itgcoscmulx - Mathbox for Glauco Siliprandi

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Theorem itgcoscmulx 46324

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

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

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

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

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

Step	Hyp	Ref	Expression
1		itgcoscmulx.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
2		itgcoscmulx.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
3	1, 2	iccssred 13362	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
4	3	resmptd 6007	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶)) = (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))
5	4	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) = ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶)))
6	5	oveq2d 7384	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (ℝ D ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶))))
7		ax-resscn 11095	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
9	8	sselda 3935	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
10		itgcoscmulx.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	10	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
12		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
13	11, 12	mulcld 11164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
14	13	sincld 16067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
15		itgcoscmulx.an0	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 0)
16	15	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐴 ≠ 0)
17	14, 11, 16	divcld 11929	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((sin‘(𝐴 · 𝑦)) / 𝐴) ∈ ℂ)
18	9, 17	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((sin‘(𝐴 · 𝑦)) / 𝐴) ∈ ℂ)
19	18	fmpttd 7069	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)):ℝ⟶ℂ)
20		ssidd 3959	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
21		eqid 2737	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22		tgioo4 24761	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
23	21, 22	dvres 25880	. . . . . . . 8 ⊢ (((ℝ ⊆ ℂ ∧ (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)):ℝ⟶ℂ) ∧ (ℝ ⊆ ℝ ∧ (𝐵[,]𝐶) ⊆ ℝ)) → (ℝ D ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶))) = ((ℝ D (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ↾ ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶))))
24	8, 19, 20, 3, 23	syl22anc 839	. . . . . . 7 ⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶))) = ((ℝ D (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ↾ ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶))))
25		reelprrecn 11130	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
27	9, 14	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
28	10	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
29	28, 9	mulcld 11164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐴 · 𝑦) ∈ ℂ)
30	29	coscld 16068	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
31	28, 30	mulcld 11164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ)
32	8	resmptd 6007	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦))))
33	32	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦))) = ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ))
34	33	oveq2d 7384	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦)))) = (ℝ D ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ)))
35	14	fmpttd 7069	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))):ℂ⟶ℂ)
36		ssidd 3959	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ⊆ ℂ)
37		dvsinax 46268	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
38	10, 37	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
39	38	dmeqd 5862	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = dom (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
40	13	coscld 16068	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
41	11, 40	mulcld 11164	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ)
42	41	ralrimiva 3130	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑦 ∈ ℂ (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ)
43		dmmptg 6208	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ ℂ (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ → dom (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) = ℂ)
44	42, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) = ℂ)
45	39, 44	eqtr2d 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℂ = dom (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))))
46	7, 45	sseqtrid 3978	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))))
47		dvres3 25882	. . . . . . . . . . . . 13 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))):ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))))) → (ℝ D ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) ↾ ℝ))
48	26, 35, 36, 46, 47	syl22anc 839	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) ↾ ℝ))
