lhe4.4ex1a - Mathbox for Steve Rodriguez

	Mathbox for Steve Rodriguez		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lhe4.4ex1a		Structured version Visualization version GIF version

Theorem lhe4.4ex1a 41947

Description: Example of the Fundamental Theorem of Calculus, part two (ftc2 25208): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 25208 as simply the "Fundamental Theorem of Calculus", then ftc1 25206 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 10976	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
2		2re 12047	. . . . 5 ⊢ 2 ∈ ℝ
3	2	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
4		1le2 12182	. . . . 5 ⊢ 1 ≤ 2
5	4	a1i 11	. . . 4 ⊢ (⊤ → 1 ≤ 2)
6		reelprrecn 10963	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
8		recn 10961	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
9		3nn0 12251	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
10		expcl 13800	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑦↑3) ∈ ℂ)
11	9, 10	mpan2 688	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → (𝑦↑3) ∈ ℂ)
12	8, 11	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦↑3) ∈ ℂ)
13		3cn 12054	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
14		3ne0 12079	. . . . . . . . . 10 ⊢ 3 ≠ 0
15		divcl 11639	. . . . . . . . . 10 ⊢ (((𝑦↑3) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑦↑3) / 3) ∈ ℂ)
16	13, 14, 15	mp3an23 1452	. . . . . . . . 9 ⊢ ((𝑦↑3) ∈ ℂ → ((𝑦↑3) / 3) ∈ ℂ)
17	12, 16	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝑦↑3) / 3) ∈ ℂ)
18		mulcl 10955	. . . . . . . . 9 ⊢ ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (3 · 𝑦) ∈ ℂ)
19	13, 8, 18	sylancr 587	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (3 · 𝑦) ∈ ℂ)
20	17, 19	subcld 11332	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦↑3) / 3) − (3 · 𝑦)) ∈ ℂ)
21	20	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (((𝑦↑3) / 3) − (3 · 𝑦)) ∈ ℂ)
22		ovexd 7310	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → ((𝑦↑2) − 3) ∈ V)
23	17	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → ((𝑦↑3) / 3) ∈ ℂ)
24		ovexd 7310	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (𝑦↑2) ∈ V)
25		divrec2 11650	. . . . . . . . . . . . 13 ⊢ (((𝑦↑3) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
26	13, 14, 25	mp3an23 1452	. . . . . . . . . . . 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 5177	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3)) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))
29	28	oveq2i 7286	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3))) = (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3))))
30	12	adantl 482	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (𝑦↑3) ∈ ℂ)
31		ovexd 7310	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (3 · (𝑦↑2)) ∈ V)
32		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ ↦ (𝑦↑3)) = (𝑦 ∈ ℂ ↦ (𝑦↑3))
33	32, 11	fmpti 6986	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℂ ↦ (𝑦↑3)):ℂ⟶ℂ
34		ssid 3943	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
35		ax-resscn 10928	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
36		ovex 7308	. . . . . . . . . . . . . . . 16 ⊢ (3 · (𝑦↑2)) ∈ V
37		3nn 12052	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℕ
38		dvexp 25117	. . . . . . . . . . . . . . . . . 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 12101	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 − 1) = 2
41	40	oveq2i 7286	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦↑(3 − 1)) = (𝑦↑2)
42	41	oveq2i 7286	. . . . . . . . . . . . . . . . . 18 ⊢ (3 · (𝑦↑(3 − 1))) = (3 · (𝑦↑2))
43	42	mpteq2i 5179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ ↦ (3 · (𝑦↑(3 − 1)))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑2)))
