dvtan - Mathbox for Brendan Leahy

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Theorem dvtan 38362

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 16150	. . . 4 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
2		cnvimass 6089	. . . . . . . . 9 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ dom cos
3		cosf 16206	. . . . . . . . . 10 ⊢ cos:ℂ⟶ℂ
4	3	fdmi 6724	. . . . . . . . 9 ⊢ dom cos = ℂ
5	2, 4	sseqtri 3988	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
6	5	sseli 3936	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
7	6	sincld 16211	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (sin‘𝑥) ∈ ℂ)
8	6	coscld 16212	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (cos‘𝑥) ∈ ℂ)
9		ffn 6712	. . . . . . . 8 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ)
10		elpreima 7060	. . . . . . . 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 4763	. . . . . . . 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 12012	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / (cos‘𝑥)) = ((sin‘𝑥) · (1 / (cos‘𝑥))))
16	15	mpteq2ia 5211	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
17	1, 16	eqtri 2789	. . 3 ⊢ tan = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥))))
18	17	oveq2i 7434	. 2 ⊢ (ℂ D tan) = (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) · (1 / (cos‘𝑥)))))
19		cnelprrecn 11211	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
20	19	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
21		difss 4093	. . . . . . . . 9 ⊢ (ℂ ∖ {0}) ⊆ ℂ
22		imass2 6109	. . . . . . . . 9 ⊢ ((ℂ ∖ {0}) ⊆ ℂ → (^◡cos “ (ℂ ∖ {0})) ⊆ (^◡cos “ ℂ))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ (^◡cos “ ℂ)
24		fimacnv 6735	. . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → (^◡cos “ ℂ) = ℂ)
25	3, 24	ax-mp 5	. . . . . . . 8 ⊢ (^◡cos “ ℂ) = ℂ
26	23, 25	sseqtri 3988	. . . . . . 7 ⊢ (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ
27	26	sseli 3936	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → 𝑥 ∈ ℂ)
28	27	sincld 16211	. . . . 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 16207	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
32	31	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
33		coscl 16208	. . . . . 6 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
34	33	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (cos‘𝑥) ∈ ℂ)
35		dvsin 26178	. . . . . 6 ⊢ (ℂ D sin) = cos
36		sinf 16205	. . . . . . . . 9 ⊢ sin:ℂ⟶ℂ
37	36	a1i 11	. . . . . . . 8 ⊢ (⊤ → sin:ℂ⟶ℂ)
38	37	feqmptd 6956	. . . . . . 7 ⊢ (⊤ → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
39	38	oveq2d 7439	. . . . . 6 ⊢ (⊤ → (ℂ D sin) = (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))))
40	3	a1i 11	. . . . . . 7 ⊢ (⊤ → cos:ℂ⟶ℂ)
41	40	feqmptd 6956	. . . . . 6 ⊢ (⊤ → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
42	35, 39, 41	3eqtr3a 2825	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
43	26	a1i 11	. . . . 5 ⊢ (⊤ → (^◡cos “ (ℂ ∖ {0})) ⊆ ℂ)
44		eqid 2766	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24976	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
46	45	toponrestid 23115	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
47		dvtanlem 38361	. . . . . 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 26159	. . . 4 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (sin‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥)))
50	8, 14	reccld 12002	. . . . 5 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / (cos‘𝑥)) ∈ ℂ)
51	50	adantl 487	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → (1 / (cos‘𝑥)) ∈ ℂ)
52		ovexd 7458	. . . 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 11574	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))) → -(sin‘𝑥) ∈ ℂ)
56		eldifi 4088	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ∈ ℂ)
57		eldifsni 4763	. . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → 𝑦 ≠ 0)
58	56, 57	reccld 12002	. . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ {0}) → (1 / 𝑦) ∈ ℂ)
59	58	adantl 487	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → (1 / 𝑦) ∈ ℂ)
60		negex 11473	. . . . . 6 ⊢ -(1 / (𝑦↑2)) ∈ V
61	60	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑦 ∈ (ℂ ∖ {0})) → -(1 / (𝑦↑2)) ∈ V)
62	32	negcld 11574	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → -(sin‘𝑥) ∈ ℂ)
63	41	oveq2d 7439	. . . . . . 7 ⊢ (⊤ → (ℂ D cos) = (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))))
64		dvcos 26179	. . . . . . 7 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥))
