dvtan - Mathbox for Brendan Leahy

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Theorem dvtan 38349

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 16131	. . . 4 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2		cnvimass 6083	. . . . . . . . 9 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ dom cos
3		cosf 16187	. . . . . . . . . 10 ⊢ cos:ℂ⟶ℂ
4	3	fdmi 6717	. . . . . . . . 9 ⊢ dom cos = ℂ
5	2, 4	sseqtri 3984	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
6	5	sseli 3932	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
7	6	sincld 16192	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
8	6	coscld 16193	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9		ffn 6705	. . . . . . . 8 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
10		elpreima 7053	. . . . . . . 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 4757	. . . . . . . 8 ⊢ ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
13	12	adantl 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ (cos‘𝑥) ∈ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
14	11, 13	sylbi 220	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ≠ 0)
15	7, 8, 14	divrecd 12000	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
16	15	mpteq2ia 5205	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
17	1, 16	eqtri 2785	. . 3 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
18	17	oveq2i 7423	. 2 ⊢ (ℂ D tan) = (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19		cnelprrecn 11199	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
20	19	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
21		difss 4089	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
22		imass2 6103	. . . . . . . . 9 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (^◡cos “ (ℂ ∖ {0})) ⊆ (^◡cos “ ℂ))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ (^◡cos “ ℂ)
24		fimacnv 6728	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → (^◡cos “ ℂ) = ℂ)
25	3, 24	ax-mp 5	. . . . . . . 8 ⊢ (^◡cos “ ℂ) = ℂ
26	23, 25	sseqtri 3984	. . . . . . 7 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
27	26	sseli 3932	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
28	27	sincld 16192	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
29	28	adantl 486	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
30	8	adantl 486	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31		sincl 16188	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
32	31	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33		coscl 16189	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
34	33	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35		dvsin 26152	. . . . . 6 ⊢ (ℂ D sin) = cos
36		sinf 16186	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
37	36	a1i 11	. . . . . . . 8 ⊢ (⊤ → sin:ℂ⟶ℂ)
38	37	feqmptd 6949	. . . . . . 7 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
39	38	oveq2d 7428	. . . . . 6 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
40	3	a1i 11	. . . . . . 7 ⊢ (⊤ → cos:ℂ⟶ℂ)
41	40	feqmptd 6949	. . . . . 6 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
42	35, 39, 41	3eqtr3a 2821	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
43	26	a1i 11	. . . . 5 ⊢ (⊤ → (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ)
44		eqid 2762	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24950	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46	45	toponrestid 23089	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
47		dvtanlem 38348	. . . . . 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 26133	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
50	8, 14	reccld 11990	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
51	50	adantl 486	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52		ovexd 7447	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) ∈ V)
53	11	simprbi 502	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
54	53	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
55	29	negcld 11562	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56		eldifi 4084	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57		eldifsni 4757	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
58	56, 57	reccld 11990	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
59	58	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60		negex 11461	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
61	60	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
62	32	negcld 11562	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
63	41	oveq2d 7428	. . . . . . 7 ⊢ (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64		dvcos 26153	. . . . . . 7 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
65	63, 64	eqtr3di 2812	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
66	20, 34, 62, 65, 43, 46, 44, 48	dvmptres 26133	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67		ax-1cn 11164	. . . . . 6 ⊢ 1 ∈ ℂ
68		dvrec 26125	. . . . . 6 ⊢ (1 ∈ ℂ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
69	67, 68	mp1i 14	. . . . 5 ⊢ (⊤ → (ℂ D (𝑦 ∈ (ℂ ∖ {0}) ↦ (1 / 𝑦))) = (𝑦 ∈ (ℂ ∖ {0}) ↦ -(1 / (𝑦↑2))))
70		oveq2 7420	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71		oveq1 7419	. . . . . . 7 ⊢ (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
72	71	oveq2d 7428	. . . . . 6 ⊢ (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
73	72	negeqd 11457	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
74	20, 20, 54, 55, 59, 61, 66, 69, 70, 73	dvmptco 26142	. . . 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 26131	. . 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 1576	. 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 7445	. . . . 5 ⊢ ((sin‘𝑥) / (cos‘𝑥)) ∈ V
78	77, 1	dmmpti 6679	. . . 4 ⊢ dom tan = (^◡cos “ (ℂ ∖ {0}))
79	78	eqcomi 2771	. . 3 ⊢ (^◡cos “ (ℂ ∖ {0})) = dom tan
80	8	sqcld 14187	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
81	7	sqcld 14187	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82		sqne0 14166	. . . . . . . . 9 ⊢ ((cos‘𝑥) ∈ ℂ → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
83	8, 82	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) ≠ 0 ↔ (cos‘𝑥) ≠ 0))
84	14, 83	mpbird 260	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ≠ 0)
85	80, 81, 80, 84	divdird 12035	. . . . . 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 11418	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87		sincossq 16238	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
88	6, 87	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
89	86, 88	eqtrd 2797	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
90	89	oveq1d 7427	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
91	85, 90	eqtr3d 2799	. . . . 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 11992	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
93	80, 84	dividd 11995	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
94	92, 93	eqtr4d 2800	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
95	7, 7, 80, 84	div23d 12034	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
96	7	sqvald 14186	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
97	96	oveq1d 7427	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
98	80, 84	reccld 11990	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
99	98, 7	mul2negd 11675	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
100	7, 80, 84	divrec2d 12001	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
101	99, 100	eqtr4d 2800	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102	101	oveq1d 7427	. . . . . . 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 2808	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
104	94, 103	oveq12d 7430	. . . . 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 12527	. . . . . 6 ⊢ 2 ∈ ℕ₀
106		expneg 14112	. . . . . 6 ⊢ (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
107	8, 105, 106	sylancl 597	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
108	91, 104, 107	3eqtr4d 2807	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109	108	rgen 3080	. . 3 ⊢ ∀𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110		mpteq12 5198	. . 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 704	. 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 2789	1 ⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 400 = wceq 1569 ⊤wtru 1570 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 Vcvv 3454 ∖ cdif 3901 ⊆ wss 3904 {csn 4588 {cpr 4590 ↦ cmpt 5191 ^◡ccnv 5659 dom cdm 5660 “ cima 5663 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 -cneg 11448 / cdiv 11877 2c2 12301 ℕ₀cn0 12510 ↑cexp 14104 sincsin 16123 cosccos 16124 tanctan 16125 TopOpenctopn 17480 ℂ_fldccnfld 21533 D cdv 26033

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-tan 16131 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-t1 23482 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037

This theorem is used by: (None)

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