44	39, 43	eqtri 2766	. . . . . . . . . . . . . . . 16 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑2)))
45	36, 44	dmmpti 6577	. . . . . . . . . . . . . . 15 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = ℂ
46	35, 45	sseqtrri 3958	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3)))
47		dvres3 25077	. . . . . . . . . . . . . 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 690	. . . . . . . . . . . . 13 ⊢ (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ)
49		resmpt 5945	. . . . . . . . . . . . . . 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 7286	. . . . . . . . . . . . 13 ⊢ (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = (ℝ D (𝑦 ∈ ℝ ↦ (𝑦↑3)))
52	44	reseq1i 5887	. . . . . . . . . . . . . 14 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ) = ((𝑦 ∈ ℂ ↦ (3 · (𝑦↑2))) ↾ ℝ)
53		resmpt 5945	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2)))
56	48, 51, 55	3eqtr3i 2774	. . . . . . . . . . . 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 10929	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
59	58, 13, 14	divcli 11717	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
60	59	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (1 / 3) ∈ ℂ)
61	7, 30, 31, 57, 60	dvmptcmul 25128	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2)))))
62	61	mptru 1546	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2))))
63		sqcl 13838	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
64		mulcl 10955	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ (𝑦↑2) ∈ ℂ) → (3 · (𝑦↑2)) ∈ ℂ)
65	13, 63, 64	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (3 · (𝑦↑2)) ∈ ℂ)
66		divrec2 11650	. . . . . . . . . . . . 13 ⊢ (((3 · (𝑦↑2)) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
67	13, 14, 66	mp3an23 1452	. . . . . . . . . . . 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 11659	. . . . . . . . . . . . 13 ⊢ (((𝑦↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
70	13, 14, 69	mp3an23 1452	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → ((1 / 3) · (3 · (𝑦↑2))) = (𝑦↑2))
73	72	mpteq2ia 5177	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2)))) = (𝑦 ∈ ℝ ↦ (𝑦↑2))
74	29, 62, 73	3eqtri 2770	. . . . . . . 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 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (3 · 𝑦) ∈ ℂ)
77		3ex 12055	. . . . . . . 8 ⊢ 3 ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 3 ∈ V)
79	8	adantl 482	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
80		1red 10976	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℝ)
81	7	dvmptid 25121	. . . . . . . . 9 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
82	13	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℂ)
83	7, 79, 80, 81, 82	dvmptcmul 25128	. . . . . . . 8 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ (3 · 1)))
84		3t1e3 12138	. . . . . . . . 9 ⊢ (3 · 1) = 3
85	84	mpteq2i 5179	. . . . . . . 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 25131	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑦↑2) − 3)))
88		1re 10975	. . . . . . . 8 ⊢ 1 ∈ ℝ
89		iccssre 13161	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → (1[,]2) ⊆ ℝ)
90	88, 2, 89	mp2an 689	. . . . . . 7 ⊢ (1[,]2) ⊆ ℝ
91	90	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,]2) ⊆ ℝ)
92		eqid 2738	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
93	92	tgioo2 23966	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
94		iccntr 23984	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(1[,]2)) = (1(,)2))
95	88, 2, 94	mp2an 689	. . . . . . 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 25126	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)))
