848	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
12	11	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
13		0z 11590	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
14		fzpr 12603	. . . . . . . . . . 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 2842	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, (0 + 1)})
17		vex 3354	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
18	17	elpr 4338	. . . . . . . . 9 ⊢ (𝑘 ∈ {0, (0 + 1)} ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
19	16, 18	bitri 264	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
20		oveq2 6801	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (2C𝑘) = (2C0))
21		bcn0 13301	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℕ₀ → (2C0) = 1)
22	1, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2C0) = 1
23	20, 22	syl6eq 2821	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (2C𝑘) = 1)
24		oveq1 6800	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
25		oveq2 6801	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (2 − 𝑘) = (2 − 0))
26	25	oveq1d 6808	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = ((2 − 0) + 1))
27		2cn 11293	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
28	27	subid1i 10555	. . . . . . . . . . . . . . . 16 ⊢ (2 − 0) = 2
29	28	oveq1i 6803	. . . . . . . . . . . . . . 15 ⊢ ((2 − 0) + 1) = (2 + 1)
30		df-3 11282	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
31	29, 30	eqtr4i 2796	. . . . . . . . . . . . . 14 ⊢ ((2 − 0) + 1) = 3
32	26, 31	syl6eq 2821	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = 3)
33	24, 32	oveq12d 6811	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 3))
34	23, 33	oveq12d 6811	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
35		bpoly0 14987	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
36	35	oveq1d 6808	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 3) = (1 / 3))
37	36	oveq2d 6809	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 · (1 / 3)))
38		3cn 11297	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
39		3ne0 11317	. . . . . . . . . . . . . 14 ⊢ 3 ≠ 0
40	38, 39	reccli 10957	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℂ
41	40	mulid2i 10245	. . . . . . . . . . . 12 ⊢ (1 · (1 / 3)) = (1 / 3)
42	37, 41	syl6eq 2821	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 / 3))
43	34, 42	sylan9eqr 2827	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
44	43, 40	syl6eqel 2858	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
45	5	eqeq2i 2783	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
46		oveq2 6801	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (2C𝑘) = (2C1))
47		bcn1 13304	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ₀ → (2C1) = 2)
48	1, 47	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2C1) = 2
49	46, 48	syl6eq 2821	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (2C𝑘) = 2)
50		oveq1 6800	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
51		oveq2 6801	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2 − 𝑘) = (2 − 1))
52	51	oveq1d 6808	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = ((2 − 1) + 1))
53		ax-1cn 10196	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
54		npcan 10492	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ) → ((2 − 1) + 1) = 2)
55	27, 53, 54	mp2an 672	. . . . . . . . . . . . . . 15 ⊢ ((2 − 1) + 1) = 2
56	52, 55	syl6eq 2821	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = 2)
57	50, 56	oveq12d 6811	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 2))
58	49, 57	oveq12d 6811	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
59	45, 58	sylbi 207	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
60		bpoly1 14988	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
61	60	oveq1d 6808	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 2) = ((𝑋 − (1 / 2)) / 2))
62	61	oveq2d 6809	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (2 · ((𝑋 − (1 / 2)) / 2)))
63		halfcn 11449	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℂ
64		subcl 10482	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
65	63, 64	mpan2 671	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
66		2ne0 11315	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
67		divcan2 10895	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
68	27, 66, 67	mp3an23 1564	. . . . . . . . . . . . 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 2805	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (𝑋 − (1 / 2)))
71	59, 70	sylan9eqr 2827	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
72	65	adantr 466	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → (𝑋 − (1 / 2)) ∈ ℂ)
73	71, 72	eqeltrd 2850	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
74	44, 73	jaodan 938	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = (0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
75	19, 74	sylan2b 581	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
76	12, 75, 59	fsump1 14695	. . . . . 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	syl6eqel 2858	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) ∈ ℂ)
78	34	fsum1 14684	. . . . . . . . 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 575	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
80	79, 42	eqtrd 2805	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
81	80, 70	oveq12d 6811	. . . . . 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 2805	. . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = ((1 / 3) + (𝑋 − (1 / 2))))
83		addsub12 10496	. . . . . . 7 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
84	40, 63, 83	mp3an13 1563	. . . . . 6 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
85	63, 40	negsubdi2i 10569	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = ((1 / 3) − (1 / 2))
86		halfthird 11886	. . . . . . . . 9 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
87	86	negeqi 10476	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = -(1 / 6)
88	85, 87	eqtr3i 2795	. . . . . . 7 ⊢ ((1 / 3) − (1 / 2)) = -(1 / 6)
89	88	oveq2i 6804	. . . . . 6 ⊢ (𝑋 + ((1 / 3) − (1 / 2))) = (𝑋 + -(1 / 6))
90	84, 89	syl6eq 2821	. . . . 5 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + -(1 / 6)))
91		6cn 11304	. . . . . . 7 ⊢ 6 ∈ ℂ
92		6re 11303	. . . . . . . 8 ⊢ 6 ∈ ℝ
93		6pos 11321	. . . . . . . 8 ⊢ 0 < 6
94	92, 93	gt0ne0ii 10766	. . . . . . 7 ⊢ 6 ≠ 0
95	91, 94	reccli 10957	. . . . . 6 ⊢ (1 / 6) ∈ ℂ
96		negsub 10531	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
97	95, 96	mpan2 671	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
98	82, 90, 97	3eqtrd 2809	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
99	8, 98	syl5eq 2817	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
100	99	oveq2d 6809	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))) = ((𝑋↑2) − (𝑋 − (1 / 6))))
101		sqcl 13132	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
102		subsub 10513	. . . 4 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
103	95, 102	mp3an3 1561	. . 3 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
104	101, 103	mpancom 668	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
105	3, 100, 104	3eqtrd 2809	1 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∨ wo 834 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 {cpr 4318 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 0cc0 10138 1c1 10139 + caddc 10141 · cmul 10143 − cmin 10468 -cneg 10469 / cdiv 10886 2c2 11272 3c3 11273 6c6 11276 ℕ₀cn0 11494 ℤcz 11579 ℤ_≥cuz 11888 ...cfz 12533 ↑cexp 13067 Ccbc 13293 Σcsu 14624 BernPoly cbp 14983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 df-bpoly 14984

This theorem is referenced by: bpoly3 14995 bpoly4 14996

W3C validator