itgsincmulx - Mathbox for Glauco Siliprandi

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Theorem itgsincmulx 38866

Description: Exercise: the integral of 𝑥 ↦ sin𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
itgsincmulx.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
itgsincmulx.an0	⊢ (𝜑 → 𝐴 ≠ 0)
itgsincmulx.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
itgsincmulx.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
itgsincmulx.blec	⊢ (𝜑 → 𝐵 ≤ 𝐶)

**Assertion**
Ref	Expression
itgsincmulx	⊢ (𝜑 → ∫(𝐵(,)𝐶)(sin‘(𝐴 · 𝑥)) d𝑥 = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝜑,𝑥

Proof of Theorem itgsincmulx Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . . . . 7 ⊢ (𝑦 ∈ ℂ ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)) = (𝑦 ∈ ℂ ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))
2		itgsincmulx.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
4		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
5	3, 4	mulcld 9939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
6	5	coscld 14700	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
7	6	negcld 10258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -(cos‘(𝐴 · 𝑦)) ∈ ℂ)
8		itgsincmulx.an0	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ≠ 0)
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐴 ≠ 0)
10	7, 3, 9	divcld 10680	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (-(cos‘(𝐴 · 𝑦)) / 𝐴) ∈ ℂ)
11		cnelprrecn 9908	. . . . . . . . . 10 ⊢ ℂ ∈ {ℝ, ℂ}
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
13	5	sincld 14699	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
14	13	negcld 10258	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -(sin‘(𝐴 · 𝑦)) ∈ ℂ)
15	3, 14	mulcld 9939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · -(sin‘(𝐴 · 𝑦))) ∈ ℂ)
16	15	negcld 10258	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -(𝐴 · -(sin‘(𝐴 · 𝑦))) ∈ ℂ)
17		dvcosax 38816	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (cos‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑦)))))
18	2, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (cos‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑦)))))
19	12, 6, 15, 18	dvmptneg 23535	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ -(cos‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ -(𝐴 · -(sin‘(𝐴 · 𝑦)))))
20	12, 7, 16, 19, 2, 8	dvmptdivc 23534	. . . . . . . 8 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ ℂ ↦ (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴)))
21	15, 3, 9	divnegd 10693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴))
22	21	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = -((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴))
23	14, 3, 9	divcan3d 10685	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = -(sin‘(𝐴 · 𝑦)))
24	23	negeqd 10154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = --(sin‘(𝐴 · 𝑦)))
25	13	negnegd 10262	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → --(sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝑦)))
26	22, 24, 25	3eqtrd 2648	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = (sin‘(𝐴 · 𝑦)))
27	26	mpteq2dva 4672	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴)) = (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))))
28	20, 27	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))))
29		itgsincmulx.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
30		itgsincmulx.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
31	1, 10, 28, 13, 29, 30	dvmptresicc 38809	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))))
32	31	fveq1d 6105	. . . . 5 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) = ((𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦)))‘𝑥))
33	32	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) = ((𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦)))‘𝑥))
34		eqidd 2611	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))))
35		oveq2 6557	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐴 · 𝑦) = (𝐴 · 𝑥))
36	35	fveq2d 6107	. . . . . 6 ⊢ (𝑦 = 𝑥 → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝑥)))
37	36	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) ∧ 𝑦 = 𝑥) → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝑥)))
38		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐵(,)𝐶))
39	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐴 ∈ ℂ)
40		ioosscn 38563	. . . . . . . 8 ⊢ (𝐵(,)𝐶) ⊆ ℂ
41	40, 38	sseldi 3566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ℂ)
42	39, 41	mulcld 9939	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (𝐴 · 𝑥) ∈ ℂ)
43	42	sincld 14699	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (sin‘(𝐴 · 𝑥)) ∈ ℂ)
44	34, 37, 38, 43	fvmptd 6197	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → ((𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦)))‘𝑥) = (sin‘(𝐴 · 𝑥)))
