2efiatan - Metamath Proof Explorer

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Theorem 2efiatan 26900

Description: Value of the exponential of an arctangent. (Contributed by Mario Carneiro, 2-Apr-2015.)

**Assertion**
Ref	Expression
2efiatan	⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (((2 · i) / (𝐴 + i)) − 1))

**Proof of Theorem 2efiatan**
Step	Hyp	Ref	Expression
1		atanval 26866	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
2	1	oveq2d 7372	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
3		2cn 12247	. . . . . 6 ⊢ 2 ∈ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → 2 ∈ ℂ)
5		ax-icn 11088	. . . . . 6 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ∈ ℂ)
7		atancl 26863	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
8	4, 6, 7	mulassd 11159	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = (2 · (i · (arctan‘𝐴))))
9		halfcl 12394	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
10	5, 9	ax-mp 5	. . . . . . . . 9 ⊢ (i / 2) ∈ ℂ
11	3, 5, 10	mulassi 11147	. . . . . . . 8 ⊢ ((2 · i) · (i / 2)) = (2 · (i · (i / 2)))
12	3, 5, 10	mul12i 11332	. . . . . . . 8 ⊢ (2 · (i · (i / 2))) = (i · (2 · (i / 2)))
13		2ne0 12276	. . . . . . . . . . 11 ⊢ 2 ≠ 0
14	5, 3, 13	divcan2i 11889	. . . . . . . . . 10 ⊢ (2 · (i / 2)) = i
15	14	oveq2i 7367	. . . . . . . . 9 ⊢ (i · (2 · (i / 2))) = (i · i)
16		ixi 11770	. . . . . . . . 9 ⊢ (i · i) = -1
17	15, 16	eqtri 2762	. . . . . . . 8 ⊢ (i · (2 · (i / 2))) = -1
18	11, 12, 17	3eqtri 2766	. . . . . . 7 ⊢ ((2 · i) · (i / 2)) = -1
19	18	oveq1i 7366	. . . . . 6 ⊢ (((2 · i) · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = (-1 · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
20		ax-1cn 11087	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		atandm2 26859	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
22	21	simp1bi 1151	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
23		mulcl 11113	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
24	5, 22, 23	sylancr 593	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
25		subcl 11383	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
26	20, 24, 25	sylancr 593	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
27	21	simp2bi 1152	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
28	26, 27	logcld 26552	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
29		addcl 11111	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
30	20, 24, 29	sylancr 593	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
31	21	simp3bi 1153	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
32	30, 31	logcld 26552	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
33	28, 32	subcld 11496	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
34	33	mulm1d 11593	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (-1 · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
35	19, 34	eqtrid 2786	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (((2 · i) · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = -((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
36		2mulicn 12392	. . . . . . 7 ⊢ (2 · i) ∈ ℂ
37	36	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (2 · i) ∈ ℂ)
38	10	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i / 2) ∈ ℂ)
39	37, 38, 33	mulassd 11159	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (((2 · i) · (i / 2)) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
40	28, 32	negsubdi2d 11512	. . . . 5 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
41	35, 39, 40	3eqtr3d 2782	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
42	2, 8, 41	3eqtr3d 2782	. . 3 ⊢ (𝐴 ∈ dom arctan → (2 · (i · (arctan‘𝐴))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
43	42	fveq2d 6831	. 2 ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
44		efsub 16058	. . 3 ⊢ (((log‘(1 + (i · 𝐴))) ∈ ℂ ∧ (log‘(1 − (i · 𝐴))) ∈ ℂ) → (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))))
45	32, 28, 44	syl2anc 590	. 2 ⊢ (𝐴 ∈ dom arctan → (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))))
46		eflog 26558	. . . . 5 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (1 + (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
47	30, 31, 46	syl2anc 590	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
48		eflog 26558	. . . . 5 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
49	26, 27, 48	syl2anc 590	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
50	47, 49	oveq12d 7374	. . 3 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
51		negsub 11433	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + -𝐴) = (i − 𝐴))
52	5, 22, 51	sylancr 593	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i + -𝐴) = (i − 𝐴))
53	6	mulridd 11153	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · 1) = i)
54	16	oveq1i 7366	. . . . . . . . 9 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴)
