lhe4.4ex1a - Mathbox for Steve Rodriguez

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Theorem lhe4.4ex1a 41620

Description: Example of the Fundamental Theorem of Calculus, part two (ftc2 24941): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 24941 as simply the "Fundamental Theorem of Calculus", then ftc1 24939 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)

**Assertion**
Ref	Expression
lhe4.4ex1a	⊢ ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3)

Proof of Theorem lhe4.4ex1a Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 10834	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
2		2re 11904	. . . . 5 ⊢ 2 ∈ ℝ
3	2	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
4		1le2 12039	. . . . 5 ⊢ 1 ≤ 2
5	4	a1i 11	. . . 4 ⊢ (⊤ → 1 ≤ 2)
6		reelprrecn 10821	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
8		recn 10819	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
9		3nn0 12108	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
10		expcl 13653	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑦↑3) ∈ ℂ)
11	9, 10	mpan2 691	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → (𝑦↑3) ∈ ℂ)
12	8, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦↑3) ∈ ℂ)
13		3cn 11911	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
14		3ne0 11936	. . . . . . . . . 10 ⊢ 3 ≠ 0
15		divcl 11496	. . . . . . . . . 10 ⊢ (((𝑦↑3) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑦↑3) / 3) ∈ ℂ)
16	13, 14, 15	mp3an23 1455	. . . . . . . . 9 ⊢ ((𝑦↑3) ∈ ℂ → ((𝑦↑3) / 3) ∈ ℂ)
17	12, 16	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝑦↑3) / 3) ∈ ℂ)
18		mulcl 10813	. . . . . . . . 9 ⊢ ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (3 · 𝑦) ∈ ℂ)
19	13, 8, 18	sylancr 590	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (3 · 𝑦) ∈ ℂ)
20	17, 19	subcld 11189	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦↑3) / 3) − (3 · 𝑦)) ∈ ℂ)
21	20	adantl 485	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (((𝑦↑3) / 3) − (3 · 𝑦)) ∈ ℂ)
22		ovexd 7248	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → ((𝑦↑2) − 3) ∈ V)
23	17	adantl 485	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → ((𝑦↑3) / 3) ∈ ℂ)
24		ovexd 7248	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (𝑦↑2) ∈ V)
25		divrec2 11507	. . . . . . . . . . . . 13 ⊢ (((𝑦↑3) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
26	13, 14, 25	mp3an23 1455	. . . . . . . . . . . 12 ⊢ ((𝑦↑3) ∈ ℂ → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
27	12, 26	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
28	27	mpteq2ia 5146	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3)) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))
29	28	oveq2i 7224	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3))) = (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3))))
30	12	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (𝑦↑3) ∈ ℂ)
31		ovexd 7248	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (3 · (𝑦↑2)) ∈ V)
32		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ ↦ (𝑦↑3)) = (𝑦 ∈ ℂ ↦ (𝑦↑3))
33	32, 11	fmpti 6929	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℂ ↦ (𝑦↑3)):ℂ⟶ℂ
34		ssid 3923	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
35		ax-resscn 10786	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
36		ovex 7246	. . . . . . . . . . . . . . . 16 ⊢ (3 · (𝑦↑2)) ∈ V
37		3nn 11909	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℕ
38		dvexp 24850	. . . . . . . . . . . . . . . . . 18 ⊢ (3 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑(3 − 1)))))
