dvcosax - Mathbox for Glauco Siliprandi

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Theorem dvcosax 45316

Description: Derivative exercise: the derivative with respect to x of cos(Ax), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
dvcosax	⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥)))) = (𝑥 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑥)))))

Distinct variable group: 𝑥,𝐴

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

Step	Hyp	Ref	Expression
1		mulcl 11228	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐴 · 𝑥) ∈ ℂ)
2		eqidd 2728	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))
3		cosf 16107	. . . . . . . 8 ⊢ cos:ℂ⟶ℂ
4	3	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → cos:ℂ⟶ℂ)
5	4	feqmptd 6970	. . . . . 6 ⊢ (𝐴 ∈ ℂ → cos = (𝑦 ∈ ℂ ↦ (cos‘𝑦)))
6		fveq2 6900	. . . . . 6 ⊢ (𝑦 = (𝐴 · 𝑥) → (cos‘𝑦) = (cos‘(𝐴 · 𝑥)))
7	1, 2, 5, 6	fmptco 7142	. . . . 5 ⊢ (𝐴 ∈ ℂ → (cos ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥))))
8	7	eqcomd 2733	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥))) = (cos ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))))
9	8	oveq2d 7440	. . 3 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥)))) = (ℂ D (cos ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))))
10		cnelprrecn 11237	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
11	10	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → ℂ ∈ {ℝ, ℂ})
12	1	fmpttd 7128	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)):ℂ⟶ℂ)
13		dvcos 25933	. . . . . . 7 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
14	13	dmeqi 5909	. . . . . 6 ⊢ dom (ℂ D cos) = dom (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
15		dmmptg 6249	. . . . . . 7 ⊢ (∀𝑥 ∈ ℂ -(sin‘𝑥) ∈ ℂ → dom (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) = ℂ)
16		sincl 16108	. . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
17	16	negcld 11594	. . . . . . 7 ⊢ (𝑥 ∈ ℂ → -(sin‘𝑥) ∈ ℂ)
18	15, 17	mprg 3063	. . . . . 6 ⊢ dom (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) = ℂ
19	14, 18	eqtri 2755	. . . . 5 ⊢ dom (ℂ D cos) = ℂ
20	19	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → dom (ℂ D cos) = ℂ)
21		simpl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ)
22		0red 11253	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 0 ∈ ℝ)
23		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
24	11, 23	dvmptc 25908	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0))
25		simpr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
26		1red 11251	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℝ)
27	11	dvmptid 25907	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
28	11, 21, 22, 24, 25, 26, 27	dvmptmul 25911	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) = (𝑥 ∈ ℂ ↦ ((0 · 𝑥) + (1 · 𝐴))))
29	28	dmeqd 5910	. . . . 5 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) = dom (𝑥 ∈ ℂ ↦ ((0 · 𝑥) + (1 · 𝐴))))
30		dmmptg 6249	. . . . . 6 ⊢ (∀𝑥 ∈ ℂ ((0 · 𝑥) + (1 · 𝐴)) ∈ V → dom (𝑥 ∈ ℂ ↦ ((0 · 𝑥) + (1 · 𝐴))) = ℂ)
31		ovex 7457	. . . . . . 7 ⊢ ((0 · 𝑥) + (1 · 𝐴)) ∈ V
32	31	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ ℂ → ((0 · 𝑥) + (1 · 𝐴)) ∈ V)
33	30, 32	mprg 3063	. . . . 5 ⊢ dom (𝑥 ∈ ℂ ↦ ((0 · 𝑥) + (1 · 𝐴))) = ℂ
34	29, 33	eqtrdi 2783	. . . 4 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) = ℂ)
35	11, 11, 4, 12, 20, 34	dvcof 25898	. . 3 ⊢ (𝐴 ∈ ℂ → (ℂ D (cos ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = (((ℂ D cos) ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))))
36		dvcos 25933	. . . . . . 7 ⊢ (ℂ D cos) = (𝑦 ∈ ℂ ↦ -(sin‘𝑦))
37	36	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D cos) = (𝑦 ∈ ℂ ↦ -(sin‘𝑦)))
38		fveq2 6900	. . . . . . 7 ⊢ (𝑦 = (𝐴 · 𝑥) → (sin‘𝑦) = (sin‘(𝐴 · 𝑥)))
