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Theorem bpoly3 16253

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 12651	. . 3 ⊢ 3 ∈ ℕ₀
2		bpolyval 16244	. . 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 12497	. . . . . . 7 ⊢ (3 − 1) = 2
5		df-2 12432	. . . . . . 7 ⊢ 2 = (1 + 1)
6	4, 5	eqtri 2786	. . . . . 6 ⊢ (3 − 1) = (1 + 1)
7	6	oveq2i 7454	. . . . 5 ⊢ (0...(3 − 1)) = (0...(1 + 1))
8	7	sumeq1i 15893	. . . 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 13034	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
10	9	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ ℂ → 1 ∈ (ℤ_≥‘0))
11		0z 12731	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
12		fzpr 13740	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
14		0p1e1 12490	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
15	14	oveq2i 7454	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = (0...1)
16	14	preq2i 4698	. . . . . . . . . . . 12 ⊢ {0, (0 + 1)} = {0, 1}
17	13, 15, 16	3eqtr3ri 2795	. . . . . . . . . . 11 ⊢ {0, 1} = (0...1)
18	5	sneqi 4595	. . . . . . . . . . 11 ⊢ {2} = {(1 + 1)}
19	17, 18	uneq12i 4168	. . . . . . . . . 10 ⊢ ({0, 1} ∪ {2}) = ((0...1) ∪ {(1 + 1)})
20		df-tp 4593	. . . . . . . . . 10 ⊢ {0, 1, 2} = ({0, 1} ∪ {2})
21		fzsuc 13732	. . . . . . . . . . 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 2797	. . . . . . . . 9 ⊢ (0...(1 + 1)) = {0, 1, 2}
24	23	eleq2i 2856	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(1 + 1)) ↔ 𝑘 ∈ {0, 1, 2})
25		vex 3458	. . . . . . . . 9 ⊢ 𝑘 ∈ V
26	25	eltp 4650	. . . . . . . 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 7451	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (3C𝑘) = (3C0))
29		bcn0 14481	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℕ₀ → (3C0) = 1)
30	1, 29	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3C0) = 1
31	28, 30	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (3C𝑘) = 1)
32		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
33		oveq2 7451	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (3 − 𝑘) = (3 − 0))
34	33	oveq1d 7458	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((3 − 𝑘) + 1) = ((3 − 0) + 1))
35		3cn 12451	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
36	35	subid1i 11657	. . . . . . . . . . . . . . 15 ⊢ (3 − 0) = 3
37	36	oveq1i 7453	. . . . . . . . . . . . . 14 ⊢ ((3 − 0) + 1) = (3 + 1)
38		df-4 12434	. . . . . . . . . . . . . 14 ⊢ 4 = (3 + 1)
39	37, 38	eqtr4i 2789	. . . . . . . . . . . . 13 ⊢ ((3 − 0) + 1) = 4
40	34, 39	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((3 − 𝑘) + 1) = 4)
41	32, 40	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 4))
42	31, 41	oveq12d 7461	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
43		bpoly0 16245	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
44	43	oveq1d 7458	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 4) = (1 / 4))
45	44	oveq2d 7459	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) = (1 · (1 / 4)))
46		4cn 12455	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
47		4ne0 12481	. . . . . . . . . . . . 13 ⊢ 4 ≠ 0
48	46, 47	reccli 12074	. . . . . . . . . . . 12 ⊢ (1 / 4) ∈ ℂ
49	48	mullidi 11341	. . . . . . . . . . 11 ⊢ (1 · (1 / 4)) = (1 / 4)
50	45, 49	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) = (1 / 4))
51	42, 50	sylan9eqr 2820	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 / 4))
52	51, 48	eqeltrdi 2872	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
53		oveq2 7451	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (3C𝑘) = (3C1))
54		bcn1 14484	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℕ₀ → (3C1) = 3)
55	1, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3C1) = 3
56	53, 55	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (3C𝑘) = 3)
57		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
58		oveq2 7451	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (3 − 𝑘) = (3 − 1))
59	58	oveq1d 7458	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((3 − 𝑘) + 1) = ((3 − 1) + 1))
60		ax-1cn 11285	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
61		npcan 11593	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ) → ((3 − 1) + 1) = 3)
62	35, 60, 61	mp2an 705	. . . . . . . . . . . . 13 ⊢ ((3 − 1) + 1) = 3
63	59, 62	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((3 − 𝑘) + 1) = 3)
64	57, 63	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 3))
65	56, 64	oveq12d 7461	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((1 BernPoly 𝑋) / 3)))
66		bpoly1 16246	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
67	66	oveq1d 7458	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 3) = ((𝑋 − (1 / 2)) / 3))