39	37, 38	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑(3 − 1))))
40		3m1e2 11958	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 − 1) = 2
41	40	oveq2i 7224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦↑(3 − 1)) = (𝑦↑2)
42	41	oveq2i 7224	. . . . . . . . . . . . . . . . . 18 ⊢ (3 · (𝑦↑(3 − 1))) = (3 · (𝑦↑2))
43	42	mpteq2i 5147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ ↦ (3 · (𝑦↑(3 − 1)))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑2)))
44	39, 43	eqtri 2765	. . . . . . . . . . . . . . . 16 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑2)))
45	36, 44	dmmpti 6522	. . . . . . . . . . . . . . 15 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = ℂ
46	35, 45	sseqtrri 3938	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3)))
47		dvres3 24810	. . . . . . . . . . . . . 14 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ (𝑦 ∈ ℂ ↦ (𝑦↑3)):ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))))) → (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ))
48	6, 33, 34, 46, 47	mp4an 693	. . . . . . . . . . . . 13 ⊢ (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ)
49		resmpt 5905	. . . . . . . . . . . . . . 15 ⊢ (ℝ ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (𝑦↑3)))
50	35, 49	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (𝑦↑3))
51	50	oveq2i 7224	. . . . . . . . . . . . 13 ⊢ (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = (ℝ D (𝑦 ∈ ℝ ↦ (𝑦↑3)))
52	44	reseq1i 5847	. . . . . . . . . . . . . 14 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ) = ((𝑦 ∈ ℂ ↦ (3 · (𝑦↑2))) ↾ ℝ)
53		resmpt 5905	. . . . . . . . . . . . . . 15 ⊢ (ℝ ⊆ ℂ → ((𝑦 ∈ ℂ ↦ (3 · (𝑦↑2))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2))))
54	35, 53	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℂ ↦ (3 · (𝑦↑2))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2)))
55	52, 54	eqtri 2765	. . . . . . . . . . . . 13 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2)))
56	48, 51, 55	3eqtr3i 2773	. . . . . . . . . . . 12 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ (𝑦↑3))) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2)))
57	56	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (𝑦↑3))) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2))))
58		ax-1cn 10787	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
59	58, 13, 14	divcli 11574	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
60	59	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (1 / 3) ∈ ℂ)
61	7, 30, 31, 57, 60	dvmptcmul 24861	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2)))))
62	61	mptru 1550	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2))))
63		sqcl 13690	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
64		mulcl 10813	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ (𝑦↑2) ∈ ℂ) → (3 · (𝑦↑2)) ∈ ℂ)
65	13, 63, 64	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (3 · (𝑦↑2)) ∈ ℂ)
66		divrec2 11507	. . . . . . . . . . . . 13 ⊢ (((3 · (𝑦↑2)) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
67	13, 14, 66	mp3an23 1455	. . . . . . . . . . . 12 ⊢ ((3 · (𝑦↑2)) ∈ ℂ → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
68	8, 65, 67	3syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
69		divcan3 11516	. . . . . . . . . . . . 13 ⊢ (((𝑦↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
70	13, 14, 69	mp3an23 1455	. . . . . . . . . . . 12 ⊢ ((𝑦↑2) ∈ ℂ → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
