bpoly2 - Metamath Proof Explorer

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Theorem bpoly2 15073

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 11559	. . 3 ⊢ 2 ∈ ℕ₀
2		bpolyval 15065	. . 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 681	. 2 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))))
4		2m1e1 11407	. . . . . . 7 ⊢ (2 − 1) = 1
5		0p1e1 11403	. . . . . . 7 ⊢ (0 + 1) = 1
6	4, 5	eqtr4i 2790	. . . . . 6 ⊢ (2 − 1) = (0 + 1)
7	6	oveq2i 6855	. . . . 5 ⊢ (0...(2 − 1)) = (0...(0 + 1))
8	7	sumeq1i 14716	. . . 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 11557	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
10		nn0uz 11925	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtri 2842	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
12	11	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
13		0z 11637	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
14		fzpr 12606	. . . . . . . . . . 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 2836	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, (0 + 1)})
17		vex 3353	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
18	17	elpr 4359	. . . . . . . . 9 ⊢ (𝑘 ∈ {0, (0 + 1)} ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
19	16, 18	bitri 266	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
20		oveq2 6852	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (2C𝑘) = (2C0))
21		bcn0 13304	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℕ₀ → (2C0) = 1)
22	1, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2C0) = 1
23	20, 22	syl6eq 2815	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (2C𝑘) = 1)
24		oveq1 6851	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
25		oveq2 6852	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (2 − 𝑘) = (2 − 0))
26	25	oveq1d 6859	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = ((2 − 0) + 1))
27		2cn 11349	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
28	27	subid1i 10609	. . . . . . . . . . . . . . . 16 ⊢ (2 − 0) = 2
29	28	oveq1i 6854	. . . . . . . . . . . . . . 15 ⊢ ((2 − 0) + 1) = (2 + 1)
30		df-3 11338	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
31	29, 30	eqtr4i 2790	. . . . . . . . . . . . . 14 ⊢ ((2 − 0) + 1) = 3
32	26, 31	syl6eq 2815	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = 3)
33	24, 32	oveq12d 6862	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 3))
34	23, 33	oveq12d 6862	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
35		bpoly0 15066	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
36	35	oveq1d 6859	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 3) = (1 / 3))
37	36	oveq2d 6860	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 · (1 / 3)))
38		3cn 11355	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
39		3ne0 11387	. . . . . . . . . . . . . 14 ⊢ 3 ≠ 0
40	38, 39	reccli 11011	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℂ
41	40	mulid2i 10301	. . . . . . . . . . . 12 ⊢ (1 · (1 / 3)) = (1 / 3)
42	37, 41	syl6eq 2815	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 / 3))
43	34, 42	sylan9eqr 2821	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
44	43, 40	syl6eqel 2852	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
45	5	eqeq2i 2777	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
46		oveq2 6852	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (2C𝑘) = (2C1))
47		bcn1 13307	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ₀ → (2C1) = 2)
48	1, 47	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2C1) = 2
49	46, 48	syl6eq 2815	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (2C𝑘) = 2)
50		oveq1 6851	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
51		oveq2 6852	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2 − 𝑘) = (2 − 1))
52	51	oveq1d 6859	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = ((2 − 1) + 1))
53		ax-1cn 10249	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
54		npcan 10546	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ) → ((2 − 1) + 1) = 2)
55	27, 53, 54	mp2an 683	. . . . . . . . . . . . . . 15 ⊢ ((2 − 1) + 1) = 2
56	52, 55	syl6eq 2815	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = 2)
57	50, 56	oveq12d 6862	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 2))
58	49, 57	oveq12d 6862	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
59	45, 58	sylbi 208	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
60		bpoly1 15067	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
61	60	oveq1d 6859	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 2) = ((𝑋 − (1 / 2)) / 2))
62	61	oveq2d 6860	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (2 · ((𝑋 − (1 / 2)) / 2)))
63		halfcn 11495	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℂ
64		subcl 10536	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
65	63, 64	mpan2 682	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
66		2ne0 11385	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
67		divcan2 10949	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
68	27, 66, 67	mp3an23 1577	. . . . . . . . . . . . 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 2799	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (𝑋 − (1 / 2)))
71	59, 70	sylan9eqr 2821	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
72	65	adantr 472	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → (𝑋 − (1 / 2)) ∈ ℂ)
73	71, 72	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
74	44, 73	jaodan 980	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = (0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
75	19, 74	sylan2b 587	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(0 + 1))) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
76	12, 75, 59	fsump1 14775	. . . . . 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 2852	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) ∈ ℂ)
78	34	fsum1 14764	. . . . . . . . 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 581	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
80	79, 42	eqtrd 2799	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
81	80, 70	oveq12d 6862	. . . . . 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 2799	. . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = ((1 / 3) + (𝑋 − (1 / 2))))
83		addsub12 10550	. . . . . . 7 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
84	40, 63, 83	mp3an13 1576	. . . . . 6 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
85	63, 40	negsubdi2i 10623	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = ((1 / 3) − (1 / 2))
86		halfthird 11887	. . . . . . . . 9 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
87	86	negeqi 10530	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = -(1 / 6)
88	85, 87	eqtr3i 2789	. . . . . . 7 ⊢ ((1 / 3) − (1 / 2)) = -(1 / 6)
89	88	oveq2i 6855	. . . . . 6 ⊢ (𝑋 + ((1 / 3) − (1 / 2))) = (𝑋 + -(1 / 6))
90	84, 89	syl6eq 2815	. . . . 5 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + -(1 / 6)))
91		6cn 11368	. . . . . . 7 ⊢ 6 ∈ ℂ
92		6re 11367	. . . . . . . 8 ⊢ 6 ∈ ℝ
93		6pos 11391	. . . . . . . 8 ⊢ 0 < 6
94	92, 93	gt0ne0ii 10820	. . . . . . 7 ⊢ 6 ≠ 0
95	91, 94	reccli 11011	. . . . . 6 ⊢ (1 / 6) ∈ ℂ
96		negsub 10585	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
97	95, 96	mpan2 682	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
98	82, 90, 97	3eqtrd 2803	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
99	8, 98	syl5eq 2811	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
100	99	oveq2d 6860	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))) = ((𝑋↑2) − (𝑋 − (1 / 6))))
101		sqcl 13135	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
102		subsub 10567	. . . 4 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
103	95, 102	mp3an3 1574	. . 3 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
104	101, 103	mpancom 679	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
105	3, 100, 104	3eqtrd 2803	1 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∨ wo 873 = wceq 1652 ∈ wcel 2155 ≠ wne 2937 {cpr 4338 ‘cfv 6070 (class class class)co 6844 ℂcc 10189 0cc0 10191 1c1 10192 + caddc 10194 · cmul 10196 − cmin 10522 -cneg 10523 / cdiv 10940 2c2 11329 3c3 11330 6c6 11333 ℕ₀cn0 11540 ℤcz 11626 ℤ_≥cuz 11889 ...cfz 12536 ↑cexp 13070 Ccbc 13296 Σcsu 14704 BernPoly cbp 15062