68	67	oveq2d 7459	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((1 BernPoly 𝑋) / 3)) = (3 · ((𝑋 − (1 / 2)) / 3)))
69		halfcn 12587	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℂ
70		subcl 11583	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
71	69, 70	mpan2 704	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
72		3ne0 12479	. . . . . . . . . . . . 13 ⊢ 3 ≠ 0
73		divcan2 12009	. . . . . . . . . . . . 13 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
74	35, 72, 73	mp3an23 1483	. . . . . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (3 · ((1 BernPoly 𝑋) / 3)) = (𝑋 − (1 / 2)))
77	65, 76	sylan9eqr 2820	. . . . . . . . 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 2864	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
80		oveq2 7451	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → (3C𝑘) = (3C2))
81		bcn2 14490	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ₀ → (3C2) = ((3 · (3 − 1)) / 2))
82	1, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (3C2) = ((3 · (3 − 1)) / 2)
83	4	oveq2i 7454	. . . . . . . . . . . . . . 15 ⊢ (3 · (3 − 1)) = (3 · 2)
84	83	oveq1i 7453	. . . . . . . . . . . . . 14 ⊢ ((3 · (3 − 1)) / 2) = ((3 · 2) / 2)
85		2cn 12445	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
86		2ne0 12476	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
87	35, 85, 86	divcan4i 12091	. . . . . . . . . . . . . 14 ⊢ ((3 · 2) / 2) = 3
88	84, 87	eqtri 2786	. . . . . . . . . . . . 13 ⊢ ((3 · (3 − 1)) / 2) = 3
89	82, 88	eqtri 2786	. . . . . . . . . . . 12 ⊢ (3C2) = 3
90	80, 89	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (3C𝑘) = 3)
91		oveq1 7450	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → (𝑘 BernPoly 𝑋) = (2 BernPoly 𝑋))
92		oveq2 7451	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 2 → (3 − 𝑘) = (3 − 2))
93	92	oveq1d 7458	. . . . . . . . . . . . 13 ⊢ (𝑘 = 2 → ((3 − 𝑘) + 1) = ((3 − 2) + 1))
94		2p1e3 12511	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
95	35, 85, 60, 94	subaddrii 11674	. . . . . . . . . . . . . . 15 ⊢ (3 − 2) = 1
96	95	oveq1i 7453	. . . . . . . . . . . . . 14 ⊢ ((3 − 2) + 1) = (1 + 1)
97	96, 5	eqtr4i 2789	. . . . . . . . . . . . 13 ⊢ ((3 − 2) + 1) = 2
98	93, 97	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → ((3 − 𝑘) + 1) = 2)
99	91, 98	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((2 BernPoly 𝑋) / 2))
100	90, 99	oveq12d 7461	. . . . . . . . . 10 ⊢ (𝑘 = 2 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((2 BernPoly 𝑋) / 2)))
101		2nn0 12650	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
102		bpolycl 16247	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (2 BernPoly 𝑋) ∈ ℂ)
103	101, 102	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) ∈ ℂ)
104		2cnne0 12582	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
105		div12 12023	. . . . . . . . . . . . 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 1482	. . . . . . . . . . . 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 12086	. . . . . . . . . . . 12 ⊢ (3 / 2) ∈ ℂ
109		mulcom 11313	. . . . . . . . . . . 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 16252	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))
112	111	oveq2d 7459	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (2 BernPoly 𝑋)) = ((3 / 2) · (((𝑋↑2) − 𝑋) + (1 / 6))))
113		sqcl 14288	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
114		6cn 12461	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℂ
115		6re 12460	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℝ
116		6pos 12483	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 6
117	115, 116	gt0ne0ii 11879	. . . . . . . . . . . . . . . 16 ⊢ 6 ≠ 0
118	114, 117	reccli 12074	. . . . . . . . . . . . . . 15 ⊢ (1 / 6) ∈ ℂ
119		subsub 11615	. . . . . . . . . . . . . . 15 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
120	118, 119	mp3an3 1480	. . . . . . . . . . . . . 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 7459	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = ((3 / 2) · (((𝑋↑2) − 𝑋) + (1 / 6))))
123		subcl 11583	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 − (1 / 6)) ∈ ℂ)
124	118, 123	mpan2 704	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 6)) ∈ ℂ)
125		subdi 11776	. . . . . . . . . . . . 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 1496	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
127	112, 122, 126	3eqtr2d 2804	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (2 BernPoly 𝑋)) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
128	107, 110, 127	3eqtrd 2802	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
129	100, 128	sylan9eqr 2820	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
