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Theorem bpoly3 14995

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 11512	. . 3 ⊢ 3 ∈ ℕ₀
2		bpolyval 14986	. . 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 670	. 2 ⊢ (𝑋 ∈ ℂ → (3 BernPoly 𝑋) = ((𝑋↑3) − Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))))
4		3m1e2 11339	. . . . . . 7 ⊢ (3 − 1) = 2
5		df-2 11281	. . . . . . 7 ⊢ 2 = (1 + 1)
6	4, 5	eqtri 2793	. . . . . 6 ⊢ (3 − 1) = (1 + 1)
7	6	oveq2i 6804	. . . . 5 ⊢ (0...(3 − 1)) = (0...(1 + 1))
8	7	sumeq1i 14636	. . . 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 11934	. . . . . . 7 ⊢ 1 ∈ (ℤ_≥‘0)
10	9	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ ℂ → 1 ∈ (ℤ_≥‘0))
11		0z 11590	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
12		fzpr 12603	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
14		0p1e1 11334	. . . . . . . . . . . . 13 ⊢ (0 + 1) = 1
15	14	oveq2i 6804	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = (0...1)
16	14	preq2i 4408	. . . . . . . . . . . 12 ⊢ {0, (0 + 1)} = {0, 1}
17	13, 15, 16	3eqtr3ri 2802	. . . . . . . . . . 11 ⊢ {0, 1} = (0...1)
18	5	sneqi 4327	. . . . . . . . . . 11 ⊢ {2} = {(1 + 1)}
19	17, 18	uneq12i 3916	. . . . . . . . . 10 ⊢ ({0, 1} ∪ {2}) = ((0...1) ∪ {(1 + 1)})
20		df-tp 4321	. . . . . . . . . 10 ⊢ {0, 1, 2} = ({0, 1} ∪ {2})
21		fzsuc 12595	. . . . . . . . . . 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 2804	. . . . . . . . 9 ⊢ (0...(1 + 1)) = {0, 1, 2}
24	23	eleq2i 2842	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(1 + 1)) ↔ 𝑘 ∈ {0, 1, 2})
25		vex 3354	. . . . . . . . 9 ⊢ 𝑘 ∈ V
26	25	eltp 4367	. . . . . . . 8 ⊢ (𝑘 ∈ {0, 1, 2} ↔ (𝑘 = 0 ∨ 𝑘 = 1 ∨ 𝑘 = 2))
27	24, 26	bitri 264	. . . . . . 7 ⊢ (𝑘 ∈ (0...(1 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = 1 ∨ 𝑘 = 2))
28		oveq2 6801	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (3C𝑘) = (3C0))
29		bcn0 13301	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℕ₀ → (3C0) = 1)
30	1, 29	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3C0) = 1
31	28, 30	syl6eq 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (3C𝑘) = 1)
32		oveq1 6800	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (𝑘 BernPoly 𝑋) = (0 BernPoly 𝑋))
33		oveq2 6801	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (3 − 𝑘) = (3 − 0))
34	33	oveq1d 6808	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((3 − 𝑘) + 1) = ((3 − 0) + 1))
35		3cn 11297	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℂ
36	35	subid1i 10555	. . . . . . . . . . . . . . 15 ⊢ (3 − 0) = 3
37	36	oveq1i 6803	. . . . . . . . . . . . . 14 ⊢ ((3 − 0) + 1) = (3 + 1)
38		df-4 11283	. . . . . . . . . . . . . 14 ⊢ 4 = (3 + 1)
39	37, 38	eqtr4i 2796	. . . . . . . . . . . . 13 ⊢ ((3 − 0) + 1) = 4
40	34, 39	syl6eq 2821	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((3 − 𝑘) + 1) = 4)
41	32, 40	oveq12d 6811	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((0 BernPoly 𝑋) / 4))
42	31, 41	oveq12d 6811	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
43		bpoly0 14987	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (0 BernPoly 𝑋) = 1)
44	43	oveq1d 6808	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((0 BernPoly 𝑋) / 4) = (1 / 4))
45	44	oveq2d 6809	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) = (1 · (1 / 4)))
46		4cn 11300	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
47		4ne0 11319	. . . . . . . . . . . . 13 ⊢ 4 ≠ 0
48	46, 47	reccli 10957	. . . . . . . . . . . 12 ⊢ (1 / 4) ∈ ℂ
49	48	mulid2i 10245	. . . . . . . . . . 11 ⊢ (1 · (1 / 4)) = (1 / 4)