49	38	reseq1d 5945	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) ↾ ℝ) = ((𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) ↾ ℝ))
50	8	resmptd 6007	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
51	48, 49, 50	3eqtrd 2776	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ)) = (𝑦 ∈ ℝ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
52	34, 51	eqtrd 2772	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℝ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
53	26, 27, 31, 52, 10, 15	dvmptdivc 25937	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ ℝ ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)))
54		iccntr 24778	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶)) = (𝐵(,)𝐶))
55	1, 2, 54	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶)) = (𝐵(,)𝐶))
56	53, 55	reseq12d 5947	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ↾ ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶))) = ((𝑦 ∈ ℝ ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)) ↾ (𝐵(,)𝐶)))
57		ioossre 13335	. . . . . . . . 9 ⊢ (𝐵(,)𝐶) ⊆ ℝ
58		resmpt 6004	. . . . . . . . 9 ⊢ ((𝐵(,)𝐶) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)) ↾ (𝐵(,)𝐶)) = (𝑦 ∈ (𝐵(,)𝐶) ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)))
59	57, 58	mp1i 13	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)) ↾ (𝐵(,)𝐶)) = (𝑦 ∈ (𝐵(,)𝐶) ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)))
60		elioore 13303	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐵(,)𝐶) → 𝑦 ∈ ℝ)
61	60	recnd 11172	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐵(,)𝐶) → 𝑦 ∈ ℂ)
62	61, 40	sylan2 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
63	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → 𝐴 ∈ ℂ)
64	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → 𝐴 ≠ 0)
65	62, 63, 64	divcan3d 11934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴) = (cos‘(𝐴 · 𝑦)))
66	65	mpteq2dva 5193	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))))
67	56, 59, 66	3eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ↾ ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))))
68	6, 24, 67	3eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))))
69	68	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))))
70		oveq2 7376	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐴 · 𝑦) = (𝐴 · 𝑥))
71	70	fveq2d 6846	. . . . . 6 ⊢ (𝑦 = 𝑥 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝑥)))
72	71	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) ∧ 𝑦 = 𝑥) → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝑥)))
73		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐵(,)𝐶))
74	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐴 ∈ ℂ)
75	57, 8	sstrid 3947	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ ℂ)
76	75	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ℂ)
77	74, 76	mulcld 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (𝐴 · 𝑥) ∈ ℂ)
78	77	coscld 16068	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (cos‘(𝐴 · 𝑥)) ∈ ℂ)
79	69, 72, 73, 78	fvmptd 6957	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) = (cos‘(𝐴 · 𝑥)))
80	79	eqcomd 2743	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (cos‘(𝐴 · 𝑥)) = ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥))
81	80	itgeq2dv 25751	. 2 ⊢ (𝜑 → ∫(𝐵(,)𝐶)(cos‘(𝐴 · 𝑥)) d𝑥 = ∫(𝐵(,)𝐶)((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) d𝑥)
82		eqidd 2738	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) = (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))
83		oveq2 7376	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝐴 · 𝑦) = (𝐴 · 𝐶))
84	83	fveq2d 6846	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝐶)))
85	84	oveq1d 7383	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐶)) / 𝐴))
86	85	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐶) → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐶)) / 𝐴))
87	1	rexrd 11194	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
88	2	rexrd 11194	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
89		itgcoscmulx.blec	. . . . . 6 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
90		ubicc2 13393	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐵[,]𝐶))
91	87, 88, 89, 90	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐵[,]𝐶))
92	2	recnd 11172	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
93	10, 92	mulcld 11164	. . . . . . 7 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
94	93	sincld 16067	. . . . . 6 ⊢ (𝜑 → (sin‘(𝐴 · 𝐶)) ∈ ℂ)
95	94, 10, 15	divcld 11929	. . . . 5 ⊢ (𝜑 → ((sin‘(𝐴 · 𝐶)) / 𝐴) ∈ ℂ)
96	82, 86, 91, 95	fvmptd 6957	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) = ((sin‘(𝐴 · 𝐶)) / 𝐴))
97		oveq2 7376	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
98	97	fveq2d 6846	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝐵)))
99	98	oveq1d 7383	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐵)) / 𝐴))
100	99	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐵)) / 𝐴))
101		lbicc2 13392	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐵 ∈ (𝐵[,]𝐶))
