dvtan - Mathbox for Brendan Leahy

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Theorem dvtan 38044

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 16034	. . . 4 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2		cnvimass 6041	. . . . . . . . 9 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ dom cos
3		cosf 16090	. . . . . . . . . 10 ⊢ cos:ℂ⟶ℂ
4	3	fdmi 6673	. . . . . . . . 9 ⊢ dom cos = ℂ
5	2, 4	sseqtri 3970	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
6	5	sseli 3918	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
7	6	sincld 16095	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
8	6	coscld 16096	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9		ffn 6662	. . . . . . . 8 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
10		elpreima 7006	. . . . . . . 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 4730	. . . . . . . 8 ⊢ ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
13	12	adantl 482	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
14	11, 13	sylbi 218	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
15	7, 8, 14	divrecd 11932	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
16	15	mpteq2ia 5174	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
17	1, 16	eqtri 2763	. . 3 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
18	17	oveq2i 7374	. 2 ⊢ (ℂ D tan) = (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19		cnelprrecn 11129	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
20	19	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
21		difss 4073	. . . . . . . . 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 3970	. . . . . . 7 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
27	26	sseli 3918	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
28	27	sincld 16095	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
29	28	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
30	8	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31		sincl 16091	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
32	31	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33		coscl 16092	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
34	33	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35		dvsin 25974	. . . . . 6 ⊢ (ℂ D sin) = cos
36		sinf 16089	. . . . . . . . 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 7379	. . . . . 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 2799	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
43	26	a1i 11	. . . . 5 ⊢ (⊤ → (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ)
44		eqid 2740	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24772	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46	45	toponrestid 22911	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
47		dvtanlem 38043	. . . . . 6 ⊢ (^◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂ_fld)
48	47	a1i 11	. . . . 5 ⊢ (⊤ → (^◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂ_fld))
49	20, 32, 34, 42, 43, 46, 44, 48	dvmptres 25955	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
50	8, 14	reccld 11922	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
51	50	adantl 482	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52		ovexd 7398	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
53	11	simprbi 498	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
54	53	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
55	29	negcld 11490	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56		eldifi 4068	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57		eldifsni 4730	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
58	56, 57	reccld 11922	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
59	58	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60		negex 11389	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
61	60	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
62	32	negcld 11490	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
63	41	oveq2d 7379	. . . . . . 7 ⊢ (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64		dvcos 25975	. . . . . . 7 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
65	63, 64	eqtr3di 2790	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
66	20, 34, 62, 65, 43, 46, 44, 48	dvmptres 25955	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67		ax-1cn 11094	. . . . . 6 ⊢ 1 ∈ ℂ
68		dvrec 25947	. . . . . 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 7371	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71		oveq1 7370	. . . . . . 7 ⊢ (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
72	71	oveq2d 7379	. . . . . 6 ⊢ (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
73	72	negeqd 11385	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
74	20, 20, 54, 55, 59, 61, 66, 69, 70, 73	dvmptco 25964	. . . 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 25953	. . 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 1554	. 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 7396	. . . . 5 ⊢ ((sin‘𝑥) / (cos‘𝑥)) ∈ V
78	77, 1	dmmpti 6636	. . . 4 ⊢ dom tan = (^◡cos “ (ℂ ∖ {0}))
79	78	eqcomi 2749	. . 3 ⊢ (^◡cos “ (ℂ ∖ {0})) = dom tan
80	8	sqcld 14104	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
81	7	sqcld 14104	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82		sqne0 14083	. . . . . . . . 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 258	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
85	80, 81, 80, 84	divdird 11967	. . . . . 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 11346	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87		sincossq 16141	. . . . . . . . 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 2775	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
90	89	oveq1d 7378	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
91	85, 90	eqtr3d 2777	. . . . 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 11924	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
93	80, 84	dividd 11927	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
94	92, 93	eqtr4d 2778	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
95	7, 7, 80, 84	div23d 11966	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
96	7	sqvald 14103	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
97	96	oveq1d 7378	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
98	80, 84	reccld 11922	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
99	98, 7	mul2negd 11603	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
100	7, 80, 84	divrec2d 11933	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
101	99, 100	eqtr4d 2778	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102	101	oveq1d 7378	. . . . . . 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 2786	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
104	94, 103	oveq12d 7381	. . . . 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 12452	. . . . . 6 ⊢ 2 ∈ ℕ₀
106		expneg 14029	. . . . . 6 ⊢ (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
107	8, 105, 106	sylancl 592	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
108	91, 104, 107	3eqtr4d 2785	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109	108	rgen 3056	. . 3 ⊢ ∀𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110		mpteq12 5167	. . 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 698	. 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 2767	1 ⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 Vcvv 3432 ∖ cdif 3887 ⊆ wss 3890 {csn 4562 {cpr 4564 ↦ cmpt 5160 ^◡ccnv 5624 dom cdm 5625 “ cima 5628 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 -cneg 11376 / cdiv 11805 2c2 12234 ℕ₀cn0 12435 ↑cexp 14021 sincsin 16026 cosccos 16027 tanctan 16028 TopOpenctopn 17382 ℂ_fldccnfld 21354 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-tan 16034 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-t1 23304 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859

This theorem is referenced by: (None)

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