itgsincmulx - Mathbox for Glauco Siliprandi

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Theorem itgsincmulx 43405

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 2738	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
5	3, 4	mulcld 10926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
6	5	coscld 15768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
7	6	negcld 11249	. . . . . . . 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 11681	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (-(cos‘(𝐴 · 𝑦)) / 𝐴) ∈ ℂ)
11		cnelprrecn 10895	. . . . . . . . . 10 ⊢ ℂ ∈ {ℝ, ℂ}
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
13	5	sincld 15767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
14	13	negcld 11249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -(sin‘(𝐴 · 𝑦)) ∈ ℂ)
15	3, 14	mulcld 10926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · -(sin‘(𝐴 · 𝑦))) ∈ ℂ)
16	15	negcld 11249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -(𝐴 · -(sin‘(𝐴 · 𝑦))) ∈ ℂ)
17		dvcosax 43357	. . . . . . . . . . 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 25035	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ -(cos‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ -(𝐴 · -(sin‘(𝐴 · 𝑦)))))
20	12, 7, 16, 19, 2, 8	dvmptdivc 25034	. . . . . . . 8 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ ℂ ↦ (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴)))
21	15, 3, 9	divnegd 11694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴))
22	21	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = -((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴))
23	14, 3, 9	divcan3d 11686	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = -(sin‘(𝐴 · 𝑦)))
24	23	negeqd 11145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → -((𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = --(sin‘(𝐴 · 𝑦)))
25	13	negnegd 11253	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → --(sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝑦)))
26	22, 24, 25	3eqtrd 2782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴) = (sin‘(𝐴 · 𝑦)))
27	26	mpteq2dva 5170	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (-(𝐴 · -(sin‘(𝐴 · 𝑦))) / 𝐴)) = (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))))
28	20, 27	eqtrd 2778	. . . . . . 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 24985	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))))
32	31	fveq1d 6758	. . . . 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 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))))
35		oveq2 7263	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐴 · 𝑦) = (𝐴 · 𝑥))
36	35	fveq2d 6760	. . . . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐵(,)𝐶))
39	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐴 ∈ ℂ)
40		ioosscn 13070	. . . . . . . 8 ⊢ (𝐵(,)𝐶) ⊆ ℂ
41	40, 38	sselid 3915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ℂ)
42	39, 41	mulcld 10926	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (𝐴 · 𝑥) ∈ ℂ)
43	42	sincld 15767	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (sin‘(𝐴 · 𝑥)) ∈ ℂ)
44	34, 37, 38, 43	fvmptd 6864	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → ((𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦)))‘𝑥) = (sin‘(𝐴 · 𝑥)))
45	33, 44	eqtr2d 2779	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (sin‘(𝐴 · 𝑥)) = ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥))
46	45	itgeq2dv 24851	. 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 25508	. . . . . 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 3939	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
52	51	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
53	50, 2, 52	constcncfg 43303	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝐴) ∈ ((𝐵(,)𝐶)–cn→ℂ))
54	50, 52	idcncfg 43304	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝑦) ∈ ((𝐵(,)𝐶)–cn→ℂ))
55	53, 54	mulcncf 24515	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵(,)𝐶)–cn→ℂ))
56	49, 55	cncfmpt1f 23983	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
57	31, 56	eqeltrd 2839	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
58		ioossicc 13094	. . . . . 6 ⊢ (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶)
59	58	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶))
60		ioombl 24634	. . . . . 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 13095	. . . . . . . . 9 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
64		ax-resscn 10859	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
65	63, 64	sstrdi 3929	. . . . . . . 8 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℂ)
66	65	sselda 3917	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝑦 ∈ ℂ)
67	62, 66	mulcld 10926	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (𝐴 · 𝑦) ∈ ℂ)
68	67	sincld 15767	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
69	65, 2, 52	constcncfg 43303	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→ℂ))
70	65, 52	idcncfg 43304	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝑦) ∈ ((𝐵[,]𝐶)–cn→ℂ))
71	69, 70	mulcncf 24515	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
72	49, 71	cncfmpt1f 23983	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
73		cniccibl 24910	. . . . . 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 1369	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ 𝐿¹)
75	59, 61, 68, 74	iblss 24874	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ 𝐿¹)
76	31, 75	eqeltrd 2839	. . 3 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))) ∈ 𝐿¹)
77		coscn 25509	. . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ)
78	77	a1i 11	. . . . . 6 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
79	78, 71	cncfmpt1f 23983	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
80	79	negcncfg 43312	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ -(cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