50	45, 49	syl6eq 2821	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) = (1 / 4))
51	42, 50	sylan9eqr 2827	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 / 4))
52	51, 48	syl6eqel 2858	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 0) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
53		oveq2 6801	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (3C𝑘) = (3C1))
54		bcn1 13304	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℕ₀ → (3C1) = 3)
55	1, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ (3C1) = 3
56	53, 55	syl6eq 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (3C𝑘) = 3)
57		oveq1 6800	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (𝑘 BernPoly 𝑋) = (1 BernPoly 𝑋))
58		oveq2 6801	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (3 − 𝑘) = (3 − 1))
59	58	oveq1d 6808	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → ((3 − 𝑘) + 1) = ((3 − 1) + 1))
60		ax-1cn 10196	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
61		npcan 10492	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℂ) → ((3 − 1) + 1) = 3)
62	35, 60, 61	mp2an 672	. . . . . . . . . . . . 13 ⊢ ((3 − 1) + 1) = 3
63	59, 62	syl6eq 2821	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((3 − 𝑘) + 1) = 3)
64	57, 63	oveq12d 6811	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((1 BernPoly 𝑋) / 3))
65	56, 64	oveq12d 6811	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((1 BernPoly 𝑋) / 3)))
66		bpoly1 14988	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (1 BernPoly 𝑋) = (𝑋 − (1 / 2)))
67	66	oveq1d 6808	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((1 BernPoly 𝑋) / 3) = ((𝑋 − (1 / 2)) / 3))
68	67	oveq2d 6809	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((1 BernPoly 𝑋) / 3)) = (3 · ((𝑋 − (1 / 2)) / 3)))
69		halfcn 11449	. . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℂ
70		subcl 10482	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝑋 − (1 / 2)) ∈ ℂ)
71	69, 70	mpan2 671	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 2)) ∈ ℂ)
72		3ne0 11317	. . . . . . . . . . . . 13 ⊢ 3 ≠ 0
73		divcan2 10895	. . . . . . . . . . . . 13 ⊢ (((𝑋 − (1 / 2)) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
74	35, 72, 73	mp3an23 1564	. . . . . . . . . . . 12 ⊢ ((𝑋 − (1 / 2)) ∈ ℂ → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
75	71, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((𝑋 − (1 / 2)) / 3)) = (𝑋 − (1 / 2)))
76	68, 75	eqtrd 2805	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (3 · ((1 BernPoly 𝑋) / 3)) = (𝑋 − (1 / 2)))
77	65, 76	sylan9eqr 2827	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (𝑋 − (1 / 2)))
78	71	adantr 466	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → (𝑋 − (1 / 2)) ∈ ℂ)
79	77, 78	eqeltrd 2850	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
80		oveq2 6801	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → (3C𝑘) = (3C2))
81		bcn2 13310	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ₀ → (3C2) = ((3 · (3 − 1)) / 2))
82	1, 81	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (3C2) = ((3 · (3 − 1)) / 2)
83	4	oveq2i 6804	. . . . . . . . . . . . . . 15 ⊢ (3 · (3 − 1)) = (3 · 2)
84	83	oveq1i 6803	. . . . . . . . . . . . . 14 ⊢ ((3 · (3 − 1)) / 2) = ((3 · 2) / 2)
85		2cn 11293	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
86		2ne0 11315	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
87	35, 85, 86	divcan4i 10974	. . . . . . . . . . . . . 14 ⊢ ((3 · 2) / 2) = 3
