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Theorem bpoly3 16150

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

**Assertion**
Ref	Expression
bpoly3	⊢ (𝑋 ∈ ℂ → (3 BernPoly 𝑋) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))

Proof of Theorem bpoly3 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3nn0 12550	. . 3 ⊢ 3 ∈ ℕ₀
2		bpolyval 16141	. . 3 ⊢ ((3 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (3 BernPoly 𝑋) = ((𝑋↑3) − Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))))
3	1, 2	mpan 703	. 2 ⊢ (𝑋 ∈ ℂ → (3 BernPoly 𝑋) = ((𝑋↑3) − Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))))
4		3m1e2 12396	. . . . . . 7 ⊢ (3 − 1) = 2
5		df-2 12331	. . . . . . 7 ⊢ 2 = (1 + 1)
6	4, 5	eqtri 2785	. . . . . 6 ⊢ (3 − 1) = (1 + 1)
7	6	oveq2i 7428	. . . . 5 ⊢ (0...(3 − 1)) = (0...(1 + 1))
8	7	sumeq1i 15788	. . . 4 ⊢ Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = Σ𝑘 ∈ (0...(1 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))
9		1eluzge0 12933	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
10	9	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ ℂ → 1 ∈ (ℤ_≥‘0))
11		0z 12630	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
12		fzpr 13638	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
14		0p1e1 12389	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
15	14	oveq2i 7428	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = (0...1)
16	14	preq2i 4701	. . . . . . . . . . . 12 ⊢ {0, (0 + 1)} = {0, 1}
17	13, 15, 16	3eqtr3ri 2794	. . . . . . . . . . 11 ⊢ {0, 1} = (0...1)
18	5	sneqi 4598	. . . . . . . . . . 11 ⊢ {2} = {(1 + 1)}
19	17, 18	uneq12i 4116	. . . . . . . . . 10 ⊢ ({0, 1} ∪ {2}) = ((0...1) ∪ {(1 + 1)})
20		df-tp 4592	. . . . . . . . . 10 ⊢ {0, 1, 2} = ({0, 1} ∪ {2})
21		fzsuc 13630	. . . . . . . . . . 11 ⊢ (1 ∈ (ℤ_≥‘0) → (0...(1 + 1)) = ((0...1) ∪ {(1 + 1)}))
22	9, 21	ax-mp 5	. . . . . . . . . 10 ⊢ (0...(1 + 1)) = ((0...1) ∪ {(1 + 1)})
23	19, 20, 22	3eqtr4ri 2796	. . . . . . . . 9 ⊢ (0...(1 + 1)) = {0, 1, 2}
24	23	eleq2i 2854	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(1 + 1)) ↔ 𝑘 ∈ {0, 1, 2})
25		vex 3457	. . . . . . . . 9 ⊢ 𝑘 ∈ V
26	25	eltp 4653	. . . . . . . 8 ⊢ (𝑘 ∈ {0, 1, 2} ↔ (𝑘 = 0 ∨ 𝑘 = 1 ∨ 𝑘 = 2))
27	24, 26	bitri 278	. . . . . . 7 ⊢ (𝑘 ∈ (0...(1 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = 1 ∨ 𝑘 = 2))
28		oveq2 7425	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (3C𝑘) = (3C0))
29		bcn0 14378	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℕ₀ → (3C0) = 1)
30	1, 29	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3C0) = 1
31	28, 30	eqtrdi 2813	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (3C𝑘) = 1)
32		oveq1 7424	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
33		oveq2 7425	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (3 − 𝑘) = (3 − 0))
34	33	oveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((3 − 𝑘) + 1) = ((3 − 0) + 1))
35		3cn 12350	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
36	35	subid1i 11558	. . . . . . . . . . . . . . 15 ⊢ (3 − 0) = 3
37	36	oveq1i 7427	. . . . . . . . . . . . . 14 ⊢ ((3 − 0) + 1) = (3 + 1)
38		df-4 12333	. . . . . . . . . . . . . 14 ⊢ 4 = (3 + 1)
39	37, 38	eqtr4i 2788	. . . . . . . . . . . . 13 ⊢ ((3 − 0) + 1) = 4
40	34, 39	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((3 − 𝑘) + 1) = 4)
41	32, 40	oveq12d 7435	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 4))
42	31, 41	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
43		bpoly0 16142	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
44	43	oveq1d 7432	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 4) = (1 / 4))
45	44	oveq2d 7433	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) = (1 · (1 / 4)))
46		4cn 12354	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
47		4ne0 12380	. . . . . . . . . . . . 13 ⊢ 4 ≠ 0
48	46, 47	reccli 11973	. . . . . . . . . . . 12 ⊢ (1 / 4) ∈ ℂ
49	48	mullidi 11242	. . . . . . . . . . 11 ⊢ (1 · (1 / 4)) = (1 / 4)
50	45, 49	eqtrdi 2813	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) = (1 / 4))
51	42, 50	sylan9eqr 2819	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 / 4))
