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Theorem dvsincos 26307

Description: Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

**Assertion**
Ref	Expression
dvsincos	⊢ ((ℂ D sin) = cos ∧ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))

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

Step	Hyp	Ref	Expression
1		cnelprrecn 11297	. . . . . 6 ⊢ ℂ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . . 5 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
3		ax-icn 11263	. . . . . . . . . 10 ⊢ i ∈ ℂ
4	3	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → i ∈ ℂ)
5		simpr 490	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
6	4, 5	mulcld 11333	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (i · 𝑥) ∈ ℂ)
7		efcl 16254	. . . . . . . 8 ⊢ ((i · 𝑥) ∈ ℂ → (exp‘(i · 𝑥)) ∈ ℂ)
8	6, 7	syl 18	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (exp‘(i · 𝑥)) ∈ ℂ)
9		ine0 11755	. . . . . . . 8 ⊢ i ≠ 0
10	9	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → i ≠ 0)
11	8, 4, 10	divcld 12097	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘(i · 𝑥)) / i) ∈ ℂ)
12		negicn 11562	. . . . . . . . . 10 ⊢ -i ∈ ℂ
13		mulcl 11288	. . . . . . . . . 10 ⊢ (( -i ∈ ℂ ∧ 𝑥 ∈ ℂ) → ( -i · 𝑥) ∈ ℂ)
14	12, 5, 13	sylancr 599	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ( -i · 𝑥) ∈ ℂ)
15		efcl 16254	. . . . . . . . 9 ⊢ (( -i · 𝑥) ∈ ℂ → (exp‘( -i · 𝑥)) ∈ ℂ)
16	14, 15	syl 18	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (exp‘( -i · 𝑥)) ∈ ℂ)
17	16, 4, 10	divcld 12097	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘( -i · 𝑥)) / i) ∈ ℂ)
18	17	negcld 11660	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -((exp‘( -i · 𝑥)) / i) ∈ ℂ)
19	11, 18	addcld 11332	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) ∈ ℂ)
20	8, 16	addcld 11332	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) ∈ ℂ)
21	8, 4	mulcld 11333	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘(i · 𝑥)) · i) ∈ ℂ)
22		efcl 16254	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → (exp‘𝑦) ∈ ℂ)
23	22	adantl 487	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑦 ∈ ℂ) → (exp‘𝑦) ∈ ℂ)
24		1cnd 11306	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ)
25	2	dvmptid 26283	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1))
26	3	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → i ∈ ℂ)
27	2, 5, 24, 25, 26	dvmptcmul 26290	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (i · 𝑥))) = (𝑥 ∈ ℂ ↦ (i · 1)))
28	3	mulridi 11317	. . . . . . . . . . 11 ⊢ (i · 1) = i
29	28	mpteq2i 5200	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (i · 1)) = (𝑥 ∈ ℂ ↦ i)
30	27, 29	eqtrdi 2811	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (i · 𝑥))) = (𝑥 ∈ ℂ ↦ i))
31		eff 16253	. . . . . . . . . . . . 13 ⊢ exp:ℂ⟶ℂ
32	31	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → exp:ℂ⟶ℂ)
33	32	feqmptd 6954	. . . . . . . . . . 11 ⊢ (⊤ → exp = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
34	33	oveq2d 7437	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D exp) = (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))))
35		dvef 26306	. . . . . . . . . . 11 ⊢ (ℂ D exp) = exp
36	35, 33	eqtrid 2807	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D exp) = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
37	34, 36	eqtr3d 2797	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ (exp‘𝑦))) = (𝑦 ∈ ℂ ↦ (exp‘𝑦)))
38		fveq2 6886	. . . . . . . . 9 ⊢ (𝑦 = (i · 𝑥) → (exp‘𝑦) = (exp‘(i · 𝑥)))
39	2, 2, 6, 4, 23, 23, 30, 37, 38, 38	dvmptco 26298	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) · i)))
40	9	a1i 11	. . . . . . . 8 ⊢ (⊤ → i ≠ 0)
41	2, 8, 21, 39, 26, 40	dvmptdivc 26291	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) / i))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) · i) / i)))
42	8, 4, 10	divcan4d 12103	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) · i) / i) = (exp‘(i · 𝑥)))
43	42	mpteq2dva 5197	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) · i) / i)) = (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))))
44	41, 43	eqtrd 2795	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) / i))) = (𝑥 ∈ ℂ ↦ (exp‘(i · 𝑥))))
45		mulcl 11288	. . . . . . . . . 10 ⊢ (((exp‘( -i · 𝑥)) ∈ ℂ ∧ -i ∈ ℂ) → ((exp‘( -i · 𝑥)) · -i) ∈ ℂ)
46	16, 12, 45	sylancl 598	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘( -i · 𝑥)) · -i) ∈ ℂ)
47	46, 4, 10	divcld 12097	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘( -i · 𝑥)) · -i) / i) ∈ ℂ)
48	12	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -i ∈ ℂ)
49	12	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → -i ∈ ℂ)