88	84, 87	eqtri 2793	. . . . . . . . . . . . 13 ⊢ ((3 · (3 − 1)) / 2) = 3
89	82, 88	eqtri 2793	. . . . . . . . . . . 12 ⊢ (3C2) = 3
90	80, 89	syl6eq 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (3C𝑘) = 3)
91		oveq1 6800	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → (𝑘 BernPoly 𝑋) = (2 BernPoly 𝑋))
92		oveq2 6801	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 2 → (3 − 𝑘) = (3 − 2))
93	92	oveq1d 6808	. . . . . . . . . . . . 13 ⊢ (𝑘 = 2 → ((3 − 𝑘) + 1) = ((3 − 2) + 1))
94		2p1e3 11353	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = 3
95	35, 85, 60, 94	subaddrii 10572	. . . . . . . . . . . . . . 15 ⊢ (3 − 2) = 1
96	95	oveq1i 6803	. . . . . . . . . . . . . 14 ⊢ ((3 − 2) + 1) = (1 + 1)
97	96, 5	eqtr4i 2796	. . . . . . . . . . . . 13 ⊢ ((3 − 2) + 1) = 2
98	93, 97	syl6eq 2821	. . . . . . . . . . . 12 ⊢ (𝑘 = 2 → ((3 − 𝑘) + 1) = 2)
99	91, 98	oveq12d 6811	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)) = ((2 BernPoly 𝑋) / 2))
100	90, 99	oveq12d 6811	. . . . . . . . . 10 ⊢ (𝑘 = 2 → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((2 BernPoly 𝑋) / 2)))
101		2nn0 11511	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
102		bpolycl 14989	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℕ₀ ∧ 𝑋 ∈ ℂ) → (2 BernPoly 𝑋) ∈ ℂ)
103	101, 102	mpan 670	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) ∈ ℂ)
104		2cnne0 11444	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
105		div12 10909	. . . . . . . . . . . . 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 1563	. . . . . . . . . . . 12 ⊢ ((2 BernPoly 𝑋) ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = ((2 BernPoly 𝑋) · (3 / 2)))
107	103, 106	syl 17	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = ((2 BernPoly 𝑋) · (3 / 2)))
108	35, 85, 86	divcli 10969	. . . . . . . . . . . 12 ⊢ (3 / 2) ∈ ℂ
109		mulcom 10224	. . . . . . . . . . . 12 ⊢ (((2 BernPoly 𝑋) ∈ ℂ ∧ (3 / 2) ∈ ℂ) → ((2 BernPoly 𝑋) · (3 / 2)) = ((3 / 2) · (2 BernPoly 𝑋)))
110	103, 108, 109	sylancl 574	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((2 BernPoly 𝑋) · (3 / 2)) = ((3 / 2) · (2 BernPoly 𝑋)))
111		bpoly2 14994	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (2 BernPoly 𝑋) = (((𝑋↑2) − 𝑋) + (1 / 6)))
112	111	oveq2d 6809	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (2 BernPoly 𝑋)) = ((3 / 2) · (((𝑋↑2) − 𝑋) + (1 / 6))))
113		sqcl 13132	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (𝑋↑2) ∈ ℂ)
114		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ ℂ)
115		6cn 11304	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℂ
116		6re 11303	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℝ
117		6pos 11321	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 6
118	116, 117	gt0ne0ii 10766	. . . . . . . . . . . . . . . 16 ⊢ 6 ≠ 0
119	115, 118	reccli 10957	. . . . . . . . . . . . . . 15 ⊢ (1 / 6) ∈ ℂ
120		subsub 10513	. . . . . . . . . . . . . . 15 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
121	119, 120	mp3an3 1561	. . . . . . . . . . . . . 14 ⊢ (((𝑋↑2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
122	113, 114, 121	syl2anc 573	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((𝑋↑2) − (𝑋 − (1 / 6))) = (((𝑋↑2) − 𝑋) + (1 / 6)))
123	122	oveq2d 6809	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = ((3 / 2) · (((𝑋↑2) − 𝑋) + (1 / 6))))
124		subcl 10482	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → (𝑋 − (1 / 6)) ∈ ℂ)
125	119, 124	mpan2 671	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (𝑋 − (1 / 6)) ∈ ℂ)
