lhe4.4ex1a - Mathbox for Steve Rodriguez

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Theorem lhe4.4ex1a 45272

Description: Example of the Fundamental Theorem of Calculus, part two (ftc2 26344): ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 26344 as simply the "Fundamental Theorem of Calculus", then ftc1 26342 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 11290	. . . 4 ⊢ (⊤ → 1 ∈ ℝ)
2		2re 12398	. . . . 5 ⊢ 2 ∈ ℝ
3	2	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
4		1le2 12535	. . . . 5 ⊢ 1 ≤ 2
5	4	a1i 11	. . . 4 ⊢ (⊤ → 1 ≤ 2)
6		reelprrecn 11273	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
7	6	a1i 11	. . . . . 6 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
8		recn 11271	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
9		3nn0 12605	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
10		expcl 14202	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 3 ∈ ℕ₀) → (𝑦↑3) ∈ ℂ)
11	9, 10	mpan2 704	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → (𝑦↑3) ∈ ℂ)
12	8, 11	syl 18	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝑦↑3) ∈ ℂ)
13		3cn 12405	. . . . . . . . . 10 ⊢ 3 ∈ ℂ
14		3ne0 12433	. . . . . . . . . 10 ⊢ 3 ≠ 0
15		divcl 11961	. . . . . . . . . 10 ⊢ (((𝑦↑3) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑦↑3) / 3) ∈ ℂ)
16	13, 14, 15	mp3an23 1482	. . . . . . . . 9 ⊢ ((𝑦↑3) ∈ ℂ → ((𝑦↑3) / 3) ∈ ℂ)
17	12, 16	syl 18	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝑦↑3) / 3) ∈ ℂ)
18		mulcl 11265	. . . . . . . . 9 ⊢ ((3 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (3 · 𝑦) ∈ ℂ)
19	13, 8, 18	sylancr 599	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → (3 · 𝑦) ∈ ℂ)
20	17, 19	subcld 11650	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → (((𝑦↑3) / 3) − (3 · 𝑦)) ∈ ℂ)
21	20	adantl 487	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (((𝑦↑3) / 3) − (3 · 𝑦)) ∈ ℂ)
22		ovexd 7447	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → ((𝑦↑2) − 3) ∈ V)
23	17	adantl 487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → ((𝑦↑3) / 3) ∈ ℂ)
24		ovexd 7447	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (𝑦↑2) ∈ V)
25		divrec2 11972	. . . . . . . . . . . . 13 ⊢ (((𝑦↑3) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
26	13, 14, 25	mp3an23 1482	. . . . . . . . . . . 12 ⊢ ((𝑦↑3) ∈ ℂ → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
27	12, 26	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((𝑦↑3) / 3) = ((1 / 3) · (𝑦↑3)))
28	27	mpteq2ia 5200	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3)) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))
29	28	oveq2i 7423	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦↑3) / 3))) = (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3))))
30	12	adantl 487	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (𝑦↑3) ∈ ℂ)
31		ovexd 7447	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (3 · (𝑦↑2)) ∈ V)
32		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℂ ↦ (𝑦↑3)) = (𝑦 ∈ ℂ ↦ (𝑦↑3))
33	32, 11	fmpti 7104	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℂ ↦ (𝑦↑3)):ℂ⟶ℂ
34		ssid 3953	. . . . . . . . . . . . . 14 ⊢ ℂ ⊆ ℂ
35		ax-resscn 11238	. . . . . . . . . . . . . . 15 ⊢ ℝ ⊆ ℂ
36		ovex 7445	. . . . . . . . . . . . . . . 16 ⊢ (3 · (𝑦↑2)) ∈ V
37		3nn 12403	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℕ
38		dvexp 26253	. . . . . . . . . . . . . . . . . 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 12451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (3 − 1) = 2
41	40	oveq2i 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦↑(3 − 1)) = (𝑦↑2)
42	41	oveq2i 7423	. . . . . . . . . . . . . . . . . 18 ⊢ (3 · (𝑦↑(3 − 1))) = (3 · (𝑦↑2))
43	42	mpteq2i 5201	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℂ ↦ (3 · (𝑦↑(3 − 1)))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑2)))
44	39, 43	eqtri 2784	. . . . . . . . . . . . . . . 16 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = (𝑦 ∈ ℂ ↦ (3 · (𝑦↑2)))
45	36, 44	dmmpti 6675	. . . . . . . . . . . . . . 15 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) = ℂ
46	35, 45	sseqtrri 3980	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3)))
47		dvres3 26213	. . . . . . . . . . . . . 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 706	. . . . . . . . . . . . 13 ⊢ (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ)
49		resmpt 6031	. . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 13 ⊢ (ℝ D ((𝑦 ∈ ℂ ↦ (𝑦↑3)) ↾ ℝ)) = (ℝ D (𝑦 ∈ ℝ ↦ (𝑦↑3)))
52	44	reseq1i 5966	. . . . . . . . . . . . . 14 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ) = ((𝑦 ∈ ℂ ↦ (3 · (𝑦↑2))) ↾ ℝ)
53		resmpt 6031	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . 13 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑3))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (3 · (𝑦↑2)))
56	48, 51, 55	3eqtr3i 2792	. . . . . . . . . . . 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 11239	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
59	58, 13, 14	divcli 12040	. . . . . . . . . . . 12 ⊢ (1 / 3) ∈ ℂ
60	59	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (1 / 3) ∈ ℂ)
61	7, 30, 31, 57, 60	dvmptcmul 26264	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2)))))
62	61	mptru 1577	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ ((1 / 3) · (𝑦↑3)))) = (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2))))
63		sqcl 14241	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑦↑2) ∈ ℂ)
64		mulcl 11265	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ (𝑦↑2) ∈ ℂ) → (3 · (𝑦↑2)) ∈ ℂ)
65	13, 63, 64	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (3 · (𝑦↑2)) ∈ ℂ)
66		divrec2 11972	. . . . . . . . . . . . 13 ⊢ (((3 · (𝑦↑2)) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
67	13, 14, 66	mp3an23 1482	. . . . . . . . . . . 12 ⊢ ((3 · (𝑦↑2)) ∈ ℂ → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
68	8, 65, 67	3syl 19	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((3 · (𝑦↑2)) / 3) = ((1 / 3) · (3 · (𝑦↑2))))
69		divcan3 11981	. . . . . . . . . . . . 13 ⊢ (((𝑦↑2) ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
70	13, 14, 69	mp3an23 1482	. . . . . . . . . . . 12 ⊢ ((𝑦↑2) ∈ ℂ → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
71	8, 63, 70	3syl 19	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ → ((3 · (𝑦↑2)) / 3) = (𝑦↑2))
72	68, 71	eqtr3d 2798	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → ((1 / 3) · (3 · (𝑦↑2))) = (𝑦↑2))
73	72	mpteq2ia 5200	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ ↦ ((1 / 3) · (3 · (𝑦↑2)))) = (𝑦 ∈ ℝ ↦ (𝑦↑2))
74	29, 62, 73	3eqtri 2788	. . . . . . . 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 487	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → (3 · 𝑦) ∈ ℂ)
77		3ex 12406	. . . . . . . 8 ⊢ 3 ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 3 ∈ V)