71	8, 63, 70	3syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
72	68, 71	eqtr3d 2779	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → ((1 / 3) · (3 · (𝑦↑2))) = (𝑦↑2))
73	72	mpteq2ia 5146	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2)))) = (𝑦 ∈ ℝ ↦ (𝑦↑2))
74	29, 62, 73	3eqtri 2769	. . . . . . . 8 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3))) = (𝑦 ∈ ℝ ↦ (𝑦↑2))
75	74	a1i 11	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3))) = (𝑦 ∈ ℝ ↦ (𝑦↑2)))
76	19	adantl 485	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (3 · 𝑦) ∈ ℂ)
77		3ex 11912	. . . . . . . 8 ⊢ 3 ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 3 ∈ V)
79	8	adantl 485	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
80		1red 10834	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℝ)
81	7	dvmptid 24854	. . . . . . . . 9 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
82	13	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℂ)
83	7, 79, 80, 81, 82	dvmptcmul 24861	. . . . . . . 8 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ (3 · 1)))
84		3t1e3 11995	. . . . . . . . 9 ⊢ (3 · 1) = 3
85	84	mpteq2i 5147	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ ↦ (3 · 1)) = (𝑦 ∈ ℝ ↦ 3)
86	83, 85	eqtrdi 2794	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ 3))
87	7, 23, 24, 75, 76, 78, 86	dvmptsub 24864	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑦↑2) − 3)))
88		1re 10833	. . . . . . . 8 ⊢ 1 ∈ ℝ
89		iccssre 13017	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → (1[,]2) ⊆ ℝ)
90	88, 2, 89	mp2an 692	. . . . . . 7 ⊢ (1[,]2) ⊆ ℝ
91	90	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,]2) ⊆ ℝ)
92		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
93	92	tgioo2 23700	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
94		iccntr 23718	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(1[,]2)) = (1(,)2))
95	88, 2, 94	mp2an 692	. . . . . . 7 ⊢ ((int‘(topGen‘ran (,)))‘(1[,]2)) = (1(,)2)
96	95	a1i 11	. . . . . 6 ⊢ (⊤ → ((int‘(topGen‘ran (,)))‘(1[,]2)) = (1(,)2))
97	7, 21, 22, 87, 91, 93, 92, 96	dvmptres2 24859	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)))
98		ioossicc 13021	. . . . . . 7 ⊢ (1(,)2) ⊆ (1[,]2)
99		resmpt 5905	. . . . . . 7 ⊢ ((1(,)2) ⊆ (1[,]2) → ((𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ↾ (1(,)2)) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)))
100	98, 99	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ↾ (1(,)2)) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3))
101	90, 35	sstri 3910	. . . . . . . . 9 ⊢ (1[,]2) ⊆ ℂ
102		resmpt 5905	. . . . . . . . 9 ⊢ ((1[,]2) ⊆ ℂ → ((𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ↾ (1[,]2)) = (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)))
103	101, 102	ax-mp 5	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ↾ (1[,]2)) = (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3))
104		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) = (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))
105		subcl 11077	. . . . . . . . . . . . . 14 ⊢ (((𝑦↑2) ∈ ℂ ∧ 3 ∈ ℂ) → ((𝑦↑2) − 3) ∈ ℂ)
106	13, 105	mpan2 691	. . . . . . . . . . . . 13 ⊢ ((𝑦↑2) ∈ ℂ → ((𝑦↑2) − 3) ∈ ℂ)
107	63, 106	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) − 3) ∈ ℂ)
108	104, 107	fmpti 6929	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ
109	34, 108, 34	3pm3.2i 1341	. . . . . . . . . 10 ⊢ (ℂ ⊆ ℂ ∧ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ ∧ ℂ ⊆ ℂ)
110		ovex 7246	. . . . . . . . . . 11 ⊢ ((2 · (𝑦↑(2 − 1))) − 0) ∈ V