126		subdi 10665	. . . . . . . . . . . . . 14 ⊢ (((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)))))
127	108, 126	mp3an1 1559	. . . . . . . . . . . . 13 ⊢ (((𝑋↑2) ∈ ℂ ∧ (𝑋 − (1 / 6)) ∈ ℂ) → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
128	113, 125, 127	syl2anc 573	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · ((𝑋↑2) − (𝑋 − (1 / 6)))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
129	112, 123, 128	3eqtr2d 2811	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (2 BernPoly 𝑋)) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
130	107, 110, 129	3eqtrd 2809	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (3 · ((2 BernPoly 𝑋) / 2)) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
131	100, 130	sylan9eqr 2827	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))))
132		mulcl 10222	. . . . . . . . . . . 12 ⊢ (((3 / 2) ∈ ℂ ∧ (𝑋↑2) ∈ ℂ) → ((3 / 2) · (𝑋↑2)) ∈ ℂ)
133	108, 113, 132	sylancr 575	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋↑2)) ∈ ℂ)
134		mulcl 10222	. . . . . . . . . . . 12 ⊢ (((3 / 2) ∈ ℂ ∧ (𝑋 − (1 / 6)) ∈ ℂ) → ((3 / 2) · (𝑋 − (1 / 6))) ∈ ℂ)
135	108, 125, 134	sylancr 575	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋 − (1 / 6))) ∈ ℂ)
136	133, 135	subcld 10594	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))) ∈ ℂ)
137	136	adantr 466	. . . . . . . . 9 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → (((3 / 2) · (𝑋↑2)) − ((3 / 2) · (𝑋 − (1 / 6)))) ∈ ℂ)
138	131, 137	eqeltrd 2850	. . . . . . . 8 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 = 2) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
139	52, 79, 138	3jaodan 1542	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = 1 ∨ 𝑘 = 2)) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
140	27, 139	sylan2b 581	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(1 + 1))) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
141	5	eqeq2i 2783	. . . . . . 7 ⊢ (𝑘 = 2 ↔ 𝑘 = (1 + 1))
142	141, 100	sylbir 225	. . . . . 6 ⊢ (𝑘 = (1 + 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((2 BernPoly 𝑋) / 2)))
143	10, 140, 142	fsump1 14695	. . . . 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))))
144	130	oveq2d 6809	. . . . 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))))))
145	15	sumeq1i 14636	. . . . . . . . 9 ⊢ Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1)))
146		0nn0 11509	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℕ₀
147		nn0uz 11924	. . . . . . . . . . . . 13 ⊢ ℕ₀ = (ℤ_≥‘0)
148	146, 147	eleqtri 2848	. . . . . . . . . . . 12 ⊢ 0 ∈ (ℤ_≥‘0)
149	148	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → 0 ∈ (ℤ_≥‘0))
150	13, 16	eqtri 2793	. . . . . . . . . . . . . 14 ⊢ (0...(0 + 1)) = {0, 1}
151	150	eleq2i 2842	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ 𝑘 ∈ {0, 1})
152	25	elpr 4338	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {0, 1} ↔ (𝑘 = 0 ∨ 𝑘 = 1))
153	151, 152	bitri 264	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0...(0 + 1)) ↔ (𝑘 = 0 ∨ 𝑘 = 1))
154	52, 79	jaodan 942	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ ℂ ∧ (𝑘 = 0 ∨ 𝑘 = 1)) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
155	153, 154	sylan2b 581	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℂ ∧ 𝑘 ∈ (0...(0 + 1))) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) ∈ ℂ)
156	14	eqeq2i 2783	. . . . . . . . . . . 12 ⊢ (𝑘 = (0 + 1) ↔ 𝑘 = 1)
157	156, 65	sylbi 207	. . . . . . . . . . 11 ⊢ (𝑘 = (0 + 1) → ((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (3 · ((1 BernPoly 𝑋) / 3)))