79	8	adantl 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
80		1red 11290	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℝ)
81	7	dvmptid 26257	. . . . . . . . 9 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
82	13	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 3 ∈ ℂ)
83	7, 79, 80, 81, 82	dvmptcmul 26264	. . . . . . . 8 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ (3 · 1)))
84		3t1e3 12488	. . . . . . . . 9 ⊢ (3 · 1) = 3
85	84	mpteq2i 5201	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ ↦ (3 · 1)) = (𝑦 ∈ ℝ ↦ 3)
86	83, 85	eqtrdi 2812	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ 3))
87	7, 23, 24, 75, 76, 78, 86	dvmptsub 26267	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑦↑2) − 3)))
88		1re 11289	. . . . . . . 8 ⊢ 1 ∈ ℝ
89		iccssre 13541	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → (1[,]2) ⊆ ℝ)
90	88, 2, 89	mp2an 705	. . . . . . 7 ⊢ (1[,]2) ⊆ ℝ
91	90	a1i 11	. . . . . 6 ⊢ (⊤ → (1[,]2) ⊆ ℝ)
92		tgioo4 25104	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
93		eqid 2761	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
94		iccntr 25121	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(1[,]2)) = (1(,)2))
95	88, 2, 94	mp2an 705	. . . . . . 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, 92, 93, 96	dvmptres2 26262	. . . . 5 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)))
98		ioossicc 13545	. . . . . . 7 ⊢ (1(,)2) ⊆ (1[,]2)
99		resmpt 6031	. . . . . . 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 3940	. . . . . . . . 9 ⊢ (1[,]2) ⊆ ℂ
102		resmpt 6031	. . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) = (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))
105		subcl 11537	. . . . . . . . . . . . . 14 ⊢ (((𝑦↑2) ∈ ℂ ∧ 3 ∈ ℂ) → ((𝑦↑2) − 3) ∈ ℂ)
106	13, 105	mpan2 704	. . . . . . . . . . . . 13 ⊢ ((𝑦↑2) ∈ ℂ → ((𝑦↑2) − 3) ∈ ℂ)
107	63, 106	syl 18	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑦↑2) − 3) ∈ ℂ)
108	104, 107	fmpti 7104	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ
109	34, 108, 34	3pm3.2i 1358	. . . . . . . . . 10 ⊢ (ℂ ⊆ ℂ ∧ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)):ℂ⟶ℂ ∧ ℂ ⊆ ℂ)
110		ovex 7445	. . . . . . . . . . 11 ⊢ ((2 · (𝑦↑(2 − 1))) − 0) ∈ V
111		cnelprrecn 11274	. . . . . . . . . . . . . 14 ⊢ ℂ ∈ {ℝ, ℂ}
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
113	63	adantl 487	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (𝑦↑2) ∈ ℂ)
114		ovexd 7447	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (2 · (𝑦↑(2 − 1))) ∈ V)
115		2nn 12397	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
116		dvexp 26253	. . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
121	120	a1i 11	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → 0 ∈ V)
122	112, 82	dvmptc 26258	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 3)) = (𝑦 ∈ ℂ ↦ 0))
123	112, 113, 114, 118, 119, 121, 122	dvmptsub 26267	. . . . . . . . . . . 12 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = (𝑦 ∈ ℂ ↦ ((2 · (𝑦↑(2 − 1))) − 0)))
124	123	mptru 1577	. . . . . . . . . . 11 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = (𝑦 ∈ ℂ ↦ ((2 · (𝑦↑(2 − 1))) − 0))
125	110, 124	dmmpti 6675	. . . . . . . . . 10 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3))) = ℂ
126		dvcn 26221	. . . . . . . . . 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 705	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ ↦ ((𝑦↑2) − 3)) ∈ (ℂ–cn→ℂ)