52	51, 48	eqeltrdi 2870	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
53		oveq2 7425	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (3C𝑘) = (3C1))
54		bcn1 14381	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℕ₀ → (3C1) = 3)
55	1, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3C1) = 3
56	53, 55	eqtrdi 2813	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (3C𝑘) = 3)
57		oveq1 7424	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
58		oveq2 7425	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (3 − 𝑘) = (3 − 1))
59	58	oveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((3 − 𝑘) + 1) = ((3 − 1) + 1))
60		ax-1cn 11186	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
61		npcan 11494	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ) → ((3 − 1) + 1) = 3)
62	35, 60, 61	mp2an 705	. . . . . . . . . . . . 13 ⊢ ((3 − 1) + 1) = 3
63	59, 62	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((3 − 𝑘) + 1) = 3)
64	57, 63	oveq12d 7435	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 3))
65	56, 64	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((1 BernPoly 𝑋) / 3)))
66		bpoly1 16143	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
67	66	oveq1d 7432	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 3) = ((𝑋 − (1 / 2)) / 3))
68	67	oveq2d 7433	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((1 BernPoly 𝑋) / 3)) = (3 · ((𝑋 − (1 / 2)) / 3)))
69		halfcn 12486	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℂ
70		subcl 11484	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
71	69, 70	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
72		3ne0 12378	. . . . . . . . . . . . 13 ⊢ 3 ≠ 0
73		divcan2 11908	. . . . . . . . . . . . 13 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
74	35, 72, 73	mp3an23 1482	. . . . . . . . . . . 12 ⊢ ((𝑋 − (1 / 2)) ∈ ℂ → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
75	71, 74	syl 18	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
76	68, 75	eqtrd 2797	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (3 · ((1 BernPoly 𝑋) / 3)) = (𝑋 − (1 / 2)))
77	65, 76	sylan9eqr 2819	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
78	71	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → (𝑋 − (1 / 2)) ∈ ℂ)
79	77, 78	eqeltrd 2862	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
80		oveq2 7425	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → (3C𝑘) = (3C2))
81		bcn2 14387	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ₀ → (3C2) = ((3 · (3 − 1)) / 2))
82	1, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (3C2) = ((3 · (3 − 1)) / 2)
83	4	oveq2i 7428	. . . . . . . . . . . . . . 15 ⊢ (3 · (3 − 1)) = (3 · 2)
84	83	oveq1i 7427	. . . . . . . . . . . . . 14 ⊢ ((3 · (3 − 1)) / 2) = ((3 · 2) / 2)
85		2cn 12344	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
86		2ne0 12375	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
87	35, 85, 86	divcan4i 11990	. . . . . . . . . . . . . 14 ⊢ ((3 · 2) / 2) = 3
88	84, 87	eqtri 2785	. . . . . . . . . . . . 13 ⊢ ((3 · (3 − 1)) / 2) = 3
89	82, 88	eqtri 2785	. . . . . . . . . . . 12 ⊢ (3C2) = 3
90	80, 89	eqtrdi 2813	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (3C𝑘) = 3)
91		oveq1 7424	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → (𝑘 BernPoly 𝑋) = (2 BernPoly 𝑋))
92		oveq2 7425	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 2 → (3 − 𝑘) = (3 − 2))
93	92	oveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝑘 = 2 → ((3 − 𝑘) + 1) = ((3 − 2) + 1))
94		2p1e3 12410	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
95	35, 85, 60, 94	subaddrii 11575	. . . . . . . . . . . . . . 15 ⊢ (3 − 2) = 1
96	95	oveq1i 7427	. . . . . . . . . . . . . 14 ⊢ ((3 − 2) + 1) = (1 + 1)
97	96, 5	eqtr4i 2788	. . . . . . . . . . . . 13 ⊢ ((3 − 2) + 1) = 2
98	93, 97	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → ((3 − 𝑘) + 1) = 2)
99	91, 98	oveq12d 7435	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((2 BernPoly 𝑋) / 2))
100	90, 99	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝑘 = 2 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((2 BernPoly 𝑋) / 2)))