158	149, 155, 157	fsump1 14695	. . . . . . . . . 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))))
159	50, 48	syl6eqel 2858	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (1 · ((0 BernPoly 𝑋) / 4)) ∈ ℂ)
160	42	fsum1 14684	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℤ ∧ (1 · ((0 BernPoly 𝑋) / 4)) ∈ ℂ) → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
161	11, 159, 160	sylancr 575	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 · ((0 BernPoly 𝑋) / 4)))
162	161, 50	eqtrd 2805	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (1 / 4))
163	162, 76	oveq12d 6811	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → (Σ𝑘 ∈ (0...0)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) + (3 · ((1 BernPoly 𝑋) / 3))) = ((1 / 4) + (𝑋 − (1 / 2))))
164	158, 163	eqtrd 2805	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(0 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = ((1 / 4) + (𝑋 − (1 / 2))))
165	145, 164	syl5eqr 2819	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...1)((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = ((1 / 4) + (𝑋 − (1 / 2))))
166	165	oveq1d 6808	. . . . . . 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))))))
167		addcl 10220	. . . . . . . . 9 ⊢ (((1 / 4) ∈ ℂ ∧ (𝑋 − (1 / 2)) ∈ ℂ) → ((1 / 4) + (𝑋 − (1 / 2))) ∈ ℂ)
168	48, 71, 167	sylancr 575	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → ((1 / 4) + (𝑋 − (1 / 2))) ∈ ℂ)
169	168, 133, 135	addsub12d 10617	. . . . . . 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))))))
170	166, 169	eqtrd 2805	. . . . . 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))))))
171	135, 168	negsubdi2d 10610	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6)))))
172		subdi 10665	. . . . . . . . . . . 12 ⊢ (((3 / 2) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 6) ∈ ℂ) → ((3 / 2) · (𝑋 − (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
173	108, 119, 172	mp3an13 1563	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · (𝑋 − (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
174		addsub12 10496	. . . . . . . . . . . 12 ⊢ (((1 / 4) ∈ ℂ ∧ 𝑋 ∈ ℂ ∧ (1 / 2) ∈ ℂ) → ((1 / 4) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 4) − (1 / 2))))
175	48, 69, 174	mp3an13 1563	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((1 / 4) + (𝑋 − (1 / 2))) = (𝑋 + ((1 / 4) − (1 / 2))))
176	173, 175	oveq12d 6811	. . . . . . . . . 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)))))
177		mulcl 10222	. . . . . . . . . . . . 13 ⊢ (((3 / 2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((3 / 2) · 𝑋) ∈ ℂ)
178	108, 177	mpan 670	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((3 / 2) · 𝑋) ∈ ℂ)
179	108, 119	mulcli 10247	. . . . . . . . . . . 12 ⊢ ((3 / 2) · (1 / 6)) ∈ ℂ
180		negsub 10531	. . . . . . . . . . . 12 ⊢ ((((3 / 2) · 𝑋) ∈ ℂ ∧ ((3 / 2) · (1 / 6)) ∈ ℂ) → (((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
181	178, 179, 180	sylancl 574	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) = (((3 / 2) · 𝑋) − ((3 / 2) · (1 / 6))))
182	181	oveq1d 6808	. . . . . . . . . 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)))))
183	69, 48	negsubdi2i 10569	. . . . . . . . . . . . . 14 ⊢ -((1 / 2) − (1 / 4)) = ((1 / 4) − (1 / 2))
184	85, 35, 85	mul12i 10433	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · (3 · 2)) = (3 · (2 · 2))
185		3t2e6 11381	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 · 2) = 6
186	185	oveq2i 6804	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · (3 · 2)) = (2 · 6)