128		rescncf 25198	. . . . . . . . 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 2858	. . . . . . 7 ⊢ (𝑦 ∈ (1[,]2) ↦ ((𝑦↑2) − 3)) ∈ ((1[,]2)–cn→ℂ)
131		rescncf 25198	. . . . . . 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 2858	. . . . 5 ⊢ (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)) ∈ ((1(,)2)–cn→ℂ)
134	97, 133	eqeltrdi 2869	. . . 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 25866	. . . . . . 7 ⊢ (1(,)2) ∈ dom vol
137	136	a1i 11	. . . . . 6 ⊢ (⊤ → (1(,)2) ∈ dom vol)
138		ovexd 7447	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ (1[,]2)) → ((𝑦↑2) − 3) ∈ V)
139		cniccibl 26141	. . . . . . . 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 1490	. . . . . . 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 26105	. . . . 5 ⊢ (⊤ → (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3)) ∈ 𝐿¹)
143	97, 142	eqeltrd 2861	. . . 4 ⊢ (⊤ → (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) ∈ 𝐿¹)
144		resmpt 6031	. . . . . . 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 2761	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) = (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
147	146, 20	fmpti 7104	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ
148		ssid 3953	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ
149	35, 147, 148	3pm3.2i 1358	. . . . . . . 8 ⊢ (ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))):ℝ⟶ℂ ∧ ℝ ⊆ ℝ)
150		ovex 7445	. . . . . . . . 9 ⊢ ((𝑦↑2) − 3) ∈ V
151	87	mptru 1577	. . . . . . . . 9 ⊢ (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ ℝ ↦ ((𝑦↑2) − 3))
152	150, 151	dmmpti 6675	. . . . . . . 8 ⊢ dom (ℝ D (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = ℝ
153		dvcn 26221	. . . . . . . 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 705	. . . . . . 7 ⊢ (𝑦 ∈ ℝ ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) ∈ (ℝ–cn→ℂ)
155		rescncf 25198	. . . . . . 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 2858	. . . . 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 26344	. . 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 1577	. 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 26078	. . 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 7419	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦↑2) = (𝑥↑2))
163	162	oveq1d 7427	. . . 4 ⊢ (𝑦 = 𝑥 → ((𝑦↑2) − 3) = ((𝑥↑2) − 3))
164	97	mptru 1577	. . . 4 ⊢ (ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))) = (𝑦 ∈ (1(,)2) ↦ ((𝑦↑2) − 3))
165		ovex 7445	. . . 4 ⊢ ((𝑥↑2) − 3) ∈ V
166	163, 164, 165	fvmpt 6985	. . 3 ⊢ (𝑥 ∈ (1(,)2) → ((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) = ((𝑥↑2) − 3))
167	161, 166	mprg 3083	. 2 ⊢ ∫(1(,)2)((ℝ D (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))))‘𝑥) d𝑥 = ∫(1(,)2)((𝑥↑2) − 3) d𝑥
168	2	leidi 11831	. . . . . . . . 9 ⊢ 2 ≤ 2
169	88, 2	elicc2i 13524	. . . . . . . . 9 ⊢ (2 ∈ (1[,]2) ↔ (2 ∈ ℝ ∧ 1 ≤ 2 ∧ 2 ≤ 2))
170	2, 4, 168, 169	mpbir3an 1360	. . . . . . . 8 ⊢ 2 ∈ (1[,]2)
171		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑦 = 2 → (𝑦↑3) = (2↑3))
172	171	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝑦 = 2 → ((𝑦↑3) / 3) = ((2↑3) / 3))
173		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑦 = 2 → (3 · 𝑦) = (3 · 2))
174	172, 173	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑦 = 2 → (((𝑦↑3) / 3) − (3 · 𝑦)) = (((2↑3) / 3) − (3 · 2)))