45	33, 44	eqtr2d 2645	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (sin‘(𝐴 · 𝑥)) = ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥))
46	45	itgeq2dv 23354	. 2 ⊢ (𝜑 → ∫(𝐵(,)𝐶)(sin‘(𝐴 · 𝑥)) d𝑥 = ∫(𝐵(,)𝐶)((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) d𝑥)
47		itgsincmulx.blec	. . 3 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
48		sincn 24002	. . . . . 6 ⊢ sin ∈ (ℂ–cn→ℂ)
49	48	a1i 11	. . . . 5 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
50	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ ℂ)
51		ssid 3587	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
52	51	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
53	50, 2, 52	constcncfg 38756	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝐴) ∈ ((𝐵(,)𝐶)–cn→ℂ))
54	50, 52	idcncfg 38757	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝑦) ∈ ((𝐵(,)𝐶)–cn→ℂ))
55	53, 54	mulcncf 23023	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵(,)𝐶)–cn→ℂ))
56	49, 55	cncfmpt1f 22524	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
57	31, 56	eqeltrd 2688	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
58		ioossicc 12130	. . . . . 6 ⊢ (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶)
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶))
60		ioombl 23140	. . . . . 6 ⊢ (𝐵(,)𝐶) ∈ dom vol
61	60	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ∈ dom vol)
62	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝐴 ∈ ℂ)
63	29, 30	iccssred 38574	. . . . . . . . 9 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
64		ax-resscn 9872	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
65	63, 64	syl6ss 3580	. . . . . . . 8 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℂ)
66	65	sselda 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝑦 ∈ ℂ)
67	62, 66	mulcld 9939	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (𝐴 · 𝑦) ∈ ℂ)
68	67	sincld 14699	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
69	65, 2, 52	constcncfg 38756	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→ℂ))
70	65, 52	idcncfg 38757	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝑦) ∈ ((𝐵[,]𝐶)–cn→ℂ))
71	69, 70	mulcncf 23023	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
72	49, 71	cncfmpt1f 22524	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
73		cniccibl 23413	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ)) → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ 𝐿¹)
74	29, 30, 72, 73	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ 𝐿¹)
75	59, 61, 68, 74	iblss 23377	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ 𝐿¹)
76	31, 75	eqeltrd 2688	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) ∈ 𝐿¹)
77		coscn 24003	. . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ)
78	77	a1i 11	. . . . . 6 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
79	78, 71	cncfmpt1f 22524	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
80	79	negcncfg 38766	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ -(cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
81	8	neneqd 2787	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐴 = 0)
82		elsng 4139	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
83	2, 82	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ {0} ↔ 𝐴 = 0))
84	81, 83	mtbird 314	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ {0})
85	2, 84	eldifd 3551	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ {0}))
86		difssd 3700	. . . . 5 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
87	65, 85, 86	constcncfg 38756	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→(ℂ ∖ {0})))
88	80, 87	divcncf 38769	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
89	29, 30, 47, 57, 76, 88	ftc2 23611	. 2 ⊢ (𝜑 → ∫(𝐵(,)𝐶)((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) d𝑥 = (((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) − ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐵)))
90		eqidd 2611	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)) = (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))
91		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑦 = 𝐶 → (𝐴 · 𝑦) = (𝐴 · 𝐶))
92	91	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝐶)))
93	92	negeqd 10154	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → -(cos‘(𝐴 · 𝑦)) = -(cos‘(𝐴 · 𝐶)))
94	93	oveq1d 6564	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (-(cos‘(𝐴 · 𝑦)) / 𝐴) = (-(cos‘(𝐴 · 𝐶)) / 𝐴))
95	94	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐶) → (-(cos‘(𝐴 · 𝑦)) / 𝐴) = (-(cos‘(𝐴 · 𝐶)) / 𝐴))
96	29	rexrd 9968	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
97	30	rexrd 9968	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
98		ubicc2 12160	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐵[,]𝐶))
99	96, 97, 47, 98	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵[,]𝐶))
