Theorem dvtan 38005

Description: Derivative of tangent. (Contributed by Brendan Leahy, 7-Aug-2018.)

Assertion

Ref	Expression
dvtan	⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))

Proof of Theorem dvtan

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-tan 16027	. . . 4 ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2		cnvimass 6041	. . . . . . . . 9 ⊢ (◡cos “ (ℂ ∖ {0})) ⊆ dom cos
3		cosf 16083	. . . . . . . . . 10 ⊢ cos:ℂ⟶ℂ
4	3	fdmi 6673	. . . . . . . . 9 ⊢ dom cos = ℂ
5	2, 4	sseqtri 3971	. . . . . . . 8 ⊢ (◡cos “ (ℂ ∖ {0})) ⊆ ℂ
6	5	sseli 3918	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
7	6	sincld 16088	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
8	6	coscld 16089	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9		ffn 6662	. . . . . . . 8 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
10		elpreima 7004	. . . . . . . 8 ⊢ (cos Fn ℂ → (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0}))))
11	3, 9, 10	mp2b 10	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↔ (𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})))
12		eldifsni 4734	. . . . . . . 8 ⊢ ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
13	12	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
14	11, 13	sylbi 217	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
15	7, 8, 14	divrecd 11925	. . . . 5 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
16	15	mpteq2ia 5181	. . . 4 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
17	1, 16	eqtri 2760	. . 3 ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
18	17	oveq2i 7371	. 2 ⊢ (ℂ D tan) = (ℂ D (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19		cnelprrecn 11122	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
20	19	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
21		difss 4077	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
22		imass2 6061	. . . . . . . . 9 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (◡cos “ (ℂ ∖ {0})) ⊆ (◡cos “ ℂ))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ (◡cos “ (ℂ ∖ {0})) ⊆ (◡cos “ ℂ)
24		fimacnv 6684	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → (◡cos “ ℂ) = ℂ)
25	3, 24	ax-mp 5	. . . . . . . 8 ⊢ (◡cos “ ℂ) = ℂ
26	23, 25	sseqtri 3971	. . . . . . 7 ⊢ (◡cos “ (ℂ ∖ {0})) ⊆ ℂ
27	26	sseli 3918	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
28	27	sincld 16088	. . . . 5 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
29	28	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (◡cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
30	8	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31		sincl 16084	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
32	31	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33		coscl 16085	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
34	33	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35		dvsin 25959	. . . . . 6 ⊢ (ℂ D sin) = cos
36		sinf 16082	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
37	36	a1i 11	. . . . . . . 8 ⊢ (⊤ → sin:ℂ⟶ℂ)
38	37	feqmptd 6902	. . . . . . 7 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
39	38	oveq2d 7376	. . . . . 6 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
40	3	a1i 11	. . . . . . 7 ⊢ (⊤ → cos:ℂ⟶ℂ)
41	40	feqmptd 6902	. . . . . 6 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
42	35, 39, 41	3eqtr3a 2796	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
43	26	a1i 11	. . . . 5 ⊢ (⊤ → (◡cos “ (ℂ ∖ {0})) ⊆ ℂ)
44		eqid 2737	. . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
45	44	cnfldtopon 24757	. . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
46	45	toponrestid 22896	. . . . 5 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
47		dvtanlem 38004	. . . . . 6 ⊢ (◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
48	47	a1i 11	. . . . 5 ⊢ (⊤ → (◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld))
49	20, 32, 34, 42, 43, 46, 44, 48	dvmptres 25940	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
50	8, 14	reccld 11915	. . . . 5 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
51	50	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (◡cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52		ovexd 7395	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (◡cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
53	11	simprbi 497	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
54	53	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
55	29	negcld 11483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (◡cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56		eldifi 4072	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57		eldifsni 4734	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
58	56, 57	reccld 11915	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
59	58	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60		negex 11382	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
61	60	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
62	32	negcld 11483	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
63	41	oveq2d 7376	. . . . . . 7 ⊢ (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64		dvcos 25960	. . . . . . 7 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
65	63, 64	eqtr3di 2787	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
66	20, 34, 62, 65, 43, 46, 44, 48	dvmptres 25940	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67		ax-1cn 11087	. . . . . 6 ⊢ 1 ∈ ℂ
68		dvrec 25932	. . . . . 6 ⊢ (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
69	67, 68	mp1i 13	. . . . 5 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
