bpoly2 - Metamath Proof Explorer

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Theorem bpoly2 15940

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 12430	. . 3 ⊢ 2 ∈ ℕ₀
2		bpolyval 15932	. . 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 688	. 2 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))))
4		2m1e1 12279	. . . . . . 7 ⊢ (2 − 1) = 1
5		0p1e1 12275	. . . . . . 7 ⊢ (0 + 1) = 1
6	4, 5	eqtr4i 2767	. . . . . 6 ⊢ (2 − 1) = (0 + 1)
7	6	oveq2i 7368	. . . . 5 ⊢ (0...(2 − 1)) = (0...(0 + 1))
8	7	sumeq1i 15583	. . . 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 12428	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
10		nn0uz 12805	. . . . . . . . 9 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtri 2836	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
12	11	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
13		0z 12510	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
14		fzpr 13496	. . . . . . . . . . 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 2829	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, (0 + 1)})
17		vex 3449	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
18	17	elpr 4609	. . . . . . . . 9 ⊢ (𝑘 ∈ {0, (0 + 1)} ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
19	16, 18	bitri 274	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = (0 + 1)))
20		oveq2 7365	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (2C𝑘) = (2C0))
21		bcn0 14210	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℕ₀ → (2C0) = 1)
22	1, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (2C0) = 1
23	20, 22	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (2C𝑘) = 1)
24		oveq1 7364	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
25		oveq2 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (2 − 𝑘) = (2 − 0))
26	25	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = ((2 − 0) + 1))
27		2cn 12228	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℂ
28	27	subid1i 11473	. . . . . . . . . . . . . . . 16 ⊢ (2 − 0) = 2
29	28	oveq1i 7367	. . . . . . . . . . . . . . 15 ⊢ ((2 − 0) + 1) = (2 + 1)
30		df-3 12217	. . . . . . . . . . . . . . 15 ⊢ 3 = (2 + 1)
31	29, 30	eqtr4i 2767	. . . . . . . . . . . . . 14 ⊢ ((2 − 0) + 1) = 3
32	26, 31	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((2 − 𝑘) + 1) = 3)
33	24, 32	oveq12d 7375	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 3))
34	23, 33	oveq12d 7375	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 3)))
35		bpoly0 15933	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
36	35	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 3) = (1 / 3))
37	36	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 · (1 / 3)))
38		3cn 12234	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℂ
39		3ne0 12259	. . . . . . . . . . . . . 14 ⊢ 3 ≠ 0
40	38, 39	reccli 11885	. . . . . . . . . . . . 13 ⊢ (1 / 3) ∈ ℂ
41	40	mulid2i 11160	. . . . . . . . . . . 12 ⊢ (1 · (1 / 3)) = (1 / 3)
42	37, 41	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) = (1 / 3))
43	34, 42	sylan9eqr 2798	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
44	43, 40	eqeltrdi 2846	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
45	5	eqeq2i 2749	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
46		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (2C𝑘) = (2C1))
47		bcn1 14213	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ₀ → (2C1) = 2)
48	1, 47	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2C1) = 2
49	46, 48	eqtrdi 2792	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (2C𝑘) = 2)
50		oveq1 7364	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
51		oveq2 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2 − 𝑘) = (2 − 1))
52	51	oveq1d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = ((2 − 1) + 1))
53		ax-1cn 11109	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
54		npcan 11410	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ) → ((2 − 1) + 1) = 2)
55	27, 53, 54	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ ((2 − 1) + 1) = 2
