2efiatan - Metamath Proof Explorer

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Theorem 2efiatan 26049

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 26015	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
2	1	oveq2d 7284	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
3		2cn 12031	. . . . . 6 ⊢ 2 ∈ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → 2 ∈ ℂ)
5		ax-icn 10914	. . . . . 6 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ∈ ℂ)
7		atancl 26012	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
8	4, 6, 7	mulassd 10982	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = (2 · (i · (arctan‘𝐴))))
9		halfcl 12181	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
10	5, 9	ax-mp 5	. . . . . . . . 9 ⊢ (i / 2) ∈ ℂ
11	3, 5, 10	mulassi 10970	. . . . . . . 8 ⊢ ((2 · i) · (i / 2)) = (2 · (i · (i / 2)))
12	3, 5, 10	mul12i 11153	. . . . . . . 8 ⊢ (2 · (i · (i / 2))) = (i · (2 · (i / 2)))
13		2ne0 12060	. . . . . . . . . . 11 ⊢ 2 ≠ 0
14	5, 3, 13	divcan2i 11701	. . . . . . . . . 10 ⊢ (2 · (i / 2)) = i
15	14	oveq2i 7279	. . . . . . . . 9 ⊢ (i · (2 · (i / 2))) = (i · i)
16		ixi 11587	. . . . . . . . 9 ⊢ (i · i) = -1
17	15, 16	eqtri 2767	. . . . . . . 8 ⊢ (i · (2 · (i / 2))) = -1
18	11, 12, 17	3eqtri 2771	. . . . . . 7 ⊢ ((2 · i) · (i / 2)) = -1
19	18	oveq1i 7278	. . . . . 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 10913	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		atandm2 26008	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
22	21	simp1bi 1143	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
23		mulcl 10939	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
24	5, 22, 23	sylancr 586	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
25		subcl 11203	. . . . . . . . . 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 1144	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
28	26, 27	logcld 25707	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
29		addcl 10937	. . . . . . . . . 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 1145	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
32	30, 31	logcld 25707	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
33	28, 32	subcld 11315	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
34	33	mulm1d 11410	. . . . . 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 2791	. . . . 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 12179	. . . . . . 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 10982	. . . . 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 11331	. . . . 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 2787	. . . 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 2787	. . 3 ⊢ (𝐴 ∈ dom arctan → (2 · (i · (arctan‘𝐴))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
43	42	fveq2d 6772	. 2 ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
44		efsub 15790	. . 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 25713	. . . . 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 25713	. . . . 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 7286	. . 3 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
51		negsub 11252	. . . . . . . 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	mulid1d 10976	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · 1) = i)
54	16	oveq1i 7278	. . . . . . . . 9 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴)
55	6, 6, 22	mulassd 10982	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → ((i · i) · 𝐴) = (i · (i · 𝐴)))
56	22	mulm1d 11410	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (-1 · 𝐴) = -𝐴)
57	54, 55, 56	3eqtr3a 2803	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · (i · 𝐴)) = -𝐴)
58	53, 57	oveq12d 7286	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) + (i · (i · 𝐴))) = (i + -𝐴))
59	6, 22, 6	pnpcan2d 11353	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i + i) − (𝐴 + i)) = (i − 𝐴))
60	52, 58, 59	3eqtr4d 2789	. . . . . 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 10983	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((i · 1) + (i · (i · 𝐴))))
63	6	2timesd 12199	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (2 · i) = (i + i))
64	63	oveq1d 7283	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((2 · i) − (𝐴 + i)) = ((i + i) − (𝐴 + i)))
65	60, 62, 64	3eqtr4d 2789	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((2 · i) − (𝐴 + i)))
66	6, 61, 24	subdid 11414	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = ((i · 1) − (i · (i · 𝐴))))
67	53, 57	oveq12d 7286	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i − -𝐴))
68		subneg 11253	. . . . . . . 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 2779	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i + 𝐴))
71		addcom 11144	. . . . . . 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 2783	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = (𝐴 + i))
74	65, 73	oveq12d 7286	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = (((2 · i) − (𝐴 + i)) / (𝐴 + i)))
75		ine0 11393	. . . . . 6 ⊢ i ≠ 0
76	75	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ≠ 0)
77	30, 26, 6, 27, 76	divcan5d 11760	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
78		addcl 10937	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 + i) ∈ ℂ)
79	22, 5, 78	sylancl 585	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ∈ ℂ)
80		subneg 11253	. . . . . . 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 26007	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
83	82	simp2bi 1144	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ≠ -i)
84		negicn 11205	. . . . . . . 8 ⊢ -i ∈ ℂ
85		subeq0 11230	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) = 0 ↔ 𝐴 = -i))
86	85	necon3bid 2989	. . . . . . . 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 256	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) ≠ 0)
89	81, 88	eqnetrrd 3013	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ≠ 0)
90	37, 79, 79, 89	divsubdird 11773	. . . 4 ⊢ (𝐴 ∈ dom arctan → (((2 · i) − (𝐴 + i)) / (𝐴 + i)) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
91	74, 77, 90	3eqtr3d 2787	. . 3 ⊢ (𝐴 ∈ dom arctan → ((1 + (i · 𝐴)) / (1 − (i · 𝐴))) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
92	79, 89	dividd 11732	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((𝐴 + i) / (𝐴 + i)) = 1)
93	92	oveq2d 7284	. . 3 ⊢ (𝐴 ∈ dom arctan → (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))) = (((2 · i) / (𝐴 + i)) − 1))
94	50, 91, 93	3eqtrd 2783	. 2 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = (((2 · i) / (𝐴 + i)) − 1))
95	43, 45, 94	3eqtrd 2783	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 205 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 dom cdm 5588 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 0cc0 10855 1c1 10856 ici 10857 + caddc 10858 · cmul 10860 − cmin 11188 -cneg 11189 / cdiv 11615 2c2 12011 expce 15752 logclog 25691 arctancatan 25995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ioc 13066 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-fl 13493 df-mod 13571 df-seq 13703 df-exp 13764 df-fac 13969 df-bc 13998 df-hash 14026 df-shft 14759 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-limsup 15161 df-clim 15178 df-rlim 15179 df-sum 15379 df-ef 15758 df-sin 15760 df-cos 15761 df-pi 15763 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-mulg 18682 df-cntz 18904 df-cmn 19369 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-fbas 20575 df-fg 20576 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-cls 22153 df-nei 22230 df-lp 22268 df-perf 22269 df-cn 22359 df-cnp 22360 df-haus 22447 df-tx 22694 df-hmeo 22887 df-fil 22978 df-fm 23070 df-flim 23071 df-flf 23072 df-xms 23454 df-ms 23455 df-tms 23456 df-cncf 24022 df-limc 25011 df-dv 25012 df-log 25693 df-atan 25998

This theorem is referenced by: tanatan 26050

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