55	6, 6, 22	mulassd 11159	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → ((i · i) · 𝐴) = (i · (i · 𝐴)))
56	22	mulm1d 11593	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (-1 · 𝐴) = -𝐴)
57	54, 55, 56	3eqtr3a 2798	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · (i · 𝐴)) = -𝐴)
58	53, 57	oveq12d 7374	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) + (i · (i · 𝐴))) = (i + -𝐴))
59	6, 22, 6	pnpcan2d 11534	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i + i) − (𝐴 + i)) = (i − 𝐴))
60	52, 58, 59	3eqtr4d 2784	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · 1) + (i · (i · 𝐴))) = ((i + i) − (𝐴 + i)))
61	20	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 1 ∈ ℂ)
62	6, 61, 24	adddid 11160	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((i · 1) + (i · (i · 𝐴))))
63	6	2timesd 12411	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (2 · i) = (i + i))
64	63	oveq1d 7371	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((2 · i) − (𝐴 + i)) = ((i + i) − (𝐴 + i)))
65	60, 62, 64	3eqtr4d 2784	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((2 · i) − (𝐴 + i)))
66	6, 61, 24	subdid 11597	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = ((i · 1) − (i · (i · 𝐴))))
67	53, 57	oveq12d 7374	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i − -𝐴))
68		subneg 11434	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i − -𝐴) = (i + 𝐴))
69	5, 22, 68	sylancr 593	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i − -𝐴) = (i + 𝐴))
70	67, 69	eqtrd 2774	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i + 𝐴))
71		addcom 11323	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + 𝐴) = (𝐴 + i))
72	5, 22, 71	sylancr 593	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i + 𝐴) = (𝐴 + i))
73	66, 70, 72	3eqtrd 2778	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = (𝐴 + i))
74	65, 73	oveq12d 7374	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = (((2 · i) − (𝐴 + i)) / (𝐴 + i)))
75		ine0 11576	. . . . . 6 ⊢ i ≠ 0
76	75	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ≠ 0)
77	30, 26, 6, 27, 76	divcan5d 11948	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
78		addcl 11111	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 + i) ∈ ℂ)
79	22, 5, 78	sylancl 592	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ∈ ℂ)
80		subneg 11434	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 − -i) = (𝐴 + i))
81	22, 5, 80	sylancl 592	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) = (𝐴 + i))
82		atandm 26858	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
83	82	simp2bi 1152	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ≠ -i)
84		negicn 11385	. . . . . . . 8 ⊢ -i ∈ ℂ
85		subeq0 11411	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) = 0 ↔ 𝐴 = -i))
86	85	necon3bid 2978	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
87	22, 84, 86	sylancl 592	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
88	83, 87	mpbird 258	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) ≠ 0)
89	81, 88	eqnetrrd 3002	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ≠ 0)
90	37, 79, 79, 89	divsubdird 11961	. . . 4 ⊢ (𝐴 ∈ dom arctan → (((2 · i) − (𝐴 + i)) / (𝐴 + i)) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
91	74, 77, 90	3eqtr3d 2782	. . 3 ⊢ (𝐴 ∈ dom arctan → ((1 + (i · 𝐴)) / (1 − (i · 𝐴))) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
92	79, 89	dividd 11920	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((𝐴 + i) / (𝐴 + i)) = 1)
93	92	oveq2d 7372	. . 3 ⊢ (𝐴 ∈ dom arctan → (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))) = (((2 · i) / (𝐴 + i)) − 1))
94	50, 91, 93	3eqtrd 2778	. 2 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = (((2 · i) / (𝐴 + i)) − 1))
95	43, 45, 94	3eqtrd 2778	1 ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (((2 · i) / (𝐴 + i)) − 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 dom cdm 5618 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 0cc0 11029 1c1 11030 ici 11031 + caddc 11032 · cmul 11034 − cmin 11368 -cneg 11369 / cdiv 11798 2c2 12227 expce 16017 logclog 26536 arctancatan 26846

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 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 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 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 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 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-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 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-pi 16028 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 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-log 26538 df-atan 26849

This theorem is referenced by: tanatan 26901

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