175		cu2 14323	. . . . . . . . . . . . 13 ⊢ (2↑3) = 8
176	175	oveq1i 7422	. . . . . . . . . . . 12 ⊢ ((2↑3) / 3) = (8 / 3)
177		3t2e6 12489	. . . . . . . . . . . 12 ⊢ (3 · 2) = 6
178	176, 177	oveq12i 7424	. . . . . . . . . . 11 ⊢ (((2↑3) / 3) − (3 · 2)) = ((8 / 3) − 6)
179		2cn 12399	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
180		6cn 12415	. . . . . . . . . . . . . . 15 ⊢ 6 ∈ ℂ
181	179, 180, 13, 14	divdiri 12055	. . . . . . . . . . . . . 14 ⊢ ((2 + 6) / 3) = ((2 / 3) + (6 / 3))
182		6p2e8 12482	. . . . . . . . . . . . . . . 16 ⊢ (6 + 2) = 8
183	180, 179, 182	addcomli 11483	. . . . . . . . . . . . . . 15 ⊢ (2 + 6) = 8
184	183	oveq1i 7422	. . . . . . . . . . . . . 14 ⊢ ((2 + 6) / 3) = (8 / 3)
185	180, 13, 179, 14	divmuli 12052	. . . . . . . . . . . . . . . 16 ⊢ ((6 / 3) = 2 ↔ (3 · 2) = 6)
186	177, 185	mpbir 234	. . . . . . . . . . . . . . 15 ⊢ (6 / 3) = 2
187	186	oveq2i 7423	. . . . . . . . . . . . . 14 ⊢ ((2 / 3) + (6 / 3)) = ((2 / 3) + 2)
188	181, 184, 187	3eqtr3i 2792	. . . . . . . . . . . . 13 ⊢ (8 / 3) = ((2 / 3) + 2)
189	188	oveq1i 7422	. . . . . . . . . . . 12 ⊢ ((8 / 3) − 6) = (((2 / 3) + 2) − 6)
190	179, 13, 14	divcli 12040	. . . . . . . . . . . . 13 ⊢ (2 / 3) ∈ ℂ
191		subsub3 11571	. . . . . . . . . . . . 13 ⊢ (((2 / 3) ∈ ℂ ∧ 6 ∈ ℂ ∧ 2 ∈ ℂ) → ((2 / 3) − (6 − 2)) = (((2 / 3) + 2) − 6))
192	190, 180, 179, 191	mp3an 1490	. . . . . . . . . . . 12 ⊢ ((2 / 3) − (6 − 2)) = (((2 / 3) + 2) − 6)
193	189, 192	eqtr4i 2787	. . . . . . . . . . 11 ⊢ ((8 / 3) − 6) = ((2 / 3) − (6 − 2))
194		4cn 12409	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
195		4p2e6 12476	. . . . . . . . . . . . . 14 ⊢ (4 + 2) = 6
196	194, 179, 195	addcomli 11483	. . . . . . . . . . . . 13 ⊢ (2 + 4) = 6
197	180, 179, 194, 196	subaddrii 11628	. . . . . . . . . . . 12 ⊢ (6 − 2) = 4
198	197	oveq2i 7423	. . . . . . . . . . 11 ⊢ ((2 / 3) − (6 − 2)) = ((2 / 3) − 4)
199	178, 193, 198	3eqtri 2788	. . . . . . . . . 10 ⊢ (((2↑3) / 3) − (3 · 2)) = ((2 / 3) − 4)
200	174, 199	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑦 = 2 → (((𝑦↑3) / 3) − (3 · 𝑦)) = ((2 / 3) − 4))
201		eqid 2761	. . . . . . . . 9 ⊢ (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦))) = (𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))
202		ovex 7445	. . . . . . . . 9 ⊢ ((2 / 3) − 4) ∈ V
203	200, 201, 202	fvmpt 6985	. . . . . . . 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 11831	. . . . . . . . 9 ⊢ 1 ≤ 1
206	88, 2	elicc2i 13524	. . . . . . . . 9 ⊢ (1 ∈ (1[,]2) ↔ (1 ∈ ℝ ∧ 1 ≤ 1 ∧ 1 ≤ 2))
207	88, 205, 4, 206	mpbir3an 1360	. . . . . . . 8 ⊢ 1 ∈ (1[,]2)
208		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑦 = 1 → (𝑦↑3) = (1↑3))
209	208	oveq1d 7427	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → ((𝑦↑3) / 3) = ((1↑3) / 3))
210		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑦 = 1 → (3 · 𝑦) = (3 · 1))
211	209, 210	oveq12d 7430	. . . . . . . . . 10 ⊢ (𝑦 = 1 → (((𝑦↑3) / 3) − (3 · 𝑦)) = (((1↑3) / 3) − (3 · 1)))
212		3z 12710	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℤ
213		1exp 14214	. . . . . . . . . . . . 13 ⊢ (3 ∈ ℤ → (1↑3) = 1)
214	212, 213	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1↑3) = 1
215	214	oveq1i 7422	. . . . . . . . . . 11 ⊢ ((1↑3) / 3) = (1 / 3)
216	215, 84	oveq12i 7424	. . . . . . . . . 10 ⊢ (((1↑3) / 3) − (3 · 1)) = ((1 / 3) − 3)