130		mulcl 11311	. . . . . . . . . . . 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 11311	. . . . . . . . . . . 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 11696	. . . . . . . . . 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 2864	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
137	52, 79, 136	3jaodan 1459	. . . . . . 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 2776	. . . . . . 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 15951	. . . . 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 7459	. . . . 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 15893	. . . . . . . . 9 ⊢ Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))
144		0nn0 12648	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ₀
145		nn0uz 13030	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
146	144, 145	eleqtri 2862	. . . . . . . . . . . 12 ⊢ 0 ∈ (ℤ_≥‘0)
147	146	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
148	13, 16	eqtri 2786	. . . . . . . . . . . . . 14 ⊢ (0...(0 + 1)) = {0, 1}
149	148	eleq2i 2856	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, 1})
150	25	elpr 4609	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 15951	. . . . . . . . . 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 2872	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) ∈ ℂ)
158	42	fsum1 15942	. . . . . . . . . . . . 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 2798	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 / 4))
161	160, 76	oveq12d 7461	. . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = ((1 / 4) + (𝑋 − (1 / 2))))
163	143, 162	eqtr3id 2812	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = ((1 / 4) + (𝑋 − (1 / 2))))
164	163	oveq1d 7458	. . . . . . 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 11309	. . . . . . . . 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 11719	. . . . . . 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 2798	. . . . . 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 11712	. . . . . . . 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 11776	. . . . . . . . . . . 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 1482	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋 − (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
172		addsub12 11597	. . . . . . . . . . . 12 ⊢ (((1 / 4) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 4) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 4) − (1 / 2))))
173	48, 69, 172	mp3an13 1482	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((1 / 4) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 4) − (1 / 2))))
174	171, 173	oveq12d 7461	. . . . . . . . . 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 11311	. . . . . . . . . . . . 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 11343	. . . . . . . . . . . 12 ⊢ ((3 / 2) · (1 / 6)) ∈ ℂ
178		negsub 11633	. . . . . . . . . . . 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 7458	. . . . . . . . . 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 11671	. . . . . . . . . . . . . 14 ⊢ -((1 / 2) − (1 / 4)) = ((1 / 4) − (1 / 2))
182	85, 35, 85	mul12i 11532	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · (3 · 2)) = (3 · (2 · 2))
183		3t2e6 12535	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 · 2) = 6
184	183	oveq2i 7454	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · (3 · 2)) = (2 · 6)
185		2t2e4 12533	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 2) = 4
186	185	oveq2i 7454	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 · (2 · 2)) = (3 · 4)
187	182, 184, 186	3eqtr3i 2794	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 6) = (3 · 4)
188	187	oveq2i 7454	. . . . . . . . . . . . . . . . 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 12046	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (4 ∈ ℂ ∧ 4 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 · 1) / (3 · 4)) = (1 / 4))
192	60, 189, 190, 191	mp3an 1491	. . . . . . . . . . . . . . . . 17 ⊢ ((3 · 1) / (3 · 4)) = (1 / 4)
193	188, 192	eqtri 2786	. . . . . . . . . . . . . . . 16 ⊢ ((3 · 1) / (2 · 6)) = (1 / 4)
194	35, 85, 60, 114, 86, 117	divmuldivi 12104	. . . . . . . . . . . . . . . 16 ⊢ ((3 / 2) · (1 / 6)) = ((3 · 1) / (2 · 6))
195		2t1e2 12532	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 1) = 2
196	195, 5	eqtri 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · 1) = (1 + 1)
197	196, 185	oveq12i 7455	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 1) / (2 · 2)) = ((1 + 1) / 4)
198		divcan5 12046	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 1) / (2 · 2)) = (1 / 2))
199	60, 104, 104, 198	mp3an 1491	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 1) / (2 · 2)) = (1 / 2)
