2efiatan - Metamath Proof Explorer

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Theorem 2efiatan 26844

Description: Value of the exponential of an artcangent. (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 26810	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
2	1	oveq2d 7369	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
3		2cn 12221	. . . . . 6 ⊢ 2 ∈ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → 2 ∈ ℂ)
5		ax-icn 11087	. . . . . 6 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ∈ ℂ)
7		atancl 26807	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
8	4, 6, 7	mulassd 11157	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = (2 · (i · (arctan‘𝐴))))
9		halfcl 12368	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
10	5, 9	ax-mp 5	. . . . . . . . 9 ⊢ (i / 2) ∈ ℂ
11	3, 5, 10	mulassi 11145	. . . . . . . 8 ⊢ ((2 · i) · (i / 2)) = (2 · (i · (i / 2)))
12	3, 5, 10	mul12i 11329	. . . . . . . 8 ⊢ (2 · (i · (i / 2))) = (i · (2 · (i / 2)))
13		2ne0 12250	. . . . . . . . . . 11 ⊢ 2 ≠ 0
14	5, 3, 13	divcan2i 11885	. . . . . . . . . 10 ⊢ (2 · (i / 2)) = i
15	14	oveq2i 7364	. . . . . . . . 9 ⊢ (i · (2 · (i / 2))) = (i · i)
16		ixi 11767	. . . . . . . . 9 ⊢ (i · i) = -1
17	15, 16	eqtri 2752	. . . . . . . 8 ⊢ (i · (2 · (i / 2))) = -1
18	11, 12, 17	3eqtri 2756	. . . . . . 7 ⊢ ((2 · i) · (i / 2)) = -1
19	18	oveq1i 7363	. . . . . 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 11086	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		atandm2 26803	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
22	21	simp1bi 1145	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
23		mulcl 11112	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
24	5, 22, 23	sylancr 587	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
25		subcl 11380	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
26	20, 24, 25	sylancr 587	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
27	21	simp2bi 1146	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
28	26, 27	logcld 26495	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
29		addcl 11110	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
30	20, 24, 29	sylancr 587	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
31	21	simp3bi 1147	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
32	30, 31	logcld 26495	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
33	28, 32	subcld 11493	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
34	33	mulm1d 11590	. . . . . 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 2776	. . . . 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 12366	. . . . . . 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 11157	. . . . 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 11509	. . . . 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 2772	. . . 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 2772	. . 3 ⊢ (𝐴 ∈ dom arctan → (2 · (i · (arctan‘𝐴))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
43	42	fveq2d 6830	. 2 ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
44		efsub 16027	. . 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 584	. 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 26501	. . . . 5 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (1 + (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
47	30, 31, 46	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
48		eflog 26501	. . . . 5 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
49	26, 27, 48	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
50	47, 49	oveq12d 7371	. . 3 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
51		negsub 11430	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + -𝐴) = (i − 𝐴))
52	5, 22, 51	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i + -𝐴) = (i − 𝐴))
53	6	mulridd 11151	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · 1) = i)
54	16	oveq1i 7363	. . . . . . . . 9 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴)
55	6, 6, 22	mulassd 11157	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → ((i · i) · 𝐴) = (i · (i · 𝐴)))
56	22	mulm1d 11590	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (-1 · 𝐴) = -𝐴)
57	54, 55, 56	3eqtr3a 2788	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · (i · 𝐴)) = -𝐴)
58	53, 57	oveq12d 7371	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) + (i · (i · 𝐴))) = (i + -𝐴))
59	6, 22, 6	pnpcan2d 11531	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i + i) − (𝐴 + i)) = (i − 𝐴))
60	52, 58, 59	3eqtr4d 2774	. . . . . 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 11158	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((i · 1) + (i · (i · 𝐴))))
63	6	2timesd 12385	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (2 · i) = (i + i))
64	63	oveq1d 7368	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((2 · i) − (𝐴 + i)) = ((i + i) − (𝐴 + i)))
65	60, 62, 64	3eqtr4d 2774	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((2 · i) − (𝐴 + i)))
66	6, 61, 24	subdid 11594	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = ((i · 1) − (i · (i · 𝐴))))
67	53, 57	oveq12d 7371	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i − -𝐴))
68		subneg 11431	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i − -𝐴) = (i + 𝐴))
69	5, 22, 68	sylancr 587	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i − -𝐴) = (i + 𝐴))
70	67, 69	eqtrd 2764	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i + 𝐴))
71		addcom 11320	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + 𝐴) = (𝐴 + i))
72	5, 22, 71	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i + 𝐴) = (𝐴 + i))
73	66, 70, 72	3eqtrd 2768	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = (𝐴 + i))
74	65, 73	oveq12d 7371	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = (((2 · i) − (𝐴 + i)) / (𝐴 + i)))
75		ine0 11573	. . . . . 6 ⊢ i ≠ 0
76	75	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ≠ 0)
77	30, 26, 6, 27, 76	divcan5d 11944	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
78		addcl 11110	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 + i) ∈ ℂ)
79	22, 5, 78	sylancl 586	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ∈ ℂ)
80		subneg 11431	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 − -i) = (𝐴 + i))
81	22, 5, 80	sylancl 586	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) = (𝐴 + i))
82		atandm 26802	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
83	82	simp2bi 1146	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ≠ -i)
84		negicn 11382	. . . . . . . 8 ⊢ -i ∈ ℂ
85		subeq0 11408	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) = 0 ↔ 𝐴 = -i))
86	85	necon3bid 2969	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
87	22, 84, 86	sylancl 586	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
88	83, 87	mpbird 257	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) ≠ 0)
89	81, 88	eqnetrrd 2993	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ≠ 0)
90	37, 79, 79, 89	divsubdird 11957	. . . 4 ⊢ (𝐴 ∈ dom arctan → (((2 · i) − (𝐴 + i)) / (𝐴 + i)) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
91	74, 77, 90	3eqtr3d 2772	. . 3 ⊢ (𝐴 ∈ dom arctan → ((1 + (i · 𝐴)) / (1 − (i · 𝐴))) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
92	79, 89	dividd 11916	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((𝐴 + i) / (𝐴 + i)) = 1)
93	92	oveq2d 7369	. . 3 ⊢ (𝐴 ∈ dom arctan → (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))) = (((2 · i) / (𝐴 + i)) − 1))
94	50, 91, 93	3eqtrd 2768	. 2 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = (((2 · i) / (𝐴 + i)) − 1))
95	43, 45, 94	3eqtrd 2768	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 dom cdm 5623 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 0cc0 11028 1c1 11029 ici 11030 + caddc 11031 · cmul 11033 − cmin 11365 -cneg 11366 / cdiv 11795 2c2 12201 expce 15986 logclog 26479 arctancatan 26790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 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 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-atan 26793

This theorem is referenced by: tanatan 26845

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