187		2t2e4 11379	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 2) = 4
188	187	oveq2i 6804	. . . . . . . . . . . . . . . . . . 19 ⊢ (3 · (2 · 2)) = (3 · 4)
189	184, 186, 188	3eqtr3i 2801	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 6) = (3 · 4)
190	189	oveq2i 6804	. . . . . . . . . . . . . . . . 17 ⊢ ((3 · 1) / (2 · 6)) = ((3 · 1) / (3 · 4))
191	46, 47	pm3.2i 447	. . . . . . . . . . . . . . . . . 18 ⊢ (4 ∈ ℂ ∧ 4 ≠ 0)
192	35, 72	pm3.2i 447	. . . . . . . . . . . . . . . . . 18 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
193		divcan5 10929	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℂ ∧ (4 ∈ ℂ ∧ 4 ≠ 0) ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((3 · 1) / (3 · 4)) = (1 / 4))
194	60, 191, 192, 193	mp3an 1572	. . . . . . . . . . . . . . . . 17 ⊢ ((3 · 1) / (3 · 4)) = (1 / 4)
195	190, 194	eqtri 2793	. . . . . . . . . . . . . . . 16 ⊢ ((3 · 1) / (2 · 6)) = (1 / 4)
196	35, 85, 60, 115, 86, 118	divmuldivi 10987	. . . . . . . . . . . . . . . 16 ⊢ ((3 / 2) · (1 / 6)) = ((3 · 1) / (2 · 6))
197		2t1e2 11378	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2 · 1) = 2
198	197, 5	eqtri 2793	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 · 1) = (1 + 1)
199	198, 187	oveq12i 6805	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 1) / (2 · 2)) = ((1 + 1) / 4)
200		divcan5 10929	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 1) / (2 · 2)) = (1 / 2))
201	60, 104, 104, 200	mp3an 1572	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 1) / (2 · 2)) = (1 / 2)
202	60, 60, 46, 47	divdiri 10984	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 + 1) / 4) = ((1 / 4) + (1 / 4))
203	199, 201, 202	3eqtr3ri 2802	. . . . . . . . . . . . . . . . 17 ⊢ ((1 / 4) + (1 / 4)) = (1 / 2)
204	69, 48, 48, 203	subaddrii 10572	. . . . . . . . . . . . . . . 16 ⊢ ((1 / 2) − (1 / 4)) = (1 / 4)
205	195, 196, 204	3eqtr4ri 2804	. . . . . . . . . . . . . . 15 ⊢ ((1 / 2) − (1 / 4)) = ((3 / 2) · (1 / 6))
206	205	negeqi 10476	. . . . . . . . . . . . . 14 ⊢ -((1 / 2) − (1 / 4)) = -((3 / 2) · (1 / 6))
207	183, 206	eqtr3i 2795	. . . . . . . . . . . . 13 ⊢ ((1 / 4) − (1 / 2)) = -((3 / 2) · (1 / 6))
208	48, 69	subcli 10559	. . . . . . . . . . . . . 14 ⊢ ((1 / 4) − (1 / 2)) ∈ ℂ
209	179	negcli 10551	. . . . . . . . . . . . . 14 ⊢ -((3 / 2) · (1 / 6)) ∈ ℂ
210	208, 209	subeq0i 10563	. . . . . . . . . . . . 13 ⊢ ((((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6))) = 0 ↔ ((1 / 4) − (1 / 2)) = -((3 / 2) · (1 / 6)))
211	207, 210	mpbir 221	. . . . . . . . . . . 12 ⊢ (((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6))) = 0
212	211	oveq2i 6804	. . . . . . . . . . 11 ⊢ ((((3 / 2) · 𝑋) − 𝑋) − (((1 / 4) − (1 / 2)) − -((3 / 2) · (1 / 6)))) = ((((3 / 2) · 𝑋) − 𝑋) − 0)
213	208	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((1 / 4) − (1 / 2)) ∈ ℂ)
214	209	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → -((3 / 2) · (1 / 6)) ∈ ℂ)
215	178, 114, 213, 214	subadd4d 10642	. . . . . . . . . . 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)))))
216		subdir 10666	. . . . . . . . . . . . . . 15 ⊢ (((3 / 2) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑋 ∈ ℂ) → (((3 / 2) − 1) · 𝑋) = (((3 / 2) · 𝑋) − (1 · 𝑋)))
217	108, 60, 216	mp3an12 1562	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) − 1) · 𝑋) = (((3 / 2) · 𝑋) − (1 · 𝑋)))
218		divsubdir 10923	. . . . . . . . . . . . . . . . . 18 ⊢ ((3 ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((3 − 2) / 2) = ((3 / 2) − (2 / 2)))
219	35, 85, 104, 218	mp3an 1572	. . . . . . . . . . . . . . . . 17 ⊢ ((3 − 2) / 2) = ((3 / 2) − (2 / 2))