102	87, 88, 89, 101	syl3anc 1374	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐵[,]𝐶))
103	1	recnd 11172	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
104	10, 103	mulcld 11164	. . . . . . 7 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
105	104	sincld 16067	. . . . . 6 ⊢ (𝜑 → (sin‘(𝐴 · 𝐵)) ∈ ℂ)
106	105, 10, 15	divcld 11929	. . . . 5 ⊢ (𝜑 → ((sin‘(𝐴 · 𝐵)) / 𝐴) ∈ ℂ)
107	82, 100, 102, 106	fvmptd 6957	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐵) = ((sin‘(𝐴 · 𝐵)) / 𝐴))
108	96, 107	oveq12d 7386	. . 3 ⊢ (𝜑 → (((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) − ((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐵)) = (((sin‘(𝐴 · 𝐶)) / 𝐴) − ((sin‘(𝐴 · 𝐵)) / 𝐴)))
109		coscn 26423	. . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ)
110	109	a1i 11	. . . . . 6 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
111	75, 10, 36	constcncfg 46227	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝐴) ∈ ((𝐵(,)𝐶)–cn→ℂ))
112	75, 36	idcncfg 46228	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝑦) ∈ ((𝐵(,)𝐶)–cn→ℂ))
113	111, 112	mulcncf 25414	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵(,)𝐶)–cn→ℂ))
114	110, 113	cncfmpt1f 24875	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
115	68, 114	eqeltrd 2837	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
116		ioossicc 13361	. . . . . . 7 ⊢ (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶)
117	116	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶))
118		ioombl 25534	. . . . . . 7 ⊢ (𝐵(,)𝐶) ∈ dom vol
119	118	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐵(,)𝐶) ∈ dom vol)
120	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝐴 ∈ ℂ)
121	3, 7	sstrdi 3948	. . . . . . . . 9 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℂ)
122	121	sselda 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝑦 ∈ ℂ)
123	120, 122	mulcld 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (𝐴 · 𝑦) ∈ ℂ)
124	123	coscld 16068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
125	121, 10, 36	constcncfg 46227	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→ℂ))
126	121, 36	idcncfg 46228	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝑦) ∈ ((𝐵[,]𝐶)–cn→ℂ))
127	125, 126	mulcncf 25414	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
128	110, 127	cncfmpt1f 24875	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
129		cniccibl 25810	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ)) → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ 𝐿¹)
130	1, 2, 128, 129	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ 𝐿¹)
131	117, 119, 124, 130	iblss 25774	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ 𝐿¹)
132	68, 131	eqeltrd 2837	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ∈ 𝐿¹)
133		sincn 26422	. . . . . . 7 ⊢ sin ∈ (ℂ–cn→ℂ)
134	133	a1i 11	. . . . . 6 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
135	134, 127	cncfmpt1f 24875	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
136		neneq 2939	. . . . . . . 8 ⊢ (𝐴 ≠ 0 → ¬ 𝐴 = 0)
137		elsni 4599	. . . . . . . . 9 ⊢ (𝐴 ∈ {0} → 𝐴 = 0)
138	137	con3i 154	. . . . . . . 8 ⊢ (¬ 𝐴 = 0 → ¬ 𝐴 ∈ {0})
139	15, 136, 138	3syl 18	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐴 ∈ {0})
140	10, 139	eldifd 3914	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ {0}))
141		difssd 4091	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
142	121, 140, 141	constcncfg 46227	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→(ℂ ∖ {0})))
143	135, 142	divcncf 25416	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
144	1, 2, 89, 115, 132, 143	ftc2 26019	. . 3 ⊢ (𝜑 → ∫(𝐵(,)𝐶)((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) d𝑥 = (((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) − ((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐵)))
145	94, 105, 10, 15	divsubdird 11968	. . 3 ⊢ (𝜑 → (((sin‘(𝐴 · 𝐶)) − (sin‘(𝐴 · 𝐵))) / 𝐴) = (((sin‘(𝐴 · 𝐶)) / 𝐴) − ((sin‘(𝐴 · 𝐵)) / 𝐴)))
146	108, 144, 145	3eqtr4d 2782	. 2 ⊢ (𝜑 → ∫(𝐵(,)𝐶)((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) d𝑥 = (((sin‘(𝐴 · 𝐶)) − (sin‘(𝐴 · 𝐵))) / 𝐴))
147	81, 146	eqtrd 2772	1 ⊢ (𝜑 → ∫(𝐵(,)𝐶)(cos‘(𝐴 · 𝑥)) d𝑥 = (((sin‘(𝐴 · 𝐶)) − (sin‘(𝐴 · 𝐵))) / 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∖ cdif 3900 ⊆ wss 3903 {csn 4582 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ran crn 5633 ↾ cres 5634 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 · cmul 11043 ℝ^*cxr 11177 ≤ cle 11179 − cmin 11376 / cdiv 11806 (,)cioo 13273 [,]cicc 13276 sincsin 15998 cosccos 15999 TopOpenctopn 17353 topGenctg 17369 ℂ_fldccnfld 21321 intcnt 22973 –cn→ccncf 24837 volcvol 25432 𝐿¹cibl 25586 ∫citg 25587 D cdv 25832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4207 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-cmp 23343 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-ovol 25433 df-vol 25434 df-mbf 25588 df-itg1 25589 df-itg2 25590 df-ibl 25591 df-itg 25592 df-0p 25639 df-limc 25835 df-dv 25836

This theorem is referenced by: sqwvfoura 46583

W3C validator