101		2nn0 12549	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
102		bpolycl 16144	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (2 BernPoly 𝑋) ∈ ℂ)
103	101, 102	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) ∈ ℂ)
104		2cnne0 12481	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
105		div12 11922	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ (2 BernPoly 𝑋) ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (3 · ((2 BernPoly 𝑋) / 2)) = ((2 BernPoly 𝑋) · (3 / 2)))
106	35, 104, 105	mp3an13 1481	. . . . . . . . . . . 12 ⊢ ((2 BernPoly 𝑋) ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = ((2 BernPoly 𝑋) · (3 / 2)))
107	103, 106	syl 18	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = ((2 BernPoly 𝑋) · (3 / 2)))
108	35, 85, 86	divcli 11985	. . . . . . . . . . . 12 ⊢ (3 / 2) ∈ ℂ
109		mulcom 11214	. . . . . . . . . . . 12 ⊢ (((2 BernPoly 𝑋) ∈ ℂ ∧ (3 / 2) ∈ ℂ) → ((2 BernPoly 𝑋) · (3 / 2)) = ((3 / 2) · (2 BernPoly 𝑋)))
110	103, 108, 109	sylancl 598	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((2 BernPoly 𝑋) · (3 / 2)) = ((3 / 2) · (2 BernPoly 𝑋)))
111		bpoly2 16149	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))
112	111	oveq2d 7433	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (2 BernPoly 𝑋)) = ((3 / 2) · (((𝑋↑2) − 𝑋) + (1 / 6))))
113		sqcl 14186	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
114		6cn 12360	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℂ
115		6re 12359	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℝ
116		6pos 12382	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 6
117	115, 116	gt0ne0ii 11778	. . . . . . . . . . . . . . . 16 ⊢ 6 ≠ 0
118	114, 117	reccli 11973	. . . . . . . . . . . . . . 15 ⊢ (1 / 6) ∈ ℂ
119		subsub 11516	. . . . . . . . . . . . . . 15 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
120	118, 119	mp3an3 1479	. . . . . . . . . . . . . 14 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
121	113, 120	mpancom 701	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
122	121	oveq2d 7433	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = ((3 / 2) · (((𝑋↑2) − 𝑋) + (1 / 6))))
123		subcl 11484	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 − (1 / 6)) ∈ ℂ)
124	118, 123	mpan2 704	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 6)) ∈ ℂ)
125		subdi 11675	. . . . . . . . . . . . 13 ⊢ (((3 / 2) ∈ ℂ ∧ (𝑋↑2) ∈ ℂ ∧ (𝑋 − (1 / 6)) ∈ ℂ) → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
126	108, 113, 124, 125	mp3an2i 1495	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
127	112, 122, 126	3eqtr2d 2803	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (2 BernPoly 𝑋)) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
128	107, 110, 127	3eqtrd 2801	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
129	100, 128	sylan9eqr 2819	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
130		mulcl 11212	. . . . . . . . . . . 12 ⊢ (((3 / 2) ∈ ℂ ∧ (𝑋↑2) ∈ ℂ) → ((3 / 2) · (𝑋↑2)) ∈ ℂ)
131	108, 113, 130	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋↑2)) ∈ ℂ)
132		mulcl 11212	. . . . . . . . . . . 12 ⊢ (((3 / 2) ∈ ℂ ∧ (𝑋 − (1 / 6)) ∈ ℂ) → ((3 / 2) · (𝑋 − (1 / 6))) ∈ ℂ)
133	108, 124, 132	sylancr 599	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋 − (1 / 6))) ∈ ℂ)
134	131, 133	subcld 11597	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))) ∈ ℂ)
135	134	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))) ∈ ℂ)
136	129, 135	eqeltrd 2862	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
137	52, 79, 136	3jaodan 1458	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = 1 ∨ 𝑘 = 2)) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
138	27, 137	sylan2b 606	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(1 + 1))) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
139	5	eqeq2i 2775	. . . . . . 7 ⊢ (𝑘 = 2 ↔ 𝑘 = (1 + 1))
140	139, 100	sylbir 238	. . . . . 6 ⊢ (𝑘 = (1 + 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((2 BernPoly 𝑋) / 2)))
141	10, 138, 140	fsump1 15846	. . . . 5 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (3 · ((2 BernPoly 𝑋) / 2))))