111		cnelprrecn 10822	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ {ℝ, ℂ}
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
113	63	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
114		ovexd 7248	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (2 · (𝑦↑(2 − 1))) ∈ V)
115		2nn 11903	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
116		dvexp 24850	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))))
117	115, 116	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1))))
118	117	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑2))) = (𝑦 ∈ ℂ ↦ (2 · (𝑦↑(2 − 1)))))
119	13	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 3 ∈ ℂ)
120		c0ex 10827	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
121	120	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ V)
122	112, 82	dvmptc 24855	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 3)) = (𝑦 ∈ ℂ ↦ 0))
123	112, 113, 114, 118, 119, 121, 122	dvmptsub 24864	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = (𝑦 ∈ ℂ ↦ ((2 · (𝑦↑(2 − 1))) − 0)))
124	123	mptru 1550	. . . . . . . . . . 11 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = (𝑦 ∈ ℂ ↦ ((2 · (𝑦↑(2 − 1))) − 0))
125	110, 124	dmmpti 6522	. . . . . . . . . 10 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = ℂ
126		dvcn 24818	. . . . . . . . . 10 ⊢ (((ℂ ⊆ ℂ ∧ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ ∧ ℂ ⊆ ℂ) ∧ dom (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = ℂ) → (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ∈ (ℂ–cn→ℂ))
127	109, 125, 126	mp2an 692	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ∈ (ℂ–cn→ℂ)
128		rescncf 23794	. . . . . . . . 9 ⊢ ((1[,]2) ⊆ ℂ → ((𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ∈ (ℂ–cn→ℂ) → ((𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ↾ (1[,]2)) ∈ ((1[,]2)–cn→ℂ)))
129	101, 127, 128	mp2 9	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ↾ (1[,]2)) ∈ ((1[,]2)–cn→ℂ)
130	103, 129	eqeltrri 2835	. . . . . . 7 ⊢ (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ ((1[,]2)–cn→ℂ)
131		rescncf 23794	. . . . . . 7 ⊢ ((1(,)2) ⊆ (1[,]2) → ((𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ ((1[,]2)–cn→ℂ) → ((𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ↾ (1(,)2)) ∈ ((1(,)2)–cn→ℂ)))
132	98, 130, 131	mp2 9	. . . . . 6 ⊢ ((𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ↾ (1(,)2)) ∈ ((1(,)2)–cn→ℂ)
133	100, 132	eqeltrri 2835	. . . . 5 ⊢ (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)) ∈ ((1(,)2)–cn→ℂ)
134	97, 133	eqeltrdi 2846	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) ∈ ((1(,)2)–cn→ℂ))
135	98	a1i 11	. . . . . 6 ⊢ (⊤ → (1(,)2) ⊆ (1[,]2))
136		ioombl 24462	. . . . . . 7 ⊢ (1(,)2) ∈ dom vol
137	136	a1i 11	. . . . . 6 ⊢ (⊤ → (1(,)2) ∈ dom vol)
138		ovexd 7248	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,]2)) → ((𝑦↑2) − 3) ∈ V)
139		cniccibl 24738	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ ∧ (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ ((1[,]2)–cn→ℂ)) → (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ 𝐿¹)
140	88, 2, 130, 139	mp3an 1463	. . . . . . 7 ⊢ (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ 𝐿¹
141	140	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ 𝐿¹)
142	135, 137, 138, 141	iblss 24702	. . . . 5 ⊢ (⊤ → (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)) ∈ 𝐿¹)
143	97, 142	eqeltrd 2838	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) ∈ 𝐿¹)
144		resmpt 5905	. . . . . . 7 ⊢ ((1[,]2) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ↾ (1[,]2)) = (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))