100		ovex 6577	. . . . . . 7 ⊢ (-(cos‘(𝐴 · 𝐶)) / 𝐴) ∈ V
101	100	a1i 11	. . . . . 6 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐶)) / 𝐴) ∈ V)
102	90, 95, 99, 101	fvmptd 6197	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) = (-(cos‘(𝐴 · 𝐶)) / 𝐴))
103		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
104	103	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝐵)))
105	104	negeqd 10154	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → -(cos‘(𝐴 · 𝑦)) = -(cos‘(𝐴 · 𝐵)))
106	105	oveq1d 6564	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (-(cos‘(𝐴 · 𝑦)) / 𝐴) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
107	106	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (-(cos‘(𝐴 · 𝑦)) / 𝐴) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
108		lbicc2 12159	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐵 ∈ (𝐵[,]𝐶))
109	96, 97, 47, 108	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐵[,]𝐶))
110		ovex 6577	. . . . . . 7 ⊢ (-(cos‘(𝐴 · 𝐵)) / 𝐴) ∈ V
111	110	a1i 11	. . . . . 6 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐵)) / 𝐴) ∈ V)
112	90, 107, 109, 111	fvmptd 6197	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐵) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
113	102, 112	oveq12d 6567	. . . 4 ⊢ (𝜑 → (((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) − ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐵)) = ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − (-(cos‘(𝐴 · 𝐵)) / 𝐴)))
114	29	recnd 9947	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
115	2, 114	mulcld 9939	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
116	115	coscld 14700	. . . . . . 7 ⊢ (𝜑 → (cos‘(𝐴 · 𝐵)) ∈ ℂ)
117	116, 2, 8	divnegd 10693	. . . . . 6 ⊢ (𝜑 → -((cos‘(𝐴 · 𝐵)) / 𝐴) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
118	117	eqcomd 2616	. . . . 5 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐵)) / 𝐴) = -((cos‘(𝐴 · 𝐵)) / 𝐴))
119	118	oveq2d 6565	. . . 4 ⊢ (𝜑 → ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − (-(cos‘(𝐴 · 𝐵)) / 𝐴)) = ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − -((cos‘(𝐴 · 𝐵)) / 𝐴)))
120	30	recnd 9947	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
121	2, 120	mulcld 9939	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
122	121	coscld 14700	. . . . . . 7 ⊢ (𝜑 → (cos‘(𝐴 · 𝐶)) ∈ ℂ)
123	122	negcld 10258	. . . . . 6 ⊢ (𝜑 → -(cos‘(𝐴 · 𝐶)) ∈ ℂ)
124	123, 2, 8	divcld 10680	. . . . 5 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐶)) / 𝐴) ∈ ℂ)
125	116, 2, 8	divcld 10680	. . . . 5 ⊢ (𝜑 → ((cos‘(𝐴 · 𝐵)) / 𝐴) ∈ ℂ)
126	124, 125	subnegd 10278	. . . 4 ⊢ (𝜑 → ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − -((cos‘(𝐴 · 𝐵)) / 𝐴)) = ((-(cos‘(𝐴 · 𝐶)) / 𝐴) + ((cos‘(𝐴 · 𝐵)) / 𝐴)))
127	113, 119, 126	3eqtrd 2648	. . 3 ⊢ (𝜑 → (((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) − ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐵)) = ((-(cos‘(𝐴 · 𝐶)) / 𝐴) + ((cos‘(𝐴 · 𝐵)) / 𝐴)))
128	124, 125	addcomd 10117	. . 3 ⊢ (𝜑 → ((-(cos‘(𝐴 · 𝐶)) / 𝐴) + ((cos‘(𝐴 · 𝐵)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) / 𝐴) + (-(cos‘(𝐴 · 𝐶)) / 𝐴)))
129	122, 2, 8	divnegd 10693	. . . . . 6 ⊢ (𝜑 → -((cos‘(𝐴 · 𝐶)) / 𝐴) = (-(cos‘(𝐴 · 𝐶)) / 𝐴))
130	129	eqcomd 2616	. . . . 5 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐶)) / 𝐴) = -((cos‘(𝐴 · 𝐶)) / 𝐴))
131	130	oveq2d 6565	. . . 4 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) + (-(cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) / 𝐴) + -((cos‘(𝐴 · 𝐶)) / 𝐴)))
132	122, 2, 8	divcld 10680	. . . . 5 ⊢ (𝜑 → ((cos‘(𝐴 · 𝐶)) / 𝐴) ∈ ℂ)
133	125, 132	negsubd 10277	. . . 4 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) + -((cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) / 𝐴) − ((cos‘(𝐴 · 𝐶)) / 𝐴)))
134	116, 122, 2, 8	divsubdird 10719	. . . . 5 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴) = (((cos‘(𝐴 · 𝐵)) / 𝐴) − ((cos‘(𝐴 · 𝐶)) / 𝐴)))
135	134	eqcomd 2616	. . . 4 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) − ((cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
136	131, 133, 135	3eqtrd 2648	. . 3 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) + (-(cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
137	127, 128, 136	3eqtrd 2648	. 2 ⊢ (𝜑 → (((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) − ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐵)) = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
138	46, 89, 137	3eqtrd 2648	1 ⊢ (𝜑 → ∫(𝐵(,)𝐶)(sin‘(𝐴 · 𝑥)) d𝑥 = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 (,)cioo 12046 [,]cicc 12049 sincsin 14633 cosccos 14634 –cn→ccncf 22487 volcvol 23039 𝐿¹cibl 23192 ∫citg 23193 D cdv 23433

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-itg2 23196 df-ibl 23197 df-itg 23198 df-0p 23243 df-limc 23436 df-dv 23437

This theorem is referenced by: sqwvfourb 39122

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