50	2, 5, 24, 25, 49	dvmptcmul 26290	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ( -i · 𝑥))) = (𝑥 ∈ ℂ ↦ ( -i · 1)))
51	12	mulridi 11317	. . . . . . . . . . . 12 ⊢ ( -i · 1) = -i
52	51	mpteq2i 5200	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ ( -i · 1)) = (𝑥 ∈ ℂ ↦ -i)
53	50, 52	eqtrdi 2811	. . . . . . . . . 10 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ( -i · 𝑥))) = (𝑥 ∈ ℂ ↦ -i))
54		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑦 = ( -i · 𝑥) → (exp‘𝑦) = (exp‘( -i · 𝑥)))
55	2, 2, 14, 48, 23, 23, 53, 37, 54, 54	dvmptco 26298	. . . . . . . . 9 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (exp‘( -i · 𝑥)))) = (𝑥 ∈ ℂ ↦ ((exp‘( -i · 𝑥)) · -i)))
56	2, 16, 46, 55, 26, 40	dvmptdivc 26291	. . . . . . . 8 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((exp‘( -i · 𝑥)) / i))) = (𝑥 ∈ ℂ ↦ (((exp‘( -i · 𝑥)) · -i) / i)))
57	2, 17, 47, 56	dvmptneg 26292	. . . . . . 7 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ -((exp‘( -i · 𝑥)) / i))) = (𝑥 ∈ ℂ ↦ -(((exp‘( -i · 𝑥)) · -i) / i)))
58	46, 4, 10	divneg2d 12111	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(((exp‘( -i · 𝑥)) · -i) / i) = (((exp‘( -i · 𝑥)) · -i) / -i))
59	3, 9	negne0i 11637	. . . . . . . . . . 11 ⊢ -i ≠ 0
60	59	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -i ≠ 0)
61	16, 48, 60	divcan4d 12103	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘( -i · 𝑥)) · -i) / -i) = (exp‘( -i · 𝑥)))
62	58, 61	eqtrd 2795	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(((exp‘( -i · 𝑥)) · -i) / i) = (exp‘( -i · 𝑥)))
63	62	mpteq2dva 5197	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ -(((exp‘( -i · 𝑥)) · -i) / i)) = (𝑥 ∈ ℂ ↦ (exp‘( -i · 𝑥))))
64	57, 63	eqtrd 2795	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ -((exp‘( -i · 𝑥)) / i))) = (𝑥 ∈ ℂ ↦ (exp‘( -i · 𝑥))))
65	2, 11, 8, 44, 18, 16, 64	dvmptadd 26286	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)))) = (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥)))))
66		2cnd 12425	. . . . 5 ⊢ (⊤ → 2 ∈ ℂ)
67		2ne0 12453	. . . . . 6 ⊢ 2 ≠ 0
68	67	a1i 11	. . . . 5 ⊢ (⊤ → 2 ≠ 0)
69	2, 19, 20, 65, 66, 68	dvmptdivc 26291	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) / 2))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2)))
70		df-sin 16241	. . . . . 6 ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i)))
71	8, 16	subcld 11673	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) ∈ ℂ)
72		2cnd 12425	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 2 ∈ ℂ)
73	67	a1i 11	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → 2 ≠ 0)
74	71, 4, 72, 10, 73	divdiv1d 12128	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (i · 2)))
75		2cn 12422	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
76	3, 75	mulcomi 11321	. . . . . . . . . 10 ⊢ (i · 2) = (2 · i)
77	76	oveq2i 7432	. . . . . . . . 9 ⊢ (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (i · 2)) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i))
78	74, 77	eqtrdi 2811	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i)))
79	8, 16, 4, 10	divsubdird 12136	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) = (((exp‘(i · 𝑥)) / i) − ((exp‘( -i · 𝑥)) / i)))
80	11, 17	negsubd 11679	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) = (((exp‘(i · 𝑥)) / i) − ((exp‘( -i · 𝑥)) / i)))
81	79, 80	eqtr4d 2798	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) = (((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)))
82	81	oveq1d 7436	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2) = ((((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) / 2))
83	78, 82	eqtr3d 2797	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i)) = ((((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) / 2))
84	83	mpteq2dva 5197	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i))) = (𝑥 ∈ ℂ ↦ ((((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) / 2)))
85	70, 84	eqtrid 2807	. . . . 5 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ ((((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) / 2)))
86	85	oveq2d 7437	. . . 4 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ ((((exp‘(i · 𝑥)) / i) + -((exp‘( -i · 𝑥)) / i)) / 2))))
87		df-cos 16242	. . . . 5 ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))
88	87	a1i 11	. . . 4 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2)))
89	69, 86, 88	3eqtr4d 2805	. . 3 ⊢ (⊤ → (ℂ D sin) = cos)
90	21, 46	addcld 11332	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) ∈ ℂ)
91	2, 8, 21, 39, 16, 46, 55	dvmptadd 26286	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ ((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))))) = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i))))