145	90, 144	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ↾ (1[,]2)) = (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
146		eqid 2737	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
147	146, 20	fmpti 6929	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ
148		ssid 3923	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
149	35, 147, 148	3pm3.2i 1341	. . . . . . . 8 ⊢ (ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ ∧ ℝ ⊆ ℝ)
150		ovex 7246	. . . . . . . . 9 ⊢ ((𝑦↑2) − 3) ∈ V
151	87	mptru 1550	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑦↑2) − 3))
152	150, 151	dmmpti 6522	. . . . . . . 8 ⊢ dom (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = ℝ
153		dvcn 24818	. . . . . . . 8 ⊢ (((ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ ∧ ℝ ⊆ ℝ) ∧ dom (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = ℝ) → (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ (ℝ–cn→ℂ))
154	149, 152, 153	mp2an 692	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ (ℝ–cn→ℂ)
155		rescncf 23794	. . . . . . 7 ⊢ ((1[,]2) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ (ℝ–cn→ℂ) → ((𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ↾ (1[,]2)) ∈ ((1[,]2)–cn→ℂ)))
156	90, 154, 155	mp2 9	. . . . . 6 ⊢ ((𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ↾ (1[,]2)) ∈ ((1[,]2)–cn→ℂ)
157	145, 156	eqeltrri 2835	. . . . 5 ⊢ (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ ((1[,]2)–cn→ℂ)
158	157	a1i 11	. . . 4 ⊢ (⊤ → (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ ((1[,]2)–cn→ℂ))
159	1, 3, 5, 134, 143, 158	ftc2 24941	. . 3 ⊢ (⊤ → ∫(1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) d𝑥 = (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)))
160	159	mptru 1550	. 2 ⊢ ∫(1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) d𝑥 = (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1))
161		itgeq2 24675	. . 3 ⊢ (∀𝑥 ∈ (1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) = ((𝑥↑2) − 3) → ∫(1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) d𝑥 = ∫(1(,)2)((𝑥↑2) − 3) d𝑥)
162		oveq1 7220	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦↑2) = (𝑥↑2))
163	162	oveq1d 7228	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦↑2) − 3) = ((𝑥↑2) − 3))
164	97	mptru 1550	. . . 4 ⊢ (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3))
165		ovex 7246	. . . 4 ⊢ ((𝑥↑2) − 3) ∈ V
166	163, 164, 165	fvmpt 6818	. . 3 ⊢ (𝑥 ∈ (1(,)2) → ((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) = ((𝑥↑2) − 3))
167	161, 166	mprg 3075	. 2 ⊢ ∫(1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) d𝑥 = ∫(1(,)2)((𝑥↑2) − 3) d𝑥
168	2	leidi 11366	. . . . . . . . 9 ⊢ 2 ≤ 2
169	88, 2	elicc2i 13001	. . . . . . . . 9 ⊢ (2 ∈ (1[,]2) ↔ (2 ∈ ℝ ∧ 1 ≤ 2 ∧ 2 ≤ 2))
170	2, 4, 168, 169	mpbir3an 1343	. . . . . . . 8 ⊢ 2 ∈ (1[,]2)
171		oveq1 7220	. . . . . . . . . . . 12 ⊢ (𝑦 = 2 → (𝑦↑3) = (2↑3))
172	171	oveq1d 7228	. . . . . . . . . . 11 ⊢ (𝑦 = 2 → ((𝑦↑3) / 3) = ((2↑3) / 3))
173		oveq2 7221	. . . . . . . . . . 11 ⊢ (𝑦 = 2 → (3 · 𝑦) = (3 · 2))
174	172, 173	oveq12d 7231	. . . . . . . . . 10 ⊢ (𝑦 = 2 → (((𝑦↑3) / 3) − (3 · 𝑦)) = (((2↑3) / 3) − (3 · 2)))
175		cu2 13769	. . . . . . . . . . . . 13 ⊢ (2↑3) = 8
176	175	oveq1i 7223	. . . . . . . . . . . 12 ⊢ ((2↑3) / 3) = (8 / 3)
177		3t2e6 11996	. . . . . . . . . . . 12 ⊢ (3 · 2) = 6
178	176, 177	oveq12i 7225	. . . . . . . . . . 11 ⊢ (((2↑3) / 3) − (3 · 2)) = ((8 / 3) − 6)