98		ioossicc 13165	. . . . . . 7 ⊢ (1(,)2) ⊆ (1[,]2)
99		resmpt 5945	. . . . . . 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 3930	. . . . . . . . 9 ⊢ (1[,]2) ⊆ ℂ
102		resmpt 5945	. . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) = (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))
105		subcl 11220	. . . . . . . . . . . . . 14 ⊢ (((𝑦↑2) ∈ ℂ ∧ 3 ∈ ℂ) → ((𝑦↑2) − 3) ∈ ℂ)
106	13, 105	mpan2 688	. . . . . . . . . . . . 13 ⊢ ((𝑦↑2) ∈ ℂ → ((𝑦↑2) − 3) ∈ ℂ)
107	63, 106	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) − 3) ∈ ℂ)
108	104, 107	fmpti 6986	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ
109	34, 108, 34	3pm3.2i 1338	. . . . . . . . . 10 ⊢ (ℂ ⊆ ℂ ∧ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ ∧ ℂ ⊆ ℂ)
110		ovex 7308	. . . . . . . . . . 11 ⊢ ((2 · (𝑦↑(2 − 1))) − 0) ∈ V
111		cnelprrecn 10964	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ {ℝ, ℂ}
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
113	63	adantl 482	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
114		ovexd 7310	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (2 · (𝑦↑(2 − 1))) ∈ V)
115		2nn 12046	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
116		dvexp 25117	. . . . . . . . . . . . . . 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 10969	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
121	120	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ V)
122	112, 82	dvmptc 25122	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 3)) = (𝑦 ∈ ℂ ↦ 0))
123	112, 113, 114, 118, 119, 121, 122	dvmptsub 25131	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = (𝑦 ∈ ℂ ↦ ((2 · (𝑦↑(2 − 1))) − 0)))
124	123	mptru 1546	. . . . . . . . . . 11 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = (𝑦 ∈ ℂ ↦ ((2 · (𝑦↑(2 − 1))) − 0))
125	110, 124	dmmpti 6577	. . . . . . . . . 10 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = ℂ
126		dvcn 25085	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ∈ (ℂ–cn→ℂ)
128		rescncf 24060	. . . . . . . . 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 2836	. . . . . . 7 ⊢ (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ ((1[,]2)–cn→ℂ)
131		rescncf 24060	. . . . . . 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 2836	. . . . 5 ⊢ (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)) ∈ ((1(,)2)–cn→ℂ)
134	97, 133	eqeltrdi 2847	. . . 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 24729	. . . . . . 7 ⊢ (1(,)2) ∈ dom vol
137	136	a1i 11	. . . . . 6 ⊢ (⊤ → (1(,)2) ∈ dom vol)
138		ovexd 7310	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,]2)) → ((𝑦↑2) − 3) ∈ V)
139		cniccibl 25005	. . . . . . . 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 1460	. . . . . . 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 24969	. . . . 5 ⊢ (⊤ → (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)) ∈ 𝐿¹)
143	97, 142	eqeltrd 2839	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) ∈ 𝐿¹)
144		resmpt 5945	. . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
147	146, 20	fmpti 6986	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ
148		ssid 3943	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
149	35, 147, 148	3pm3.2i 1338	. . . . . . . 8 ⊢ (ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ ∧ ℝ ⊆ ℝ)
150		ovex 7308	. . . . . . . . 9 ⊢ ((𝑦↑2) − 3) ∈ V
151	87	mptru 1546	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑦↑2) − 3))
152	150, 151	dmmpti 6577	. . . . . . . 8 ⊢ dom (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = ℝ
153		dvcn 25085	. . . . . . . 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 689	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ (ℝ–cn→ℂ)