70		oveq2 7368	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71		oveq1 7367	. . . . . . 7 ⊢ (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
72	71	oveq2d 7376	. . . . . 6 ⊢ (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
73	72	negeqd 11378	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
74	20, 20, 54, 55, 59, 61, 66, 69, 70, 73	dvmptco 25949	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (1 / (cos‘𝑥)))) = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥))))
75	20, 29, 30, 49, 51, 52, 74	dvmptmul 25938	. . 3 ⊢ (⊤ → (ℂ D (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))) = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))))
76	75	mptru 1549	. 2 ⊢ (ℂ D (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))) = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))))
77		ovex 7393	. . . . 5 ⊢ ((sin‘𝑥) / (cos‘𝑥)) ∈ V
78	77, 1	dmmpti 6636	. . . 4 ⊢ dom tan = (◡cos “ (ℂ ∖ {0}))
79	78	eqcomi 2746	. . 3 ⊢ (◡cos “ (ℂ ∖ {0})) = dom tan
80	8	sqcld 14097	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
81	7	sqcld 14097	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82		sqne0 14076	. . . . . . . . 9 ⊢ ((cos‘𝑥) ∈ ℂ → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
83	8, 82	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
84	14, 83	mpbird 257	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
85	80, 81, 80, 84	divdird 11960	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))))
86	80, 81	addcomd 11339	. . . . . . . 8 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87		sincossq 16134	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
88	6, 87	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
89	86, 88	eqtrd 2772	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
90	89	oveq1d 7375	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
91	85, 90	eqtr3d 2774	. . . . 5 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))) = (1 / ((cos‘𝑥)↑2)))
92	8, 14	recidd 11917	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
93	80, 84	dividd 11920	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
94	92, 93	eqtr4d 2775	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
95	7, 7, 80, 84	div23d 11959	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
96	7	sqvald 14096	. . . . . . . 8 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
97	96	oveq1d 7375	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
98	80, 84	reccld 11915	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
99	98, 7	mul2negd 11596	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
100	7, 80, 84	divrec2d 11926	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
101	99, 100	eqtr4d 2775	. . . . . . . 8 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102	101	oveq1d 7375	. . . . . . 7 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
103	95, 97, 102	3eqtr4rd 2783	. . . . . 6 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
104	94, 103	oveq12d 7378	. . . . 5 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) + (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2))))
105		2nn0 12445	. . . . . 6 ⊢ 2 ∈ ℕ0
106		expneg 14022	. . . . . 6 ⊢ (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ0) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
107	8, 105, 106	sylancl 587	. . . . 5 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
108	91, 104, 107	3eqtr4d 2782	. . . 4 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109	108	rgen 3054	. . 3 ⊢ ∀𝑥 ∈ (◡cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110		mpteq12 5174	. . 3 ⊢ (((◡cos “ (ℂ ∖ {0})) = dom tan ∧ ∀𝑥 ∈ (◡cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)) → (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2)))
111	79, 109, 110	mp2an 693	. 2 ⊢ (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦ (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)))) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
112	18, 76, 111	3eqtri 2764	1 ⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  {csn 4568  {cpr 4570   ↦ cmpt 5167  ^◡ccnv 5623  dom cdm 5624   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034  -cneg 11369   / cdiv 11798  2c2 12227  ℕ₀cn0 12428  ↑cexp 14014  sincsin 16019  cosccos 16020  tanctan 16021  TopOpenctopn 17375  ℂ_fldccnfld 21344   D cdv 25840

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-addf 11108

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-fi 9317  df-sup 9348  df-inf 9349  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-xmul 13056  df-ico 13295  df-icc 13296  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-fac 14227  df-bc 14256  df-hash 14284  df-shft 15020  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15424  df-clim 15441  df-rlim 15442  df-sum 15640  df-ef 16023  df-sin 16025  df-cos 16026  df-tan 16027  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-rest 17376  df-topn 17377  df-0g 17395  df-gsum 17396  df-topgen 17397  df-pt 17398  df-prds 17401  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-mulg 19035  df-cntz 19283  df-cmn 19748  df-psmet 21336  df-xmet 21337  df-met 21338  df-bl 21339  df-mopn 21340  df-fbas 21341  df-fg 21342  df-cnfld 21345  df-top 22869  df-topon 22886  df-topsp 22908  df-bases 22921  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-lp 23111  df-perf 23112  df-cn 23202  df-cnp 23203  df-t1 23289  df-haus 23290  df-tx 23537  df-hmeo 23730  df-fil 23821  df-fm 23913  df-flim 23914  df-flf 23915  df-xms 24295  df-ms 24296  df-tms 24297  df-cncf 24855  df-limc 25843  df-dv 25844

This theorem is referenced by: (None)