179		2cn 11905	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
180		6cn 11921	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
181	179, 180, 13, 14	divdiri 11589	. . . . . . . . . . . . . 14 ⊢ ((2 + 6) / 3) = ((2 / 3) + (6 / 3))
182		6p2e8 11989	. . . . . . . . . . . . . . . 16 ⊢ (6 + 2) = 8
183	180, 179, 182	addcomli 11024	. . . . . . . . . . . . . . 15 ⊢ (2 + 6) = 8
184	183	oveq1i 7223	. . . . . . . . . . . . . 14 ⊢ ((2 + 6) / 3) = (8 / 3)
185	180, 13, 179, 14	divmuli 11586	. . . . . . . . . . . . . . . 16 ⊢ ((6 / 3) = 2 ↔ (3 · 2) = 6)
186	177, 185	mpbir 234	. . . . . . . . . . . . . . 15 ⊢ (6 / 3) = 2
187	186	oveq2i 7224	. . . . . . . . . . . . . 14 ⊢ ((2 / 3) + (6 / 3)) = ((2 / 3) + 2)
188	181, 184, 187	3eqtr3i 2773	. . . . . . . . . . . . 13 ⊢ (8 / 3) = ((2 / 3) + 2)
189	188	oveq1i 7223	. . . . . . . . . . . 12 ⊢ ((8 / 3) − 6) = (((2 / 3) + 2) − 6)
190	179, 13, 14	divcli 11574	. . . . . . . . . . . . 13 ⊢ (2 / 3) ∈ ℂ
191		subsub3 11110	. . . . . . . . . . . . 13 ⊢ (((2 / 3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 2 ∈ ℂ) → ((2 / 3) − (6 − 2)) = (((2 / 3) + 2) − 6))
192	190, 180, 179, 191	mp3an 1463	. . . . . . . . . . . 12 ⊢ ((2 / 3) − (6 − 2)) = (((2 / 3) + 2) − 6)
193	189, 192	eqtr4i 2768	. . . . . . . . . . 11 ⊢ ((8 / 3) − 6) = ((2 / 3) − (6 − 2))
194		4cn 11915	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
195		4p2e6 11983	. . . . . . . . . . . . . 14 ⊢ (4 + 2) = 6
196	194, 179, 195	addcomli 11024	. . . . . . . . . . . . 13 ⊢ (2 + 4) = 6
197	180, 179, 194, 196	subaddrii 11167	. . . . . . . . . . . 12 ⊢ (6 − 2) = 4
198	197	oveq2i 7224	. . . . . . . . . . 11 ⊢ ((2 / 3) − (6 − 2)) = ((2 / 3) − 4)
199	178, 193, 198	3eqtri 2769	. . . . . . . . . 10 ⊢ (((2↑3) / 3) − (3 · 2)) = ((2 / 3) − 4)
200	174, 199	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑦 = 2 → (((𝑦↑3) / 3) − (3 · 𝑦)) = ((2 / 3) − 4))
201		eqid 2737	. . . . . . . . 9 ⊢ (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) = (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
202		ovex 7246	. . . . . . . . 9 ⊢ ((2 / 3) − 4) ∈ V
203	200, 201, 202	fvmpt 6818	. . . . . . . 8 ⊢ (2 ∈ (1[,]2) → ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) = ((2 / 3) − 4))
204	170, 203	ax-mp 5	. . . . . . 7 ⊢ ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) = ((2 / 3) − 4)
205	88	leidi 11366	. . . . . . . . 9 ⊢ 1 ≤ 1
206	88, 2	elicc2i 13001	. . . . . . . . 9 ⊢ (1 ∈ (1[,]2) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 ≤ 2))
207	88, 205, 4, 206	mpbir3an 1343	. . . . . . . 8 ⊢ 1 ∈ (1[,]2)
208		oveq1 7220	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → (𝑦↑3) = (1↑3))
209	208	oveq1d 7228	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑦↑3) / 3) = ((1↑3) / 3))
210		oveq2 7221	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (3 · 𝑦) = (3 · 1))
211	209, 210	oveq12d 7231	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (((𝑦↑3) / 3) − (3 · 𝑦)) = (((1↑3) / 3) − (3 · 1)))
212		3z 12210	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
213		1exp 13664	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℤ → (1↑3) = 1)
214	212, 213	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1↑3) = 1
215	214	oveq1i 7223	. . . . . . . . . . 11 ⊢ ((1↑3) / 3) = (1 / 3)
216	215, 84	oveq12i 7225	. . . . . . . . . 10 ⊢ (((1↑3) / 3) − (3 · 1)) = ((1 / 3) − 3)
217	211, 216	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑦 = 1 → (((𝑦↑3) / 3) − (3 · 𝑦)) = ((1 / 3) − 3))
218		ovex 7246	. . . . . . . . 9 ⊢ ((1 / 3) − 3) ∈ V