142	128	oveq2d 7433	. . . . 5 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (3 · ((2 BernPoly 𝑋) / 2))) = (Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6))))))
143	15	sumeq1i 15788	. . . . . . . . 9 ⊢ Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))
144		0nn0 12547	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ₀
145		nn0uz 12929	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
146	144, 145	eleqtri 2860	. . . . . . . . . . . 12 ⊢ 0 ∈ (ℤ_≥‘0)
147	146	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
148	13, 16	eqtri 2785	. . . . . . . . . . . . . 14 ⊢ (0...(0 + 1)) = {0, 1}
149	148	eleq2i 2854	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, 1})
150	25	elpr 4612	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {0, 1} ↔ (𝑘 = 0 ∨ 𝑘 = 1))
151	149, 150	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = 1))
152	52, 79	jaodan 972	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = 1)) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
153	151, 152	sylan2b 606	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(0 + 1))) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
154	14	eqeq2i 2775	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
155	154, 65	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((1 BernPoly 𝑋) / 3)))
156	147, 153, 155	fsump1 15846	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (3 · ((1 BernPoly 𝑋) / 3))))
157	50, 48	eqeltrdi 2870	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) ∈ ℂ)
158	42	fsum1 15837	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ (1 · ((0 BernPoly 𝑋) / 4)) ∈ ℂ) → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
159	11, 157, 158	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
160	159, 50	eqtrd 2797	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 / 4))
161	160, 76	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (3 · ((1 BernPoly 𝑋) / 3))) = ((1 / 4) + (𝑋 − (1 / 2))))
162	156, 161	eqtrd 2797	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = ((1 / 4) + (𝑋 − (1 / 2))))
163	143, 162	eqtr3id 2811	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = ((1 / 4) + (𝑋 − (1 / 2))))
164	163	oveq1d 7432	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6))))) = (((1 / 4) + (𝑋 − (1 / 2))) + (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6))))))
165		addcl 11210	. . . . . . . . 9 ⊢ (((1 / 4) ∈ ℂ ∧ (𝑋 − (1 / 2)) ∈ ℂ) → ((1 / 4) + (𝑋 − (1 / 2))) ∈ ℂ)
166	48, 71, 165	sylancr 599	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((1 / 4) + (𝑋 − (1 / 2))) ∈ ℂ)
167	166, 131, 133	addsub12d 11620	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (((1 / 4) + (𝑋 − (1 / 2))) + (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6))))) = (((3 / 2) · (𝑋↑2)) + (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6))))))
168	164, 167	eqtrd 2797	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6))))) = (((3 / 2) · (𝑋↑2)) + (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6))))))
169	133, 166	negsubdi2d 11613	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6)))))
170		subdi 11675	. . . . . . . . . . . 12 ⊢ (((3 / 2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((3 / 2) · (𝑋 − (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
171	108, 118, 170	mp3an13 1481	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋 − (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
172		addsub12 11498	. . . . . . . . . . . 12 ⊢ (((1 / 4) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 4) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 4) − (1 / 2))))
173	48, 69, 172	mp3an13 1481	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((1 / 4) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 4) − (1 / 2))))
174	171, 173	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = ((((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))))
175		mulcl 11212	. . . . . . . . . . . . 13 ⊢ (((3 / 2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((3 / 2) · 𝑋) ∈ ℂ)
176	108, 175	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · 𝑋) ∈ ℂ)