39	38	negeqd 11490	. . . . . 6 ⊢ (𝑦 = (𝐴 · 𝑥) → -(sin‘𝑦) = -(sin‘(𝐴 · 𝑥)))
40	1, 2, 37, 39	fmptco 7142	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((ℂ D cos) ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))))
41	40	oveq1d 7439	. . . 4 ⊢ (𝐴 ∈ ℂ → (((ℂ D cos) ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = ((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))))
42		cnex 11225	. . . . . . 7 ⊢ ℂ ∈ V
43	42	mptex 7239	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∈ V
44		ovex 7457	. . . . . 6 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) ∈ V
45		offval3 7990	. . . . . 6 ⊢ (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∈ V ∧ (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) ∈ V) → ((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = (𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) ↦ (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦))))
46	43, 44, 45	mp2an 690	. . . . 5 ⊢ ((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = (𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) ↦ (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦)))
47	46	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = (𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) ↦ (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦))))
48	1	sincld 16112	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (sin‘(𝐴 · 𝑥)) ∈ ℂ)
49	48	negcld 11594	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → -(sin‘(𝐴 · 𝑥)) ∈ ℂ)
50	49	ralrimiva 3142	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ∀𝑥 ∈ ℂ -(sin‘(𝐴 · 𝑥)) ∈ ℂ)
51		dmmptg 6249	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℂ -(sin‘(𝐴 · 𝑥)) ∈ ℂ → dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) = ℂ)
52	50, 51	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) = ℂ)
53	52, 34	ineq12d 4213	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = (ℂ ∩ ℂ))
54		inidm 4219	. . . . . 6 ⊢ (ℂ ∩ ℂ) = ℂ
55	53, 54	eqtrdi 2783	. . . . 5 ⊢ (𝐴 ∈ ℂ → (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = ℂ)
56		simpr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))))) → 𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))))
57	55	adantr 479	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))))) → (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = ℂ)
58	56, 57	eleqtrd 2830	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))))) → 𝑦 ∈ ℂ)
59		eqidd 2728	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))))
60		oveq2 7432	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝐴 · 𝑥) = (𝐴 · 𝑦))
61	60	fveq2d 6904	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (sin‘(𝐴 · 𝑥)) = (sin‘(𝐴 · 𝑦)))
62	61	negeqd 11490	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → -(sin‘(𝐴 · 𝑥)) = -(sin‘(𝐴 · 𝑦)))
63	62	adantl 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 = 𝑦) → -(sin‘(𝐴 · 𝑥)) = -(sin‘(𝐴 · 𝑦)))
64		id 22	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
65		negex 11494	. . . . . . . . . . 11 ⊢ -(sin‘(𝐴 · 𝑦)) ∈ V
66	65	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → -(sin‘(𝐴 · 𝑦)) ∈ V)
67	59, 63, 64, 66	fvmptd 7015	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ → ((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) = -(sin‘(𝐴 · 𝑦)))
68	67	adantl 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) = -(sin‘(𝐴 · 𝑦)))
69	28	adantr 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) = (𝑥 ∈ ℂ ↦ ((0 · 𝑥) + (1 · 𝐴))))
70		oveq2 7432	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (0 · 𝑥) = (0 · 𝑦))
71	70	oveq1d 7439	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((0 · 𝑥) + (1 · 𝐴)) = ((0 · 𝑦) + (1 · 𝐴)))
72		mul02 11428	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (0 · 𝑦) = 0)