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2070 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4932 ax-sep 4943 ax-nul 4951 ax-pow 5003 ax-pr 5064 ax-un 7149 ax-inf2 8755 ax-cnex 10247 ax-resscn 10248 ax-1cn 10249 ax-icn 10250 ax-addcl 10251 ax-addrcl 10252 ax-mulcl 10253 ax-mulrcl 10254 ax-mulcom 10255 ax-addass 10256 ax-mulass 10257 ax-distr 10258 ax-i2m1 10259 ax-1ne0 10260 ax-1rid 10261 ax-rnegex 10262 ax-rrecex 10263 ax-cnre 10264 ax-pre-lttri 10265 ax-pre-lttrn 10266 ax-pre-ltadd 10267 ax-pre-mulgt0 10268 ax-pre-sup 10269

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2063 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3599 df-csb 3694 df-dif 3737 df-un 3739 df-in 3741 df-ss 3748 df-pss 3750 df-nul 4082 df-if 4246 df-pw 4319 df-sn 4337 df-pr 4339 df-tp 4341 df-op 4343 df-uni 4597 df-int 4636 df-iun 4680 df-br 4812 df-opab 4874 df-mpt 4891 df-tr 4914 df-id 5187 df-eprel 5192 df-po 5200 df-so 5201 df-fr 5238 df-se 5239 df-we 5240 df-xp 5285 df-rel 5286 df-cnv 5287 df-co 5288 df-dm 5289 df-rn 5290 df-res 5291 df-ima 5292 df-pred 5867 df-ord 5913 df-on 5914 df-lim 5915 df-suc 5916 df-iota 6033 df-fun 6072 df-fn 6073 df-f 6074 df-f1 6075 df-fo 6076 df-f1o 6077 df-fv 6078 df-isom 6079 df-riota 6805 df-ov 6847 df-oprab 6848 df-mpt2 6849 df-om 7266 df-1st 7368 df-2nd 7369 df-wrecs 7612 df-recs 7674 df-rdg 7712 df-1o 7766 df-oadd 7770 df-er 7949 df-en 8163 df-dom 8164 df-sdom 8165 df-fin 8166 df-sup 8557 df-oi 8624 df-card 9018 df-pnf 10332 df-mnf 10333 df-xr 10334 df-ltxr 10335 df-le 10336 df-sub 10524 df-neg 10525 df-div 10941 df-nn 11277 df-2 11337 df-3 11338 df-4 11339 df-5 11340 df-6 11341 df-n0 11541 df-z 11627 df-uz 11890 df-rp 12032 df-fz 12537 df-fzo 12677 df-seq 13012 df-exp 13071 df-fac 13268 df-bc 13297 df-hash 13325 df-cj 14127 df-re 14128 df-im 14129 df-sqrt 14263 df-abs 14264 df-clim 14507 df-sum 14705 df-bpoly 15063

This theorem is referenced by: bpoly3 15074 bpoly4 15075

W3C validator