177	108, 118	mulcli 11244	. . . . . . . . . . . 12 ⊢ ((3 / 2) · (1 / 6)) ∈ ℂ
178		negsub 11534	. . . . . . . . . . . 12 ⊢ ((((3 / 2) · 𝑋) ∈ ℂ ∧ ((3 / 2) · (1 / 6)) ∈ ℂ) → (((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
179	176, 177, 178	sylancl 598	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
180	179	oveq1d 7432	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))) = ((((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))))
181	69, 48	negsubdi2i 11572	. . . . . . . . . . . . . 14 ⊢ -((1 / 2) − (1 / 4)) = ((1 / 4) − (1 / 2))
182	85, 35, 85	mul12i 11433	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · (3 · 2)) = (3 · (2 · 2))
183		3t2e6 12434	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 · 2) = 6
184	183	oveq2i 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · (3 · 2)) = (2 · 6)
185		2t2e4 12432	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 2) = 4
186	185	oveq2i 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 · (2 · 2)) = (3 · 4)
187	182, 184, 186	3eqtr3i 2793	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 6) = (3 · 4)
188	187	oveq2i 7428	. . . . . . . . . . . . . . . . 17 ⊢ ((3 · 1) / (2 · 6)) = ((3 · 1) / (3 · 4))
189	46, 47	pm3.2i 476	. . . . . . . . . . . . . . . . . 18 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
190	35, 72	pm3.2i 476	. . . . . . . . . . . . . . . . . 18 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
191		divcan5 11945	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (4 ∈ ℂ ∧ 4 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 · 1) / (3 · 4)) = (1 / 4))
192	60, 189, 190, 191	mp3an 1490	. . . . . . . . . . . . . . . . 17 ⊢ ((3 · 1) / (3 · 4)) = (1 / 4)
193	188, 192	eqtri 2785	. . . . . . . . . . . . . . . 16 ⊢ ((3 · 1) / (2 · 6)) = (1 / 4)
194	35, 85, 60, 114, 86, 117	divmuldivi 12003	. . . . . . . . . . . . . . . 16 ⊢ ((3 / 2) · (1 / 6)) = ((3 · 1) / (2 · 6))
195		2t1e2 12431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 1) = 2
196	195, 5	eqtri 2785	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · 1) = (1 + 1)
197	196, 185	oveq12i 7429	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 1) / (2 · 2)) = ((1 + 1) / 4)
198		divcan5 11945	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 1) / (2 · 2)) = (1 / 2))
199	60, 104, 104, 198	mp3an 1490	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 1) / (2 · 2)) = (1 / 2)
200	60, 60, 46, 47	divdiri 12000	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 + 1) / 4) = ((1 / 4) + (1 / 4))
201	197, 199, 200	3eqtr3ri 2794	. . . . . . . . . . . . . . . . 17 ⊢ ((1 / 4) + (1 / 4)) = (1 / 2)
202	69, 48, 48, 201	subaddrii 11575	. . . . . . . . . . . . . . . 16 ⊢ ((1 / 2) − (1 / 4)) = (1 / 4)
203	193, 194, 202	3eqtr4ri 2796	. . . . . . . . . . . . . . 15 ⊢ ((1 / 2) − (1 / 4)) = ((3 / 2) · (1 / 6))
204	203	negeqi 11478	. . . . . . . . . . . . . 14 ⊢ -((1 / 2) − (1 / 4)) = -((3 / 2) · (1 / 6))
205	181, 204	eqtr3i 2787	. . . . . . . . . . . . 13 ⊢ ((1 / 4) − (1 / 2)) = -((3 / 2) · (1 / 6))
206	48, 69	subcli 11562	. . . . . . . . . . . . . 14 ⊢ ((1 / 4) − (1 / 2)) ∈ ℂ
207	177	negcli 11554	. . . . . . . . . . . . . 14 ⊢ -((3 / 2) · (1 / 6)) ∈ ℂ
208	206, 207	subeq0i 11566	. . . . . . . . . . . . 13 ⊢ ((((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6))) = 0 ↔ ((1 / 4) − (1 / 2)) = -((3 / 2) · (1 / 6)))
209	205, 208	mpbir 234	. . . . . . . . . . . 12 ⊢ (((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6))) = 0
210	209	oveq2i 7428	. . . . . . . . . . 11 ⊢ ((((3 / 2) · 𝑋) − 𝑋) − (((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6)))) = ((((3 / 2) · 𝑋) − 𝑋) − 0)
211		id 23	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
212	206	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((1 / 4) − (1 / 2)) ∈ ℂ)
213	207	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → -((3 / 2) · (1 / 6)) ∈ ℂ)
214	176, 211, 212, 213	subadd4d 11645	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − (((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6)))) = ((((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))))
