itgcoscmulx - Mathbox for Glauco Siliprandi

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Theorem itgcoscmulx 44770

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 13413	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℝ)
4	3	resmptd 6040	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶)) = (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))
5	4	eqcomd 2738	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) = ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶)))
6	5	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (ℝ D ((𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ↾ (𝐵[,]𝐶))))
7		ax-resscn 11169	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
9	8	sselda 3982	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
10		itgcoscmulx.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
11	10	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
12		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
13	11, 12	mulcld 11236	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
14	13	sincld 16075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
15		itgcoscmulx.an0	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≠ 0)
16	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐴 ≠ 0)
17	14, 11, 16	divcld 11992	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((sin‘(𝐴 · 𝑦)) / 𝐴) ∈ ℂ)
18	9, 17	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((sin‘(𝐴 · 𝑦)) / 𝐴) ∈ ℂ)
19	18	fmpttd 7116	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)):ℝ⟶ℂ)
20		ssidd 4005	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ)
21		eqid 2732	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22		tgioo4 44371	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
23	21, 22	dvres 25435	. . . . . . . 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 837	. . . . . . 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 11204	. . . . . . . . . . 11 ⊢ ℝ ∈ {ℝ, ℂ}
26	25	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
27	9, 14	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
28	10	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
29	28, 9	mulcld 11236	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐴 · 𝑦) ∈ ℂ)
30	29	coscld 16076	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
31	28, 30	mulcld 11236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ)
32	8	resmptd 6040	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ) = (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦))))
33	32	eqcomd 2738	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦))) = ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ))
34	33	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (sin‘(𝐴 · 𝑦)))) = (ℝ D ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ)))
35	14	fmpttd 7116	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))):ℂ⟶ℂ)
36		ssidd 4005	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ⊆ ℂ)
37		dvsinax 44714	. . . . . . . . . . . . . . . . 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 5905	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = dom (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
40	13	coscld 16076	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
41	11, 40	mulcld 11236	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ)
42	41	ralrimiva 3146	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑦 ∈ ℂ (𝐴 · (cos‘(𝐴 · 𝑦))) ∈ ℂ)
43		dmmptg 6241	. . . . . . . . . . . . . . . 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 4034	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))))
47		dvres3 25437	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D ((𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) ↾ ℝ)) = ((ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) ↾ ℝ))
49	38	reseq1d 5980	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) ↾ ℝ) = ((𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) ↾ ℝ))
50	8	resmptd 6040	. . . . . . . . . . . 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 25489	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ ℝ ↦ ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴)))
54		iccntr 24344	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶)) = (𝐵(,)𝐶))
55	1, 2, 54	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐵[,]𝐶)) = (𝐵(,)𝐶))
56	53, 55	reseq12d 5982	. . . . . . . 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 13387	. . . . . . . . 9 ⊢ (𝐵(,)𝐶) ⊆ ℝ
58		resmpt 6037	. . . . . . . . 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 13356	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐵(,)𝐶) → 𝑦 ∈ ℝ)
61	60	recnd 11244	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐵(,)𝐶) → 𝑦 ∈ ℂ)
62	61, 40	sylan2 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
63	10	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → 𝐴 ∈ ℂ)
64	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → 𝐴 ≠ 0)
65	62, 63, 64	divcan3d 11997	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵(,)𝐶)) → ((𝐴 · (cos‘(𝐴 · 𝑦))) / 𝐴) = (cos‘(𝐴 · 𝑦)))
66	65	mpteq2dva 5248	. . . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) = (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))))
70		oveq2 7419	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐴 · 𝑦) = (𝐴 · 𝑥))
71	70	fveq2d 6895	. . . . . 6 ⊢ (𝑦 = 𝑥 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝑥)))
72	71	adantl 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) ∧ 𝑦 = 𝑥) → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝑥)))
73		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐵(,)𝐶))
74	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝐴 ∈ ℂ)
75	57, 8	sstrid 3993	. . . . . . . 8 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ ℂ)
76	75	sselda 3982	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ℂ)
77	74, 76	mulcld 11236	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (𝐴 · 𝑥) ∈ ℂ)
78	77	coscld 16076	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (cos‘(𝐴 · 𝑥)) ∈ ℂ)
79	69, 72, 73, 78	fvmptd 7005	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) = (cos‘(𝐴 · 𝑥)))
80	79	eqcomd 2738	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → (cos‘(𝐴 · 𝑥)) = ((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥))