65	63, 64	eqtr3di 2816	. . . . . 6 ⊢ (⊤ → (ℂ D (𝑥 ∈ ℂ ↦ (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)))
66	20, 34, 62, 65, 43, 46, 44, 48	dvmptres 26159	. . . . 5 ⊢ (⊤ → (ℂ D (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ (cos‘𝑥))) = (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) ↦ -(sin‘𝑥)))
67		ax-1cn 11176	. . . . . 6 ⊢ 1 ∈ ℂ
68		dvrec 26151	. . . . . 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 7431	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → (1 / 𝑦) = (1 / (cos‘𝑥)))
71		oveq1 7430	. . . . . . 7 ⊢ (𝑦 = (cos‘𝑥) → (𝑦↑2) = ((cos‘𝑥)↑2))
72	71	oveq2d 7439	. . . . . 6 ⊢ (𝑦 = (cos‘𝑥) → (1 / (𝑦↑2)) = (1 / ((cos‘𝑥)↑2)))
73	72	negeqd 11469	. . . . 5 ⊢ (𝑦 = (cos‘𝑥) → -(1 / (𝑦↑2)) = -(1 / ((cos‘𝑥)↑2)))
74	20, 20, 54, 55, 59, 61, 66, 69, 70, 73	dvmptco 26168	. . . 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 26157	. . 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 7456	. . . . 5 ⊢ ((sin‘𝑥) / (cos‘𝑥)) ∈ V
78	77, 1	dmmpti 6686	. . . 4 ⊢ dom tan = (^◡cos “ (ℂ ∖ {0}))
79	78	eqcomi 2775	. . 3 ⊢ (^◡cos “ (ℂ ∖ {0})) = dom tan
80	8	sqcld 14200	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥)↑2) ∈ ℂ)
81	7	sqcld 14200	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) ∈ ℂ)
82		sqne0 14179	. . . . . . . . 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 12047	. . . . . 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 11430	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)))
87		sincossq 16257	. . . . . . . . 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 2801	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) = 1)
90	89	oveq1d 7438	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((((cos‘𝑥)↑2) + ((sin‘𝑥)↑2)) / ((cos‘𝑥)↑2)) = (1 / ((cos‘𝑥)↑2)))
91	85, 90	eqtr3d 2803	. . . . 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 12004	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = 1)
93	80, 84	dividd 12007	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)) = 1)
94	92, 93	eqtr4d 2804	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((cos‘𝑥) · (1 / (cos‘𝑥))) = (((cos‘𝑥)↑2) / ((cos‘𝑥)↑2)))
95	7, 7, 80, 84	div23d 12046	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
96	7	sqvald 14199	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥)↑2) = ((sin‘𝑥) · (sin‘𝑥)))
97	96	oveq1d 7438	. . . . . . 7 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)) = (((sin‘𝑥) · (sin‘𝑥)) / ((cos‘𝑥)↑2)))
98	80, 84	reccld 12002	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (1 / ((cos‘𝑥)↑2)) ∈ ℂ)
99	98, 7	mul2negd 11687	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
100	7, 80, 84	divrec2d 12013	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((sin‘𝑥) / ((cos‘𝑥)↑2)) = ((1 / ((cos‘𝑥)↑2)) · (sin‘𝑥)))
101	99, 100	eqtr4d 2804	. . . . . . . 8 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) = ((sin‘𝑥) / ((cos‘𝑥)↑2)))
102	101	oveq1d 7438	. . . . . . 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 2812	. . . . . 6 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥)) = (((sin‘𝑥)↑2) / ((cos‘𝑥)↑2)))
104	94, 103	oveq12d 7441	. . . . 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 12539	. . . . . 6 ⊢ 2 ∈ ℕ₀
106		expneg 14125	. . . . . 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 2811	. . . 4 ⊢ (𝑥 ∈ (^◡cos “ (ℂ ∖ {0})) → (((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2))
109	108	rgen 3084	. . 3 ⊢ ∀𝑥 ∈ (^◡cos “ (ℂ ∖ {0}))(((cos‘𝑥) · (1 / (cos‘𝑥))) + ((-(1 / ((cos‘𝑥)↑2)) · -(sin‘𝑥)) · (sin‘𝑥))) = ((cos‘𝑥)↑-2)
110		mpteq12 5204	. . 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 2793	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 2146 ≠ wne 2961 ∀wral 3082 Vcvv 3458 ∖ cdif 3905 ⊆ wss 3908 {csn 4594 {cpr 4596 ↦ cmpt 5197 ^◡ccnv 5665 dom cdm 5666 “ cima 5669 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 -cneg 11460 / cdiv 11889 2c2 12313 ℕ₀cn0 12522 ↑cexp 14117 sincsin 16142 cosccos 16143 tanctan 16144 TopOpenctopn 17499 ℂ_fldccnfld 21559 D cdv 26059

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-cos 16149 df-tan 16150 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-lp 23330 df-perf 23331 df-cn 23421 df-cnp 23422 df-t1 23508 df-haus 23509 df-tx 23756 df-hmeo 23949 df-fil 24040 df-fm 24132 df-flim 24133 df-flf 24134 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074 df-limc 26062 df-dv 26063

This theorem is used by: (None)

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