219	217, 201, 218	fvmpt 6818	. . . . . . . 8 ⊢ (1 ∈ (1[,]2) → ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1) = ((1 / 3) − 3))
220	207, 219	ax-mp 5	. . . . . . 7 ⊢ ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1) = ((1 / 3) − 3)
221	204, 220	oveq12i 7225	. . . . . 6 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = (((2 / 3) − 4) − ((1 / 3) − 3))
222		sub4 11123	. . . . . . 7 ⊢ ((((2 / 3) ∈ ℂ ∧ 4 ∈ ℂ) ∧ ((1 / 3) ∈ ℂ ∧ 3 ∈ ℂ)) → (((2 / 3) − 4) − ((1 / 3) − 3)) = (((2 / 3) − (1 / 3)) − (4 − 3)))
223	190, 194, 59, 13, 222	mp4an 693	. . . . . 6 ⊢ (((2 / 3) − 4) − ((1 / 3) − 3)) = (((2 / 3) − (1 / 3)) − (4 − 3))
224	13, 14	pm3.2i 474	. . . . . . . . 9 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
225		divsubdir 11526	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((2 − 1) / 3) = ((2 / 3) − (1 / 3)))
226	179, 58, 224, 225	mp3an 1463	. . . . . . . 8 ⊢ ((2 − 1) / 3) = ((2 / 3) − (1 / 3))
227		2m1e1 11956	. . . . . . . . 9 ⊢ (2 − 1) = 1
228	227	oveq1i 7223	. . . . . . . 8 ⊢ ((2 − 1) / 3) = (1 / 3)
229	226, 228	eqtr3i 2767	. . . . . . 7 ⊢ ((2 / 3) − (1 / 3)) = (1 / 3)
230		3p1e4 11975	. . . . . . . 8 ⊢ (3 + 1) = 4
231	194, 13, 58, 230	subaddrii 11167	. . . . . . 7 ⊢ (4 − 3) = 1
232	229, 231	oveq12i 7225	. . . . . 6 ⊢ (((2 / 3) − (1 / 3)) − (4 − 3)) = ((1 / 3) − 1)
233	221, 223, 232	3eqtri 2769	. . . . 5 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = ((1 / 3) − 1)
234	13, 14	dividi 11565	. . . . . 6 ⊢ (3 / 3) = 1
235	234	oveq2i 7224	. . . . 5 ⊢ ((1 / 3) − (3 / 3)) = ((1 / 3) − 1)
236	233, 235	eqtr4i 2768	. . . 4 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = ((1 / 3) − (3 / 3))
237		divsubdir 11526	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 3 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 − 3) / 3) = ((1 / 3) − (3 / 3)))
238	58, 13, 224, 237	mp3an 1463	. . . 4 ⊢ ((1 − 3) / 3) = ((1 / 3) − (3 / 3))
239	236, 238	eqtr4i 2768	. . 3 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = ((1 − 3) / 3)
240		divneg 11524	. . . . 5 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → -(2 / 3) = (-2 / 3))
241	179, 13, 14, 240	mp3an 1463	. . . 4 ⊢ -(2 / 3) = (-2 / 3)
242	13, 58	negsubdi2i 11164	. . . . . 6 ⊢ -(3 − 1) = (1 − 3)
243	40	negeqi 11071	. . . . . 6 ⊢ -(3 − 1) = -2
244	242, 243	eqtr3i 2767	. . . . 5 ⊢ (1 − 3) = -2
245	244	oveq1i 7223	. . . 4 ⊢ ((1 − 3) / 3) = (-2 / 3)
246	241, 245	eqtr4i 2768	. . 3 ⊢ -(2 / 3) = ((1 − 3) / 3)
247	239, 246	eqtr4i 2768	. 2 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = -(2 / 3)
248	160, 167, 247	3eqtr3i 2773	1 ⊢ ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 ∧ w3a 1089 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 Vcvv 3408 ⊆ wss 3866 {cpr 4543 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ran crn 5552 ↾ cres 5553 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 ≤ cle 10868 − cmin 11062 -cneg 11063 / cdiv 11489 ℕcn 11830 2c2 11885 3c3 11886 4c4 11887 6c6 11889 8c8 11891 ℕ₀cn0 12090 ℤcz 12176 (,)cioo 12935 [,]cicc 12938 ↑cexp 13635 TopOpenctopn 16926 topGenctg 16942 ℂ_fldccnfld 20363 intcnt 21914 –cn→ccncf 23773 volcvol 24360 𝐿¹cibl 24514 ∫citg 24515 D cdv 24760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-symdif 4157 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-cmp 22284 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-ovol 24361 df-vol 24362 df-mbf 24516 df-itg1 24517 df-itg2 24518 df-ibl 24519 df-itg 24520 df-0p 24567 df-limc 24763 df-dv 24764

This theorem is referenced by: (None)

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