81	80	itgeq2dv 25306	. 2 ⊢ (𝜑 → ∫(𝐵(,)𝐶)(cos‘(𝐴 · 𝑥)) d𝑥 = ∫(𝐵(,)𝐶)((ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))‘𝑥) d𝑥)
82		eqidd 2733	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) = (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)))
83		oveq2 7419	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝐴 · 𝑦) = (𝐴 · 𝐶))
84	83	fveq2d 6895	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝐶)))
85	84	oveq1d 7426	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐶)) / 𝐴))
86	85	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐶) → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐶)) / 𝐴))
87	1	rexrd 11266	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
88	2	rexrd 11266	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
89		itgcoscmulx.blec	. . . . . 6 ⊢ (𝜑 → 𝐵 ≤ 𝐶)
90		ubicc2 13444	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐶 ∈ (𝐵[,]𝐶))
91	87, 88, 89, 90	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (𝐵[,]𝐶))
92	2	recnd 11244	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
93	10, 92	mulcld 11236	. . . . . . 7 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ)
94	93	sincld 16075	. . . . . 6 ⊢ (𝜑 → (sin‘(𝐴 · 𝐶)) ∈ ℂ)
95	94, 10, 15	divcld 11992	. . . . 5 ⊢ (𝜑 → ((sin‘(𝐴 · 𝐶)) / 𝐴) ∈ ℂ)
96	82, 86, 91, 95	fvmptd 7005	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐶) = ((sin‘(𝐴 · 𝐶)) / 𝐴))
97		oveq2 7419	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 · 𝑦) = (𝐴 · 𝐵))
98	97	fveq2d 6895	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝐵)))
99	98	oveq1d 7426	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐵)) / 𝐴))
100	99	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → ((sin‘(𝐴 · 𝑦)) / 𝐴) = ((sin‘(𝐴 · 𝐵)) / 𝐴))
101		lbicc2 13443	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐵 ∈ (𝐵[,]𝐶))
102	87, 88, 89, 101	syl3anc 1371	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐵[,]𝐶))
103	1	recnd 11244	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℂ)
104	10, 103	mulcld 11236	. . . . . . 7 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
105	104	sincld 16075	. . . . . 6 ⊢ (𝜑 → (sin‘(𝐴 · 𝐵)) ∈ ℂ)
106	105, 10, 15	divcld 11992	. . . . 5 ⊢ (𝜑 → ((sin‘(𝐴 · 𝐵)) / 𝐴) ∈ ℂ)
107	82, 100, 102, 106	fvmptd 7005	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))‘𝐵) = ((sin‘(𝐴 · 𝐵)) / 𝐴))
108	96, 107	oveq12d 7429	. . 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 25964	. . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ)
110	109	a1i 11	. . . . . 6 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
111	75, 10, 36	constcncfg 44673	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝐴) ∈ ((𝐵(,)𝐶)–cn→ℂ))
112	75, 36	idcncfg 44674	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ 𝑦) ∈ ((𝐵(,)𝐶)–cn→ℂ))
113	111, 112	mulcncf 24970	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵(,)𝐶)–cn→ℂ))
114	110, 113	cncfmpt1f 24437	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
115	68, 114	eqeltrd 2833	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ∈ ((𝐵(,)𝐶)–cn→ℂ))
116		ioossicc 13412	. . . . . . 7 ⊢ (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶)
117	116	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵[,]𝐶))
118		ioombl 25089	. . . . . . 7 ⊢ (𝐵(,)𝐶) ∈ dom vol
119	118	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐵(,)𝐶) ∈ dom vol)
120	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝐴 ∈ ℂ)
121	3, 7	sstrdi 3994	. . . . . . . . 9 ⊢ (𝜑 → (𝐵[,]𝐶) ⊆ ℂ)
122	121	sselda 3982	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → 𝑦 ∈ ℂ)
123	120, 122	mulcld 11236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (𝐴 · 𝑦) ∈ ℂ)
124	123	coscld 16076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵[,]𝐶)) → (cos‘(𝐴 · 𝑦)) ∈ ℂ)
125	121, 10, 36	constcncfg 44673	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→ℂ))
126	121, 36	idcncfg 44674	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝑦) ∈ ((𝐵[,]𝐶)–cn→ℂ))
127	125, 126	mulcncf 24970	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (𝐴 · 𝑦)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
128	110, 127	cncfmpt1f 24437	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
129		cniccibl 25365	. . . . . . 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 1371	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ 𝐿¹)
131	117, 119, 124, 130	iblss 25329	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵(,)𝐶) ↦ (cos‘(𝐴 · 𝑦))) ∈ 𝐿¹)
132	68, 131	eqeltrd 2833	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴))) ∈ 𝐿¹)
133		sincn 25963	. . . . . . 7 ⊢ sin ∈ (ℂ–cn→ℂ)
134	133	a1i 11	. . . . . 6 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
135	134, 127	cncfmpt1f 24437	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ (sin‘(𝐴 · 𝑦))) ∈ ((𝐵[,]𝐶)–cn→ℂ))
136		neneq 2946	. . . . . . . 8 ⊢ (𝐴 ≠ 0 → ¬ 𝐴 = 0)
137		elsni 4645	. . . . . . . . 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 3959	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ {0}))
141		difssd 4132	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
142	121, 140, 141	constcncfg 44673	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ 𝐴) ∈ ((𝐵[,]𝐶)–cn→(ℂ ∖ {0})))
143	135, 142	divcncf 24971	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝐵[,]𝐶) ↦ ((sin‘(𝐴 · 𝑦)) / 𝐴)) ∈ ((𝐵[,]𝐶)–cn→ℂ))
144	1, 2, 89, 115, 132, 143	ftc2 25568	. . 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 12031	. . 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 · cmul 11117 ℝ^*cxr 11249 ≤ cle 11251 − cmin 11446 / cdiv 11873 (,)cioo 13326 [,]cicc 13329 sincsin 16009 cosccos 16010 TopOpenctopn 17369 topGenctg 17385 ℂ_fldccnfld 20950 intcnt 22528 –cn→ccncf 24399 volcvol 24987 𝐿¹cibl 25141 ∫citg 25142 D cdv 25387

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-symdif 4242 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-mulg 18953 df-cntz 19183 df-cmn 19652 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-ovol 24988 df-vol 24989 df-mbf 25143 df-itg1 25144 df-itg2 25145 df-ibl 25146 df-itg 25147 df-0p 25194 df-limc 25390 df-dv 25391

This theorem is referenced by: sqwvfoura 45029

W3C validator