155		rescncf 24060	. . . . . . 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 2836	. . . . 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 25208	. . 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 1546	. 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 24942	. . 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 7282	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦↑2) = (𝑥↑2))
163	162	oveq1d 7290	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦↑2) − 3) = ((𝑥↑2) − 3))
164	97	mptru 1546	. . . 4 ⊢ (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3))
165		ovex 7308	. . . 4 ⊢ ((𝑥↑2) − 3) ∈ V
166	163, 164, 165	fvmpt 6875	. . 3 ⊢ (𝑥 ∈ (1(,)2) → ((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) = ((𝑥↑2) − 3))
167	161, 166	mprg 3078	. 2 ⊢ ∫(1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) d𝑥 = ∫(1(,)2)((𝑥↑2) − 3) d𝑥
168	2	leidi 11509	. . . . . . . . 9 ⊢ 2 ≤ 2
169	88, 2	elicc2i 13145	. . . . . . . . 9 ⊢ (2 ∈ (1[,]2) ↔ (2 ∈ ℝ ∧ 1 ≤ 2 ∧ 2 ≤ 2))
170	2, 4, 168, 169	mpbir3an 1340	. . . . . . . 8 ⊢ 2 ∈ (1[,]2)
171		oveq1 7282	. . . . . . . . . . . 12 ⊢ (𝑦 = 2 → (𝑦↑3) = (2↑3))
172	171	oveq1d 7290	. . . . . . . . . . 11 ⊢ (𝑦 = 2 → ((𝑦↑3) / 3) = ((2↑3) / 3))
173		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑦 = 2 → (3 · 𝑦) = (3 · 2))
174	172, 173	oveq12d 7293	. . . . . . . . . 10 ⊢ (𝑦 = 2 → (((𝑦↑3) / 3) − (3 · 𝑦)) = (((2↑3) / 3) − (3 · 2)))
175		cu2 13917	. . . . . . . . . . . . 13 ⊢ (2↑3) = 8
176	175	oveq1i 7285	. . . . . . . . . . . 12 ⊢ ((2↑3) / 3) = (8 / 3)
177		3t2e6 12139	. . . . . . . . . . . 12 ⊢ (3 · 2) = 6
178	176, 177	oveq12i 7287	. . . . . . . . . . 11 ⊢ (((2↑3) / 3) − (3 · 2)) = ((8 / 3) − 6)
179		2cn 12048	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
180		6cn 12064	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
181	179, 180, 13, 14	divdiri 11732	. . . . . . . . . . . . . 14 ⊢ ((2 + 6) / 3) = ((2 / 3) + (6 / 3))
182		6p2e8 12132	. . . . . . . . . . . . . . . 16 ⊢ (6 + 2) = 8
183	180, 179, 182	addcomli 11167	. . . . . . . . . . . . . . 15 ⊢ (2 + 6) = 8
184	183	oveq1i 7285	. . . . . . . . . . . . . 14 ⊢ ((2 + 6) / 3) = (8 / 3)
185	180, 13, 179, 14	divmuli 11729	. . . . . . . . . . . . . . . 16 ⊢ ((6 / 3) = 2 ↔ (3 · 2) = 6)
186	177, 185	mpbir 230	. . . . . . . . . . . . . . 15 ⊢ (6 / 3) = 2
187	186	oveq2i 7286	. . . . . . . . . . . . . 14 ⊢ ((2 / 3) + (6 / 3)) = ((2 / 3) + 2)
188	181, 184, 187	3eqtr3i 2774	. . . . . . . . . . . . 13 ⊢ (8 / 3) = ((2 / 3) + 2)
189	188	oveq1i 7285	. . . . . . . . . . . 12 ⊢ ((8 / 3) − 6) = (((2 / 3) + 2) − 6)
190	179, 13, 14	divcli 11717	. . . . . . . . . . . . 13 ⊢ (2 / 3) ∈ ℂ
191		subsub3 11253	. . . . . . . . . . . . 13 ⊢ (((2 / 3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 2 ∈ ℂ) → ((2 / 3) − (6 − 2)) = (((2 / 3) + 2) − 6))
192	190, 180, 179, 191	mp3an 1460	. . . . . . . . . . . 12 ⊢ ((2 / 3) − (6 − 2)) = (((2 / 3) + 2) − 6)
193	189, 192	eqtr4i 2769	. . . . . . . . . . 11 ⊢ ((8 / 3) − 6) = ((2 / 3) − (6 − 2))
194		4cn 12058	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
195		4p2e6 12126	. . . . . . . . . . . . . 14 ⊢ (4 + 2) = 6
196	194, 179, 195	addcomli 11167	. . . . . . . . . . . . 13 ⊢ (2 + 4) = 6
197	180, 179, 194, 196	subaddrii 11310	. . . . . . . . . . . 12 ⊢ (6 − 2) = 4
198	197	oveq2i 7286	. . . . . . . . . . 11 ⊢ ((2 / 3) − (6 − 2)) = ((2 / 3) − 4)
199	178, 193, 198	3eqtri 2770	. . . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) = (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
202		ovex 7308	. . . . . . . . 9 ⊢ ((2 / 3) − 4) ∈ V
203	200, 201, 202	fvmpt 6875	. . . . . . . 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 11509	. . . . . . . . 9 ⊢ 1 ≤ 1