215		subdir 11676	. . . . . . . . . . . . . . 15 ⊢ (((3 / 2) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (((3 / 2) − 1) · 𝑋) = (((3 / 2) · 𝑋) − (1 · 𝑋)))
216	108, 60, 215	mp3an12 1480	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) − 1) · 𝑋) = (((3 / 2) · 𝑋) − (1 · 𝑋)))
217		divsubdir 11936	. . . . . . . . . . . . . . . . . 18 ⊢ ((3 ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((3 − 2) / 2) = ((3 / 2) − (2 / 2)))
218	35, 85, 104, 217	mp3an 1490	. . . . . . . . . . . . . . . . 17 ⊢ ((3 − 2) / 2) = ((3 / 2) − (2 / 2))
219	95	oveq1i 7427	. . . . . . . . . . . . . . . . 17 ⊢ ((3 − 2) / 2) = (1 / 2)
220		2div2e1 12409	. . . . . . . . . . . . . . . . . 18 ⊢ (2 / 2) = 1
221	220	oveq2i 7428	. . . . . . . . . . . . . . . . 17 ⊢ ((3 / 2) − (2 / 2)) = ((3 / 2) − 1)
222	218, 219, 221	3eqtr3ri 2794	. . . . . . . . . . . . . . . 16 ⊢ ((3 / 2) − 1) = (1 / 2)
223	222	oveq1i 7427	. . . . . . . . . . . . . . 15 ⊢ (((3 / 2) − 1) · 𝑋) = ((1 / 2) · 𝑋)
224	223	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) − 1) · 𝑋) = ((1 / 2) · 𝑋))
225		mullid 11235	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ℂ → (1 · 𝑋) = 𝑋)
226	225	oveq2d 7433	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) − (1 · 𝑋)) = (((3 / 2) · 𝑋) − 𝑋))
227	216, 224, 226	3eqtr3rd 2806	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) − 𝑋) = ((1 / 2) · 𝑋))
228	227	oveq1d 7432	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − 0) = (((1 / 2) · 𝑋) − 0))
229		mulcl 11212	. . . . . . . . . . . . . 14 ⊢ (((1 / 2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((1 / 2) · 𝑋) ∈ ℂ)
230	69, 229	mpan 703	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 / 2) · 𝑋) ∈ ℂ)
231	230	subid1d 11586	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (((1 / 2) · 𝑋) − 0) = ((1 / 2) · 𝑋))
232	228, 231	eqtrd 2797	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − 0) = ((1 / 2) · 𝑋))
233	210, 214, 232	3eqtr3a 2821	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))) = ((1 / 2) · 𝑋))
234	174, 180, 233	3eqtr2d 2803	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = ((1 / 2) · 𝑋))
235	234	negeqd 11479	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = -((1 / 2) · 𝑋))
236	169, 235	eqtr3d 2799	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6)))) = -((1 / 2) · 𝑋))
237	236	oveq2d 7433	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) + (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6))))) = (((3 / 2) · (𝑋↑2)) + -((1 / 2) · 𝑋)))
238	131, 230	negsubd 11603	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) + -((1 / 2) · 𝑋)) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
239	168, 237, 238	3eqtrd 2801	. . . . 5 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6))))) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
240	141, 142, 239	3eqtrd 2801	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
241	8, 240	eqtrid 2809	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
242	241	oveq2d 7433	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑3) − Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))) = ((𝑋↑3) − (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋))))
243		expcl 14147	. . . 4 ⊢ ((𝑋 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑋↑3) ∈ ℂ)
244	1, 243	mpan2 704	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑3) ∈ ℂ)
245	244, 131, 230	subsubd 11625	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑3) − (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋))) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))
246	3, 242, 245	3eqtrd 2801	1 ⊢ (𝑋 ∈ ℂ → (3 BernPoly 𝑋) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∪ cun 3900 {csn 4587 {cpr 4589 {ctp 4591 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 0cc0 11128 1c1 11129 + caddc 11131 · cmul 11133 − cmin 11469 -cneg 11470 / cdiv 11899 2c2 12323 3c3 12324 4c4 12325 6c6 12327 ℕ₀cn0 12532 ℤcz 12619 ℤ_≥cuz 12891 ...cfz 13565 ↑cexp 14129 Ccbc 14370 Σcsu 15777 BernPoly cbp 16138

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-fac 14342 df-bc 14371 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-sum 15778 df-bpoly 16139

This theorem is used by: bpoly4 16151

W3C validator