81	8	neneqd 2947	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐴 = 0)
82		elsng 4572	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ {0} ↔ 𝐴 = 0))
83	2, 82	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ {0} ↔ 𝐴 = 0))
84	81, 83	mtbird 324	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ {0})
85	2, 84	eldifd 3894	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ {0}))
86		difssd 4063	. . . . 5 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
87	65, 85, 86	constcncfg 43303	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→(ℂ ∖ {0})))
88	80, 87	divcncf 24516	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
89	29, 30, 47, 57, 76, 88	ftc2 25113	. 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 2739	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)) = (𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴)))
91		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑦 = 𝐶 → (𝐴 · 𝑦) = (𝐴 · 𝐶))
92	91	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑦 = 𝐶 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝐶)))
93	92	negeqd 11145	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → -(cos‘(𝐴 · 𝑦)) = -(cos‘(𝐴 · 𝐶)))
94	93	oveq1d 7270	. . . . . . 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 10956	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
97	30	rexrd 10956	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
98		ubicc2 13126	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐵[,]𝐶))
99	96, 97, 47, 98	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐵[,]𝐶))
100		ovexd 7290	. . . . . 6 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐶)) / 𝐴) ∈ V)
101	90, 95, 99, 100	fvmptd 6864	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) = (-(cos‘(𝐴 · 𝐶)) / 𝐴))
102		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
103	102	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝐵)))
104	103	negeqd 11145	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → -(cos‘(𝐴 · 𝑦)) = -(cos‘(𝐴 · 𝐵)))
105	104	oveq1d 7270	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (-(cos‘(𝐴 · 𝑦)) / 𝐴) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
106	105	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (-(cos‘(𝐴 · 𝑦)) / 𝐴) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
107		lbicc2 13125	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐵 ∈ (𝐵[,]𝐶))
108	96, 97, 47, 107	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (𝐵[,]𝐶))
109		ovexd 7290	. . . . . 6 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐵)) / 𝐴) ∈ V)
110	90, 106, 108, 109	fvmptd 6864	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ (-(cos‘(𝐴 · 𝑦)) / 𝐴))‘𝐵) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
111	101, 110	oveq12d 7273	. . . 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)))
112	29	recnd 10934	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℂ)
113	2, 112	mulcld 10926	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
114	113	coscld 15768	. . . . . . 7 ⊢ (𝜑 → (cos‘(𝐴 · 𝐵)) ∈ ℂ)
115	114, 2, 8	divnegd 11694	. . . . . 6 ⊢ (𝜑 → -((cos‘(𝐴 · 𝐵)) / 𝐴) = (-(cos‘(𝐴 · 𝐵)) / 𝐴))
116	115	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐵)) / 𝐴) = -((cos‘(𝐴 · 𝐵)) / 𝐴))
117	116	oveq2d 7271	. . . 4 ⊢ (𝜑 → ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − (-(cos‘(𝐴 · 𝐵)) / 𝐴)) = ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − -((cos‘(𝐴 · 𝐵)) / 𝐴)))
118	30	recnd 10934	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℂ)
119	2, 118	mulcld 10926	. . . . . . . 8 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
120	119	coscld 15768	. . . . . . 7 ⊢ (𝜑 → (cos‘(𝐴 · 𝐶)) ∈ ℂ)
121	120	negcld 11249	. . . . . 6 ⊢ (𝜑 → -(cos‘(𝐴 · 𝐶)) ∈ ℂ)
122	121, 2, 8	divcld 11681	. . . . 5 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐶)) / 𝐴) ∈ ℂ)
123	114, 2, 8	divcld 11681	. . . . 5 ⊢ (𝜑 → ((cos‘(𝐴 · 𝐵)) / 𝐴) ∈ ℂ)
124	122, 123	subnegd 11269	. . . 4 ⊢ (𝜑 → ((-(cos‘(𝐴 · 𝐶)) / 𝐴) − -((cos‘(𝐴 · 𝐵)) / 𝐴)) = ((-(cos‘(𝐴 · 𝐶)) / 𝐴) + ((cos‘(𝐴 · 𝐵)) / 𝐴)))
125	111, 117, 124	3eqtrd 2782	. . 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)))
126	122, 123	addcomd 11107	. . 3 ⊢ (𝜑 → ((-(cos‘(𝐴 · 𝐶)) / 𝐴) + ((cos‘(𝐴 · 𝐵)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) / 𝐴) + (-(cos‘(𝐴 · 𝐶)) / 𝐴)))
127	120, 2, 8	divnegd 11694	. . . . . 6 ⊢ (𝜑 → -((cos‘(𝐴 · 𝐶)) / 𝐴) = (-(cos‘(𝐴 · 𝐶)) / 𝐴))
128	127	eqcomd 2744	. . . . 5 ⊢ (𝜑 → (-(cos‘(𝐴 · 𝐶)) / 𝐴) = -((cos‘(𝐴 · 𝐶)) / 𝐴))
129	128	oveq2d 7271	. . . 4 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) + (-(cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) / 𝐴) + -((cos‘(𝐴 · 𝐶)) / 𝐴)))
130	120, 2, 8	divcld 11681	. . . . 5 ⊢ (𝜑 → ((cos‘(𝐴 · 𝐶)) / 𝐴) ∈ ℂ)
131	123, 130	negsubd 11268	. . . 4 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) + -((cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) / 𝐴) − ((cos‘(𝐴 · 𝐶)) / 𝐴)))
132	114, 120, 2, 8	divsubdird 11720	. . . . 5 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴) = (((cos‘(𝐴 · 𝐵)) / 𝐴) − ((cos‘(𝐴 · 𝐶)) / 𝐴)))
133	132	eqcomd 2744	. . . 4 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) − ((cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
134	129, 131, 133	3eqtrd 2782	. . 3 ⊢ (𝜑 → (((cos‘(𝐴 · 𝐵)) / 𝐴) + (-(cos‘(𝐴 · 𝐶)) / 𝐴)) = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
135	125, 126, 134	3eqtrd 2782	. 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))
136	46, 89, 135	3eqtrd 2782	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 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 {cpr 4560 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 + caddc 10805 · cmul 10807 ℝ^*cxr 10939 ≤ cle 10941 − cmin 11135 -cneg 11136 / cdiv 11562 (,)cioo 13008 [,]cicc 13011 sincsin 15701 cosccos 15702 –cn→ccncf 23945 volcvol 24532 𝐿¹cibl 24686 ∫citg 24687 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-limc 24935 df-dv 24936

This theorem is referenced by: sqwvfourb 43660

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