dvtan - Mathbox for Brendan Leahy

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Theorem dvtan 38556

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 16217	. . . 4 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2		cnvimass 6076	. . . . . . . . 9 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ dom cos
3		cosf 16273	. . . . . . . . . 10 ⊢ cos:ℂ⟶ℂ
4	3	fdmi 6713	. . . . . . . . 9 ⊢ dom cos = ℂ
5	2, 4	sseqtri 3979	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
6	5	sseli 3927	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
7	6	sincld 16278	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
8	6	coscld 16279	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9		ffn 6701	. . . . . . . 8 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
10		elpreima 7049	. . . . . . . 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 4753	. . . . . . . 8 ⊢ ((cos‘𝑥) ∈ (ℂ ∖ {0}) → (cos‘𝑥) ≠ 0)
13	12	adantl 487	. . . . . . 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 12077	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
16	15	mpteq2ia 5200	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
17	1, 16	eqtri 2784	. . 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 11274	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
20	19	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
21		difss 4083	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
22		imass2 6096	. . . . . . . . 9 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (^◡cos “ (ℂ ∖ {0})) ⊆ (^◡cos “ ℂ))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ (^◡cos “ ℂ)
24		fimacnv 6724	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → (^◡cos “ ℂ) = ℂ)
25	3, 24	ax-mp 5	. . . . . . . 8 ⊢ (^◡cos “ ℂ) = ℂ
26	23, 25	sseqtri 3979	. . . . . . 7 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
27	26	sseli 3927	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
28	27	sincld 16278	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
29	28	adantl 487	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (sin‘𝑥) ∈ ℂ)
30	8	adantl 487	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ ℂ)
31		sincl 16274	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
32	31	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33		coscl 16275	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
34	33	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35		dvsin 26282	. . . . . 6 ⊢ (ℂ D sin) = cos
36		sinf 16272	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
37	36	a1i 11	. . . . . . . 8 ⊢ (⊤ → sin:ℂ⟶ℂ)
38	37	feqmptd 6945	. . . . . . 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 6945	. . . . . 6 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
42	35, 39, 41	3eqtr3a 2820	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
43	26	a1i 11	. . . . 5 ⊢ (⊤ → (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ)
44		eqid 2761	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 25081	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46	45	toponrestid 23219	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
47		dvtanlem 38555	. . . . . 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 26263	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
50	8, 14	reccld 12067	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
51	50	adantl 487	. . . 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 503	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
54	53	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (cos‘𝑥) ∈ (ℂ ∖ {0}))
55	29	negcld 11637	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56		eldifi 4078	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57		eldifsni 4753	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
58	56, 57	reccld 12067	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
59	58	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60		negex 11536	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
61	60	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
62	32	negcld 11637	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
63	41	oveq2d 7428	. . . . . . 7 ⊢ (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64		dvcos 26283	. . . . . . 7 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
65	63, 64	eqtr3di 2811	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
66	20, 34, 62, 65, 43, 46, 44, 48	dvmptres 26263	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67		ax-1cn 11239	. . . . . 6 ⊢ 1 ∈ ℂ
68		dvrec 26255	. . . . . 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 11532	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
74	20, 20, 54, 55, 59, 61, 66, 69, 70, 73	dvmptco 26272	. . . 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 26261	. . 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 1577	. 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 6675	. . . 4 ⊢ dom tan = (^◡cos “ (ℂ ∖ {0}))
79	78	eqcomi 2770	. . 3 ⊢ (^◡cos “ (ℂ ∖ {0})) = dom tan
80	8	sqcld 14267	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
81	7	sqcld 14267	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82		sqne0 14246	. . . . . . . . 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 12112	. . . . . 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 11493	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87		sincossq 16324	. . . . . . . . 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 2796	. . . . . . 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 2798	. . . . 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 12069	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
93	80, 84	dividd 12072	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
94	92, 93	eqtr4d 2799	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
95	7, 7, 80, 84	div23d 12111	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
96	7	sqvald 14266	. . . . . . . 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 12067	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
99	98, 7	mul2negd 11752	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
100	7, 80, 84	divrec2d 12078	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
101	99, 100	eqtr4d 2799	. . . . . . . 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 2807	. . . . . 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 12604	. . . . . 6 ⊢ 2 ∈ ℕ₀
106		expneg 14192	. . . . . 6 ⊢ (((cos‘𝑥) ∈ ℂ ∧ 2 ∈ ℕ₀) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
107	8, 105, 106	sylancl 598	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑-2) = (1 / ((cos‘𝑥)↑2)))
108	91, 104, 107	3eqtr4d 2806	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109	108	rgen 3079	. . 3 ⊢ ∀𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110		mpteq12 5193	. . 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 705	. 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 2788	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 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 Vcvv 3451 ∖ cdif 3896 ⊆ wss 3899 {csn 4584 {cpr 4586 ↦ cmpt 5186 ^◡ccnv 5650 dom cdm 5651 “ cima 5654 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 -cneg 11523 / cdiv 11954 2c2 12378 ℕ₀cn0 12587 ↑cexp 14184 sincsin 16209 cosccos 16210 tanctan 16211 TopOpenctopn 17572 ℂ_fldccnfld 21658 D cdv 26163

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-tan 16217 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-t1 23612 df-haus 23613 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167

This theorem is used by: (None)

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