92	2, 20, 90, 91, 66, 68	dvmptdivc 26291	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))) = (𝑥 ∈ ℂ ↦ ((((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) / 2)))
93	88	oveq2d 7437	. . . 4 ⊢ (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘( -i · 𝑥))) / 2))))
94	71, 4, 10	divcld 12097	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) ∈ ℂ)
95	94, 72, 73	divnegd 12110	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2) = ( -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2))
96		sinval 16296	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i)))
97	96	adantl 487	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / (2 · i)))
98	97, 78	eqtr4d 2798	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) = ((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2))
99	98	negeqd 11555	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) = -((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2))
100	3	negnegi 11632	. . . . . . . . . 10 ⊢ - -i = i
101	100	oveq2i 7432	. . . . . . . . 9 ⊢ (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · - -i) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · i)
102		mulneg2 11757	. . . . . . . . . 10 ⊢ ((((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) ∈ ℂ ∧ -i ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · - -i) = -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · -i))
103	71, 12, 102	sylancl 598	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · - -i) = -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · -i))
104	101, 103	eqtr3id 2809	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · i) = -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · -i))
105		mulcl 11288	. . . . . . . . . . 11 ⊢ (((exp‘( -i · 𝑥)) ∈ ℂ ∧ i ∈ ℂ) → ((exp‘( -i · 𝑥)) · i) ∈ ℂ)
106	16, 3, 105	sylancl 598	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘( -i · 𝑥)) · i) ∈ ℂ)
107	21, 106	negsubd 11679	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) · i) + -((exp‘( -i · 𝑥)) · i)) = (((exp‘(i · 𝑥)) · i) − ((exp‘( -i · 𝑥)) · i)))
108		mulneg2 11757	. . . . . . . . . . 11 ⊢ (((exp‘( -i · 𝑥)) ∈ ℂ ∧ i ∈ ℂ) → ((exp‘( -i · 𝑥)) · -i) = -((exp‘( -i · 𝑥)) · i))
109	16, 3, 108	sylancl 598	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((exp‘( -i · 𝑥)) · -i) = -((exp‘( -i · 𝑥)) · i))
110	109	oveq2d 7437	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) = (((exp‘(i · 𝑥)) · i) + -((exp‘( -i · 𝑥)) · i)))
111	8, 16, 4	subdird 11777	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · i) = (((exp‘(i · 𝑥)) · i) − ((exp‘( -i · 𝑥)) · i)))
112	107, 110, 111	3eqtr4d 2805	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · i))
113	71, 4, 10	divrecd 12100	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · (1 / i)))
114		irec 14349	. . . . . . . . . . 11 ⊢ (1 / i) = -i
115	114	oveq2i 7432	. . . . . . . . . 10 ⊢ (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · (1 / i)) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · -i)
116	113, 115	eqtrdi 2811	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) = (((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · -i))
117	116	negeqd 11555	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) = -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) · -i))
118	104, 112, 117	3eqtr4d 2805	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) = -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i))
119	118	oveq1d 7436	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → ((((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) / 2) = ( -(((exp‘(i · 𝑥)) − (exp‘( -i · 𝑥))) / i) / 2))
120	95, 99, 119	3eqtr4d 2805	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) = ((((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) / 2))
121	120	mpteq2dva 5197	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) = (𝑥 ∈ ℂ ↦ ((((exp‘(i · 𝑥)) · i) + ((exp‘( -i · 𝑥)) · -i)) / 2)))
122	92, 93, 121	3eqtr4d 2805	. . 3 ⊢ (⊤ → (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
123	89, 122	jca 521	. 2 ⊢ (⊤ → ((ℂ D sin) = cos ∧ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))))
124	123	mptru 1576	1 ⊢ ((ℂ D sin) = cos ∧ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1569 ⊤wtru 1570 ∈ wcel 2144 ≠ wne 2955 {cpr 4586 ↦ cmpt 5185 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 ici 11206 + caddc 11207 · cmul 11209 − cmin 11545 -cneg 11546 / cdiv 11977 2c2 12401 expce 16233 sincsin 16235 cosccos 16236 D cdv 26189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193

This theorem is used by: dvsin 26308 dvcos 26309

W3C validator