206	88, 2	elicc2i 13145	. . . . . . . . 9 ⊢ (1 ∈ (1[,]2) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 ≤ 2))
207	88, 205, 4, 206	mpbir3an 1340	. . . . . . . 8 ⊢ 1 ∈ (1[,]2)
208		oveq1 7282	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → (𝑦↑3) = (1↑3))
209	208	oveq1d 7290	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑦↑3) / 3) = ((1↑3) / 3))
210		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (3 · 𝑦) = (3 · 1))
211	209, 210	oveq12d 7293	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (((𝑦↑3) / 3) − (3 · 𝑦)) = (((1↑3) / 3) − (3 · 1)))
212		3z 12353	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
213		1exp 13812	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℤ → (1↑3) = 1)
214	212, 213	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1↑3) = 1
215	214	oveq1i 7285	. . . . . . . . . . 11 ⊢ ((1↑3) / 3) = (1 / 3)
216	215, 84	oveq12i 7287	. . . . . . . . . 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 7308	. . . . . . . . 9 ⊢ ((1 / 3) − 3) ∈ V
219	217, 201, 218	fvmpt 6875	. . . . . . . 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 7287	. . . . . 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 11266	. . . . . . 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 690	. . . . . 6 ⊢ (((2 / 3) − 4) − ((1 / 3) − 3)) = (((2 / 3) − (1 / 3)) − (4 − 3))
224	13, 14	pm3.2i 471	. . . . . . . . 9 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
225		divsubdir 11669	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((2 − 1) / 3) = ((2 / 3) − (1 / 3)))
226	179, 58, 224, 225	mp3an 1460	. . . . . . . 8 ⊢ ((2 − 1) / 3) = ((2 / 3) − (1 / 3))
227		2m1e1 12099	. . . . . . . . 9 ⊢ (2 − 1) = 1
228	227	oveq1i 7285	. . . . . . . 8 ⊢ ((2 − 1) / 3) = (1 / 3)
229	226, 228	eqtr3i 2768	. . . . . . 7 ⊢ ((2 / 3) − (1 / 3)) = (1 / 3)
230		3p1e4 12118	. . . . . . . 8 ⊢ (3 + 1) = 4
231	194, 13, 58, 230	subaddrii 11310	. . . . . . 7 ⊢ (4 − 3) = 1
232	229, 231	oveq12i 7287	. . . . . 6 ⊢ (((2 / 3) − (1 / 3)) − (4 − 3)) = ((1 / 3) − 1)
233	221, 223, 232	3eqtri 2770	. . . . 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 11708	. . . . . 6 ⊢ (3 / 3) = 1
235	234	oveq2i 7286	. . . . 5 ⊢ ((1 / 3) − (3 / 3)) = ((1 / 3) − 1)
236	233, 235	eqtr4i 2769	. . . 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 11669	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 3 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 − 3) / 3) = ((1 / 3) − (3 / 3)))
238	58, 13, 224, 237	mp3an 1460	. . . 4 ⊢ ((1 − 3) / 3) = ((1 / 3) − (3 / 3))
239	236, 238	eqtr4i 2769	. . 3 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = ((1 − 3) / 3)
240		divneg 11667	. . . . 5 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → -(2 / 3) = (-2 / 3))
241	179, 13, 14, 240	mp3an 1460	. . . 4 ⊢ -(2 / 3) = (-2 / 3)
242	13, 58	negsubdi2i 11307	. . . . . 6 ⊢ -(3 − 1) = (1 − 3)
243	40	negeqi 11214	. . . . . 6 ⊢ -(3 − 1) = -2
244	242, 243	eqtr3i 2768	. . . . 5 ⊢ (1 − 3) = -2
245	244	oveq1i 7285	. . . 4 ⊢ ((1 − 3) / 3) = (-2 / 3)
246	241, 245	eqtr4i 2769	. . 3 ⊢ -(2 / 3) = ((1 − 3) / 3)
247	239, 246	eqtr4i 2769	. 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 2774	1 ⊢ ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 ∧ w3a 1086 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ⊆ wss 3887 {cpr 4563 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 2c2 12028 3c3 12029 4c4 12030 6c6 12032 8c8 12034 ℕ₀cn0 12233 ℤcz 12319 (,)cioo 13079 [,]cicc 13082 ↑cexp 13782 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 intcnt 22168 –cn→ccncf 24039 volcvol 24627 𝐿¹cibl 24781 ∫citg 24782 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-ibl 24786 df-itg 24787 df-0p 24834 df-limc 25030 df-dv 25031

This theorem is referenced by: (None)

W3C validator