217	211, 216	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑦 = 1 → (((𝑦↑3) / 3) − (3 · 𝑦)) = ((1 / 3) − 3))
218		ovex 7445	. . . . . . . . 9 ⊢ ((1 / 3) − 3) ∈ V
219	217, 201, 218	fvmpt 6985	. . . . . . . 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 7424	. . . . . 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 11584	. . . . . . 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 706	. . . . . 6 ⊢ (((2 / 3) − 4) − ((1 / 3) − 3)) = (((2 / 3) − (1 / 3)) − (4 − 3))
224	13, 14	pm3.2i 476	. . . . . . . . 9 ⊢ (3 ∈ ℂ ∧ 3 ≠ 0)
225		divsubdir 11991	. . . . . . . . 9 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((2 − 1) / 3) = ((2 / 3) − (1 / 3)))
226	179, 58, 224, 225	mp3an 1490	. . . . . . . 8 ⊢ ((2 − 1) / 3) = ((2 / 3) − (1 / 3))
227		2m1e1 12448	. . . . . . . . 9 ⊢ (2 − 1) = 1
228	227	oveq1i 7422	. . . . . . . 8 ⊢ ((2 − 1) / 3) = (1 / 3)
229	226, 228	eqtr3i 2786	. . . . . . 7 ⊢ ((2 / 3) − (1 / 3)) = (1 / 3)
230		3p1e4 12468	. . . . . . . 8 ⊢ (3 + 1) = 4
231	194, 13, 58, 230	subaddrii 11628	. . . . . . 7 ⊢ (4 − 3) = 1
232	229, 231	oveq12i 7424	. . . . . 6 ⊢ (((2 / 3) − (1 / 3)) − (4 − 3)) = ((1 / 3) − 1)
233	221, 223, 232	3eqtri 2788	. . . . 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 12031	. . . . . 6 ⊢ (3 / 3) = 1
235	234	oveq2i 7423	. . . . 5 ⊢ ((1 / 3) − (3 / 3)) = ((1 / 3) − 1)
236	233, 235	eqtr4i 2787	. . . 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 11991	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 3 ∈ ℂ ∧ (3 ∈ ℂ ∧ 3 ≠ 0)) → ((1 − 3) / 3) = ((1 / 3) − (3 / 3)))
238	58, 13, 224, 237	mp3an 1490	. . . 4 ⊢ ((1 − 3) / 3) = ((1 / 3) − (3 / 3))
239	236, 238	eqtr4i 2787	. . 3 ⊢ (((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘2) − ((𝑦 ∈ (1[,]2) ↦ (((𝑦↑3) / 3) − (3 · 𝑦)))‘1)) = ((1 − 3) / 3)
240		divneg 11989	. . . . 5 ⊢ ((2 ∈ ℂ ∧ 3 ∈ ℂ ∧ 3 ≠ 0) → -(2 / 3) = (-2 / 3))
241	179, 13, 14, 240	mp3an 1490	. . . 4 ⊢ -(2 / 3) = (-2 / 3)
242	13, 58	negsubdi2i 11625	. . . . . 6 ⊢ -(3 − 1) = (1 − 3)
243	40	negeqi 11531	. . . . . 6 ⊢ -(3 − 1) = -2
244	242, 243	eqtr3i 2786	. . . . 5 ⊢ (1 − 3) = -2
245	244	oveq1i 7422	. . . 4 ⊢ ((1 − 3) / 3) = (-2 / 3)
246	241, 245	eqtr4i 2787	. . 3 ⊢ -(2 / 3) = ((1 − 3) / 3)
247	239, 246	eqtr4i 2787	. 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 2792	1 ⊢ ∫(1(,)2)((𝑥↑2) − 3) d𝑥 = -(2 / 3)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2956 Vcvv 3451 ⊆ wss 3899 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ran crn 5652 ↾ cres 5653 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 ≤ cle 11325 − cmin 11522 -cneg 11523 / cdiv 11954 ℕcn 12316 2c2 12378 3c3 12379 4c4 12380 6c6 12382 8c8 12384 ℕ₀cn0 12587 ℤcz 12674 (,)cioo 13457 [,]cicc 13460 ↑cexp 14184 TopOpenctopn 17572 topGenctg 17588 ℂ_fldccnfld 21658 intcnt 23315 –cn→ccncf 25177 volcvol 25764 𝐿¹cibl 25918 ∫citg 25919 D cdv 26163

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cc 10494 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-omul 8465 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-dju 9963 df-card 10001 df-acn 10004 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-hash 14455 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-cmp 23685 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-ovol 25765 df-vol 25766 df-mbf 25920 df-itg1 25921 df-itg2 25922 df-ibl 25923 df-itg 25924 df-0p 25971 df-limc 26166 df-dv 26167

This theorem is used by: (None)

W3C validator