220	95	oveq1i 6803	. . . . . . . . . . . . . . . . 17 ⊢ ((3 − 2) / 2) = (1 / 2)
221		2div2e1 11352	. . . . . . . . . . . . . . . . . 18 ⊢ (2 / 2) = 1
222	221	oveq2i 6804	. . . . . . . . . . . . . . . . 17 ⊢ ((3 / 2) − (2 / 2)) = ((3 / 2) − 1)
223	219, 220, 222	3eqtr3ri 2802	. . . . . . . . . . . . . . . 16 ⊢ ((3 / 2) − 1) = (1 / 2)
224	223	oveq1i 6803	. . . . . . . . . . . . . . 15 ⊢ (((3 / 2) − 1) · 𝑋) = ((1 / 2) · 𝑋)
225	224	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) − 1) · 𝑋) = ((1 / 2) · 𝑋))
226		mulid2 10240	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ ℂ → (1 · 𝑋) = 𝑋)
227	226	oveq2d 6809	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) − (1 · 𝑋)) = (((3 / 2) · 𝑋) − 𝑋))
228	217, 225, 227	3eqtr3rd 2814	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · 𝑋) − 𝑋) = ((1 / 2) · 𝑋))
229	228	oveq1d 6808	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − 0) = (((1 / 2) · 𝑋) − 0))
230		mulcl 10222	. . . . . . . . . . . . . 14 ⊢ (((1 / 2) ∈ ℂ ∧ 𝑋 ∈ ℂ) → ((1 / 2) · 𝑋) ∈ ℂ)
231	69, 230	mpan 670	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ ℂ → ((1 / 2) · 𝑋) ∈ ℂ)
232	231	subid1d 10583	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (((1 / 2) · 𝑋) − 0) = ((1 / 2) · 𝑋))
233	229, 232	eqtrd 2805	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) − 𝑋) − 0) = ((1 / 2) · 𝑋))
234	212, 215, 233	3eqtr3a 2829	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℂ → ((((3 / 2) · 𝑋) + -((3 / 2) · (1 / 6))) − (𝑋 + ((1 / 4) − (1 / 2)))) = ((1 / 2) · 𝑋))
235	176, 182, 234	3eqtr2d 2811	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = ((1 / 2) · 𝑋))
236	235	negeqd 10477	. . . . . . . 8 ⊢ (𝑋 ∈ ℂ → -(((3 / 2) · (𝑋 − (1 / 6))) − ((1 / 4) + (𝑋 − (1 / 2)))) = -((1 / 2) · 𝑋))
237	171, 236	eqtr3d 2807	. . . . . . 7 ⊢ (𝑋 ∈ ℂ → (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6)))) = -((1 / 2) · 𝑋))
238	237	oveq2d 6809	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) + (((1 / 4) + (𝑋 − (1 / 2))) − ((3 / 2) · (𝑋 − (1 / 6))))) = (((3 / 2) · (𝑋↑2)) + -((1 / 2) · 𝑋)))
239	133, 231	negsubd 10600	. . . . . 6 ⊢ (𝑋 ∈ ℂ → (((3 / 2) · (𝑋↑2)) + -((1 / 2) · 𝑋)) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
240	170, 238, 239	3eqtrd 2809	. . . . 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)))
241	143, 144, 240	3eqtrd 2809	. . . 4 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(1 + 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
242	8, 241	syl5eq 2817	. . 3 ⊢ (𝑋 ∈ ℂ → Σ𝑘 ∈ (0...(3 − 1))((3C𝑘) · ((𝑘 BernPoly 𝑋) / ((3 − 𝑘) + 1))) = (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋)))
243	242	oveq2d 6809	. 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))))
244		expcl 13085	. . . 4 ⊢ ((𝑋 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑋↑3) ∈ ℂ)
245	1, 244	mpan2 671	. . 3 ⊢ (𝑋 ∈ ℂ → (𝑋↑3) ∈ ℂ)
246	245, 133, 231	subsubd 10622	. 2 ⊢ (𝑋 ∈ ℂ → ((𝑋↑3) − (((3 / 2) · (𝑋↑2)) − ((1 / 2) · 𝑋))) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))
247	3, 243, 246	3eqtrd 2809	1 ⊢ (𝑋 ∈ ℂ → (3 BernPoly 𝑋) = (((𝑋↑3) − ((3 / 2) · (𝑋↑2))) + ((1 / 2) · 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∨ wo 836 ∨ w3o 1070 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∪ cun 3721 {csn 4316 {cpr 4318 {ctp 4320 ‘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 4c4 11274 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 837 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: bpoly4 14996

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