2efiatan - Metamath Proof Explorer

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Theorem 2efiatan 26979

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 26945	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
2	1	oveq2d 7464	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
3		2cn 12368	. . . . . 6 ⊢ 2 ∈ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → 2 ∈ ℂ)
5		ax-icn 11243	. . . . . 6 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ∈ ℂ)
7		atancl 26942	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
8	4, 6, 7	mulassd 11313	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = (2 · (i · (arctan‘𝐴))))
9		halfcl 12518	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
10	5, 9	ax-mp 5	. . . . . . . . 9 ⊢ (i / 2) ∈ ℂ
11	3, 5, 10	mulassi 11301	. . . . . . . 8 ⊢ ((2 · i) · (i / 2)) = (2 · (i · (i / 2)))
12	3, 5, 10	mul12i 11485	. . . . . . . 8 ⊢ (2 · (i · (i / 2))) = (i · (2 · (i / 2)))
13		2ne0 12397	. . . . . . . . . . 11 ⊢ 2 ≠ 0
14	5, 3, 13	divcan2i 12037	. . . . . . . . . 10 ⊢ (2 · (i / 2)) = i
15	14	oveq2i 7459	. . . . . . . . 9 ⊢ (i · (2 · (i / 2))) = (i · i)
16		ixi 11919	. . . . . . . . 9 ⊢ (i · i) = -1
17	15, 16	eqtri 2768	. . . . . . . 8 ⊢ (i · (2 · (i / 2))) = -1
18	11, 12, 17	3eqtri 2772	. . . . . . 7 ⊢ ((2 · i) · (i / 2)) = -1
19	18	oveq1i 7458	. . . . . 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 11242	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		atandm2 26938	. . . . . . . . . . . 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 11268	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
24	5, 22, 23	sylancr 586	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
25		subcl 11535	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
26	20, 24, 25	sylancr 586	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
27	21	simp2bi 1146	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
28	26, 27	logcld 26630	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
29		addcl 11266	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
30	20, 24, 29	sylancr 586	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
31	21	simp3bi 1147	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
32	30, 31	logcld 26630	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
33	28, 32	subcld 11647	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
34	33	mulm1d 11742	. . . . . 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 2792	. . . . 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 12516	. . . . . . 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 11313	. . . . 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 11663	. . . . 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 2788	. . . 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 2788	. . 3 ⊢ (𝐴 ∈ dom arctan → (2 · (i · (arctan‘𝐴))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
43	42	fveq2d 6924	. 2 ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
44		efsub 16148	. . 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 583	. 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 26636	. . . . 5 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (1 + (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
47	30, 31, 46	syl2anc 583	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
48		eflog 26636	. . . . 5 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
49	26, 27, 48	syl2anc 583	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
50	47, 49	oveq12d 7466	. . 3 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
51		negsub 11584	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + -𝐴) = (i − 𝐴))
52	5, 22, 51	sylancr 586	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i + -𝐴) = (i − 𝐴))
53	6	mulridd 11307	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · 1) = i)
54	16	oveq1i 7458	. . . . . . . . 9 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴)
55	6, 6, 22	mulassd 11313	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → ((i · i) · 𝐴) = (i · (i · 𝐴)))
56	22	mulm1d 11742	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (-1 · 𝐴) = -𝐴)
57	54, 55, 56	3eqtr3a 2804	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · (i · 𝐴)) = -𝐴)
58	53, 57	oveq12d 7466	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) + (i · (i · 𝐴))) = (i + -𝐴))
59	6, 22, 6	pnpcan2d 11685	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i + i) − (𝐴 + i)) = (i − 𝐴))
60	52, 58, 59	3eqtr4d 2790	. . . . . 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 11314	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((i · 1) + (i · (i · 𝐴))))
63	6	2timesd 12536	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (2 · i) = (i + i))
64	63	oveq1d 7463	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((2 · i) − (𝐴 + i)) = ((i + i) − (𝐴 + i)))
65	60, 62, 64	3eqtr4d 2790	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((2 · i) − (𝐴 + i)))
66	6, 61, 24	subdid 11746	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = ((i · 1) − (i · (i · 𝐴))))
67	53, 57	oveq12d 7466	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i − -𝐴))
68		subneg 11585	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i − -𝐴) = (i + 𝐴))
69	5, 22, 68	sylancr 586	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i − -𝐴) = (i + 𝐴))
70	67, 69	eqtrd 2780	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i + 𝐴))
71		addcom 11476	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + 𝐴) = (𝐴 + i))
72	5, 22, 71	sylancr 586	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i + 𝐴) = (𝐴 + i))
73	66, 70, 72	3eqtrd 2784	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = (𝐴 + i))
74	65, 73	oveq12d 7466	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = (((2 · i) − (𝐴 + i)) / (𝐴 + i)))
75		ine0 11725	. . . . . 6 ⊢ i ≠ 0
76	75	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ≠ 0)
77	30, 26, 6, 27, 76	divcan5d 12096	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
78		addcl 11266	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 + i) ∈ ℂ)
79	22, 5, 78	sylancl 585	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ∈ ℂ)
80		subneg 11585	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 − -i) = (𝐴 + i))
81	22, 5, 80	sylancl 585	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) = (𝐴 + i))
82		atandm 26937	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
83	82	simp2bi 1146	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ≠ -i)
84		negicn 11537	. . . . . . . 8 ⊢ -i ∈ ℂ
85		subeq0 11562	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) = 0 ↔ 𝐴 = -i))
86	85	necon3bid 2991	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
87	22, 84, 86	sylancl 585	. . . . . . 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 3015	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ≠ 0)
90	37, 79, 79, 89	divsubdird 12109	. . . 4 ⊢ (𝐴 ∈ dom arctan → (((2 · i) − (𝐴 + i)) / (𝐴 + i)) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
91	74, 77, 90	3eqtr3d 2788	. . 3 ⊢ (𝐴 ∈ dom arctan → ((1 + (i · 𝐴)) / (1 − (i · 𝐴))) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
92	79, 89	dividd 12068	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((𝐴 + i) / (𝐴 + i)) = 1)
93	92	oveq2d 7464	. . 3 ⊢ (𝐴 ∈ dom arctan → (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))) = (((2 · i) / (𝐴 + i)) − 1))
94	50, 91, 93	3eqtrd 2784	. 2 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = (((2 · i) / (𝐴 + i)) − 1))
95	43, 45, 94	3eqtrd 2784	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 1537 ∈ wcel 2108 ≠ wne 2946 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 ici 11186 + caddc 11187 · cmul 11189 − cmin 11520 -cneg 11521 / cdiv 11947 2c2 12348 expce 16109 logclog 26614 arctancatan 26925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-atan 26928

This theorem is referenced by: tanatan 26980

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