73		mullid 11249	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
74	72, 73	oveqan12rd 7444	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 · 𝑦) + (1 · 𝐴)) = (0 + 𝐴))
75		addlid 11433	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
76	75	adantr 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 + 𝐴) = 𝐴)
77	74, 76	eqtrd 2767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 · 𝑦) + (1 · 𝐴)) = 𝐴)
78	71, 77	sylan9eqr 2789	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ 𝑥 = 𝑦) → ((0 · 𝑥) + (1 · 𝐴)) = 𝐴)
79		simpr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
80		simpl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
81	69, 78, 79, 80	fvmptd 7015	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦) = 𝐴)
82	68, 81	oveq12d 7442	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦)) = (-(sin‘(𝐴 · 𝑦)) · 𝐴))
83		mulcl 11228	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
84	83	sincld 16112	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (sin‘(𝐴 · 𝑦)) ∈ ℂ)
85	84	negcld 11594	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → -(sin‘(𝐴 · 𝑦)) ∈ ℂ)
86	85, 80	mulcomd 11271	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (-(sin‘(𝐴 · 𝑦)) · 𝐴) = (𝐴 · -(sin‘(𝐴 · 𝑦))))
87	82, 86	eqtrd 2767	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦)) = (𝐴 · -(sin‘(𝐴 · 𝑦))))
88	58, 87	syldan 589	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))))) → (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦)) = (𝐴 · -(sin‘(𝐴 · 𝑦))))
89	55, 88	mpteq12dva 5239	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ (dom (𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥))) ∩ dom (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) ↦ (((𝑥 ∈ ℂ ↦ -(sin‘(𝐴 · 𝑥)))‘𝑦) · ((ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))‘𝑦))) = (𝑦 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑦)))))
90	41, 47, 89	3eqtrd 2771	. . 3 ⊢ (𝐴 ∈ ℂ → (((ℂ D cos) ∘ (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))) ∘_f · (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)))) = (𝑦 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑦)))))
91	9, 35, 90	3eqtrd 2771	. 2 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥)))) = (𝑦 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑦)))))
92		oveq2 7432	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐴 · 𝑦) = (𝐴 · 𝑥))
93	92	fveq2d 6904	. . . . 5 ⊢ (𝑦 = 𝑥 → (sin‘(𝐴 · 𝑦)) = (sin‘(𝐴 · 𝑥)))
94	93	negeqd 11490	. . . 4 ⊢ (𝑦 = 𝑥 → -(sin‘(𝐴 · 𝑦)) = -(sin‘(𝐴 · 𝑥)))
95	94	oveq2d 7440	. . 3 ⊢ (𝑦 = 𝑥 → (𝐴 · -(sin‘(𝐴 · 𝑦))) = (𝐴 · -(sin‘(𝐴 · 𝑥))))
96	95	cbvmptv 5263	. 2 ⊢ (𝑦 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑦)))) = (𝑥 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑥))))
97	91, 96	eqtrdi 2783	1 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘(𝐴 · 𝑥)))) = (𝑥 ∈ ℂ ↦ (𝐴 · -(sin‘(𝐴 · 𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 Vcvv 3471 ∩ cin 3946 {cpr 4632 ↦ cmpt 5233 dom cdm 5680 ∘ ccom 5684 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ∘_f cof 7687 ℂcc 11142 ℝcr 11143 0cc0 11144 1c1 11145 + caddc 11147 · cmul 11149 -cneg 11481 sincsin 16045 cosccos 16046 D cdv 25810

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ico 13368 df-icc 13369 df-fz 13523 df-fzo 13666 df-fl 13795 df-seq 14005 df-exp 14065 df-fac 14271 df-bc 14300 df-hash 14328 df-shft 15052 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-limsup 15453 df-clim 15470 df-rlim 15471 df-sum 15671 df-ef 16049 df-sin 16051 df-cos 16052 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-fbas 21281 df-fg 21282 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-limc 25813 df-dv 25814

This theorem is referenced by: itgsincmulx 45364

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