56	52, 55	eqtrdi 2792	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → ((2 − 𝑘) + 1) = 2)
57	50, 56	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 2))
58	49, 57	oveq12d 7375	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
59	45, 58	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (2 · ((1 BernPoly 𝑋) / 2)))
60		bpoly1 15934	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
61	60	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 2) = ((𝑋 − (1 / 2)) / 2))
62	61	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (2 · ((𝑋 − (1 / 2)) / 2)))
63		halfcn 12368	. . . . . . . . . . . . . 14 ⊢ (1 / 2) ∈ ℂ
64		subcl 11400	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
65	63, 64	mpan2 689	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
66		2ne0 12257	. . . . . . . . . . . . . 14 ⊢ 2 ≠ 0
67		divcan2 11821	. . . . . . . . . . . . . 14 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · ((𝑋 − (1 / 2)) / 2)) = (𝑋 − (1 / 2)))
68	27, 66, 67	mp3an23 1453	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (2 · ((1 BernPoly 𝑋) / 2)) = (𝑋 − (1 / 2)))
71	59, 70	sylan9eqr 2798	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
72	65	adantr 481	. . . . . . . . . 10 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → (𝑋 − (1 / 2)) ∈ ℂ)
73	71, 72	eqeltrd 2838	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = (0 + 1)) → ((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) ∈ ℂ)
74	44, 73	jaodan 956	. . . . . . . 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 15641	. . . . . 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 2846	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 3)) ∈ ℂ)
78	34	fsum1 15632	. . . . . . . . 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 2776	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (1 / 3))
81	80, 70	oveq12d 7375	. . . . . 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 2776	. . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = ((1 / 3) + (𝑋 − (1 / 2))))
83		addsub12 11414	. . . . . . 7 ⊢ (((1 / 3) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
84	40, 63, 83	mp3an13 1452	. . . . . 6 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 3) − (1 / 2))))
85	63, 40	negsubdi2i 11487	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = ((1 / 3) − (1 / 2))
86		halfthird 12761	. . . . . . . . 9 ⊢ ((1 / 2) − (1 / 3)) = (1 / 6)
87	86	negeqi 11394	. . . . . . . 8 ⊢ -((1 / 2) − (1 / 3)) = -(1 / 6)
88	85, 87	eqtr3i 2766	. . . . . . 7 ⊢ ((1 / 3) − (1 / 2)) = -(1 / 6)
89	88	oveq2i 7368	. . . . . 6 ⊢ (𝑋 + ((1 / 3) − (1 / 2))) = (𝑋 + -(1 / 6))
90	84, 89	eqtrdi 2792	. . . . 5 ⊢ (𝑋 ∈ ℂ → ((1 / 3) + (𝑋 − (1 / 2))) = (𝑋 + -(1 / 6)))
91		6cn 12244	. . . . . . 7 ⊢ 6 ∈ ℂ
92		6re 12243	. . . . . . . 8 ⊢ 6 ∈ ℝ
93		6pos 12263	. . . . . . . 8 ⊢ 0 < 6
94	92, 93	gt0ne0ii 11691	. . . . . . 7 ⊢ 6 ≠ 0
95	91, 94	reccli 11885	. . . . . 6 ⊢ (1 / 6) ∈ ℂ
96		negsub 11449	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
97	95, 96	mpan2 689	. . . . 5 ⊢ (𝑋 ∈ ℂ → (𝑋 + -(1 / 6)) = (𝑋 − (1 / 6)))
98	82, 90, 97	3eqtrd 2780	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
99	8, 98	eqtrid 2788	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1))) = (𝑋 − (1 / 6)))
100	99	oveq2d 7373	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − Σ𝑘 ∈ (0...(2 − 1))((2C𝑘) · ((𝑘 BernPoly 𝑋) / ((2 − 𝑘) + 1)))) = ((𝑋↑2) − (𝑋 − (1 / 6))))
101		sqcl 14023	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
102		subsub 11431	. . . 4 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
103	95, 102	mp3an3 1450	. . 3 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
104	101, 103	mpancom 686	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
105	3, 100, 104	3eqtrd 2780	1 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 {cpr 4588 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 − cmin 11385 -cneg 11386 / cdiv 11812 2c2 12208 3c3 12209 6c6 12212 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ↑cexp 13967 Ccbc 14202 Σcsu 15570 BernPoly cbp 15929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-bpoly 15930

This theorem is referenced by: bpoly3 15941 bpoly4 15942

W3C validator