200	60, 60, 46, 47	divdiri 12101	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 + 1) / 4) = ((1 / 4) + (1 / 4))
201	197, 199, 200	3eqtr3ri 2795	. . . . . . . . . . . . . . . . 17 ⊢ ((1 / 4) + (1 / 4)) = (1 / 2)
202	69, 48, 48, 201	subaddrii 11674	. . . . . . . . . . . . . . . 16 ⊢ ((1 / 2) − (1 / 4)) = (1 / 4)
203	193, 194, 202	3eqtr4ri 2797	. . . . . . . . . . . . . . 15 ⊢ ((1 / 2) − (1 / 4)) = ((3 / 2) · (1 / 6))
204	203	negeqi 11577	. . . . . . . . . . . . . 14 ⊢ -((1 / 2) − (1 / 4)) = -((3 / 2) · (1 / 6))
205	181, 204	eqtr3i 2788	. . . . . . . . . . . . 13 ⊢ ((1 / 4) − (1 / 2)) = -((3 / 2) · (1 / 6))
206	48, 69	subcli 11661	. . . . . . . . . . . . . 14 ⊢ ((1 / 4) − (1 / 2)) ∈ ℂ
207	177	negcli 11653	. . . . . . . . . . . . . 14 ⊢ -((3 / 2) · (1 / 6)) ∈ ℂ
208	206, 207	subeq0i 11665	. . . . . . . . . . . . 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 7454	. . . . . . . . . . 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 11744	. . . . . . . . . . 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 11777	. . . . . . . . . . . . . . 15 ⊢ (((3 / 2) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (((3 / 2) − 1) · 𝑋) = (((3 / 2) · 𝑋) − (1 · 𝑋)))
216	108, 60, 215	mp3an12 1481	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) − 1) · 𝑋) = (((3 / 2) · 𝑋) − (1 · 𝑋)))
217		divsubdir 12037	. . . . . . . . . . . . . . . . . 18 ⊢ ((3 ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((3 − 2) / 2) = ((3 / 2) − (2 / 2)))
218	35, 85, 104, 217	mp3an 1491	. . . . . . . . . . . . . . . . 17 ⊢ ((3 − 2) / 2) = ((3 / 2) − (2 / 2))
219	95	oveq1i 7453	. . . . . . . . . . . . . . . . 17 ⊢ ((3 − 2) / 2) = (1 / 2)
220		2div2e1 12510	. . . . . . . . . . . . . . . . . 18 ⊢ (2 / 2) = 1
221	220	oveq2i 7454	. . . . . . . . . . . . . . . . 17 ⊢ ((3 / 2) − (2 / 2)) = ((3 / 2) − 1)
222	218, 219, 221	3eqtr3ri 2795	. . . . . . . . . . . . . . . 16 ⊢ ((3 / 2) − 1) = (1 / 2)
223	222	oveq1i 7453	. . . . . . . . . . . . . . 15 ⊢ (((3 / 2) − 1) · 𝑋) = ((1 / 2) · 𝑋)
224	223	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) − 1) · 𝑋) = ((1 / 2) · 𝑋))
225		mullid 11334	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ℂ → (1 · 𝑋) = 𝑋)
226	225	oveq2d 7459	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) − (1 · 𝑋)) = (((3 / 2) · 𝑋) − 𝑋))
227	216, 224, 226	3eqtr3rd 2807	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) − 𝑋) = ((1 / 2) · 𝑋))
228	227	oveq1d 7458	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − 0) = (((1 / 2) · 𝑋) − 0))
229		mulcl 11311	. . . . . . . . . . . . . 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 11685	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (((1 / 2) · 𝑋) − 0) = ((1 / 2) · 𝑋))
232	228, 231	eqtrd 2798	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − 0) = ((1 / 2) · 𝑋))
233	210, 214, 232	3eqtr3a 2822	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))) = ((1 / 2) · 𝑋))
234	174, 180, 233	3eqtr2d 2804	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = ((1 / 2) · 𝑋))
235	234	negeqd 11578	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = -((1 / 2) · 𝑋))
236	169, 235	eqtr3d 2800	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6)))) = -((1 / 2) · 𝑋))
237	236	oveq2d 7459	. . . . . 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 11702	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) + -((1 / 2) · 𝑋)) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
239	168, 237, 238	3eqtrd 2802	. . . . 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 2802	. . . 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 2810	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
242	241	oveq2d 7459	. 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 14249	. . . 4 ⊢ ((𝑋 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑋↑3) ∈ ℂ)
244	1, 243	mpan2 704	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑3) ∈ ℂ)
245	244, 131, 230	subsubd 11724	. 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 2802	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 2147 ≠ wne 2959 ∪ cun 4152 {csn 4588 {cpr 4590 {ctp 4592 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 0cc0 11227 1c1 11228 + caddc 11230 · cmul 11232 − cmin 11568 -cneg 11569 / cdiv 12000 2c2 12424 3c3 12425 4c4 12426 6c6 12428 ℕ₀cn0 12633 ℤcz 12720 ℤ_≥cuz 12992 ...cfz 13666 ↑cexp 14231 Ccbc 14473 Σcsu 15882 BernPoly cbp 16241

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-sup 9461 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-n0 12634 df-z 12721 df-uz 12993 df-rp 13148 df-fz 13667 df-fzo 13816 df-seq 14172 df-exp 14232 df-fac 14445 df-bc 14474 df-hash 14502 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-clim 15684 df-sum 15883 df-bpoly 16242

This theorem is used by: bpoly4 16254

W3C validator