2efiatan - Metamath Proof Explorer

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Theorem 2efiatan 26971

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 26937	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) = ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴))))))
2	1	oveq2d 7407	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = ((2 · i) · ((i / 2) · ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))))
3		2cn 12287	. . . . . 6 ⊢ 2 ∈ ℂ
4	3	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → 2 ∈ ℂ)
5		ax-icn 11126	. . . . . 6 ⊢ i ∈ ℂ
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ∈ ℂ)
7		atancl 26934	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (arctan‘𝐴) ∈ ℂ)
8	4, 6, 7	mulassd 11199	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((2 · i) · (arctan‘𝐴)) = (2 · (i · (arctan‘𝐴))))
9		halfcl 12441	. . . . . . . . . 10 ⊢ (i ∈ ℂ → (i / 2) ∈ ℂ)
10	5, 9	ax-mp 5	. . . . . . . . 9 ⊢ (i / 2) ∈ ℂ
11	3, 5, 10	mulassi 11187	. . . . . . . 8 ⊢ ((2 · i) · (i / 2)) = (2 · (i · (i / 2)))
12	3, 5, 10	mul12i 11372	. . . . . . . 8 ⊢ (2 · (i · (i / 2))) = (i · (2 · (i / 2)))
13		2ne0 12318	. . . . . . . . . . 11 ⊢ 2 ≠ 0
14	5, 3, 13	divcan2i 11928	. . . . . . . . . 10 ⊢ (2 · (i / 2)) = i
15	14	oveq2i 7402	. . . . . . . . 9 ⊢ (i · (2 · (i / 2))) = (i · i)
16		ixi 11810	. . . . . . . . 9 ⊢ (i · i) = -1
17	15, 16	eqtri 2784	. . . . . . . 8 ⊢ (i · (2 · (i / 2))) = -1
18	11, 12, 17	3eqtri 2788	. . . . . . 7 ⊢ ((2 · i) · (i / 2)) = -1
19	18	oveq1i 7401	. . . . . 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 11125	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		atandm2 26930	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
22	21	simp1bi 1157	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
23		mulcl 11151	. . . . . . . . . . 11 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
24	5, 22, 23	sylancr 596	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
25		subcl 11423	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
26	20, 24, 25	sylancr 596	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
27	21	simp2bi 1158	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
28	26, 27	logcld 26623	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
29		addcl 11149	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
30	20, 24, 29	sylancr 596	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
31	21	simp3bi 1159	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
32	30, 31	logcld 26623	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
33	28, 32	subcld 11536	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))) ∈ ℂ)
34	33	mulm1d 11633	. . . . . 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 2808	. . . . 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 12439	. . . . . . 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 11199	. . . . 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 11552	. . . . 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 2804	. . . 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 2804	. . 3 ⊢ (𝐴 ∈ dom arctan → (2 · (i · (arctan‘𝐴))) = ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))))
43	42	fveq2d 6866	. 2 ⊢ (𝐴 ∈ dom arctan → (exp‘(2 · (i · (arctan‘𝐴)))) = (exp‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
44		efsub 16123	. . 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 593	. 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 26629	. . . . 5 ⊢ (((1 + (i · 𝐴)) ∈ ℂ ∧ (1 + (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
47	30, 31, 46	syl2anc 593	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 + (i · 𝐴)))) = (1 + (i · 𝐴)))
48		eflog 26629	. . . . 5 ⊢ (((1 − (i · 𝐴)) ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0) → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
49	26, 27, 48	syl2anc 593	. . . 4 ⊢ (𝐴 ∈ dom arctan → (exp‘(log‘(1 − (i · 𝐴)))) = (1 − (i · 𝐴)))
50	47, 49	oveq12d 7409	. . 3 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
51		negsub 11473	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + -𝐴) = (i − 𝐴))
52	5, 22, 51	sylancr 596	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i + -𝐴) = (i − 𝐴))
53	6	mulridd 11193	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · 1) = i)
54	16	oveq1i 7401	. . . . . . . . 9 ⊢ ((i · i) · 𝐴) = (-1 · 𝐴)
55	6, 6, 22	mulassd 11199	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → ((i · i) · 𝐴) = (i · (i · 𝐴)))
56	22	mulm1d 11633	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → (-1 · 𝐴) = -𝐴)
57	54, 55, 56	3eqtr3a 2820	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan → (i · (i · 𝐴)) = -𝐴)
58	53, 57	oveq12d 7409	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) + (i · (i · 𝐴))) = (i + -𝐴))
59	6, 22, 6	pnpcan2d 11574	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i + i) − (𝐴 + i)) = (i − 𝐴))
60	52, 58, 59	3eqtr4d 2806	. . . . . 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 11200	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((i · 1) + (i · (i · 𝐴))))
63	6	2timesd 12458	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (2 · i) = (i + i))
64	63	oveq1d 7406	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((2 · i) − (𝐴 + i)) = ((i + i) − (𝐴 + i)))
65	60, 62, 64	3eqtr4d 2806	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 + (i · 𝐴))) = ((2 · i) − (𝐴 + i)))
66	6, 61, 24	subdid 11637	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = ((i · 1) − (i · (i · 𝐴))))
67	53, 57	oveq12d 7409	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i − -𝐴))
68		subneg 11474	. . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i − -𝐴) = (i + 𝐴))
69	5, 22, 68	sylancr 596	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (i − -𝐴) = (i + 𝐴))
70	67, 69	eqtrd 2796	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((i · 1) − (i · (i · 𝐴))) = (i + 𝐴))
71		addcom 11363	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i + 𝐴) = (𝐴 + i))
72	5, 22, 71	sylancr 596	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i + 𝐴) = (𝐴 + i))
73	66, 70, 72	3eqtrd 2800	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (i · (1 − (i · 𝐴))) = (𝐴 + i))
74	65, 73	oveq12d 7409	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = (((2 · i) − (𝐴 + i)) / (𝐴 + i)))
75		ine0 11616	. . . . . 6 ⊢ i ≠ 0
76	75	a1i 11	. . . . 5 ⊢ (𝐴 ∈ dom arctan → i ≠ 0)
77	30, 26, 6, 27, 76	divcan5d 11987	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((i · (1 + (i · 𝐴))) / (i · (1 − (i · 𝐴)))) = ((1 + (i · 𝐴)) / (1 − (i · 𝐴))))
78		addcl 11149	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 + i) ∈ ℂ)
79	22, 5, 78	sylancl 595	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ∈ ℂ)
80		subneg 11474	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ) → (𝐴 − -i) = (𝐴 + i))
81	22, 5, 80	sylancl 595	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) = (𝐴 + i))
82		atandm 26929	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i))
83	82	simp2bi 1158	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ≠ -i)
84		negicn 11425	. . . . . . . 8 ⊢ -i ∈ ℂ
85		subeq0 11451	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) = 0 ↔ 𝐴 = -i))
86	85	necon3bid 3000	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ -i ∈ ℂ) → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
87	22, 84, 86	sylancl 595	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → ((𝐴 − -i) ≠ 0 ↔ 𝐴 ≠ -i))
88	83, 87	mpbird 259	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (𝐴 − -i) ≠ 0)
89	81, 88	eqnetrrd 3024	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (𝐴 + i) ≠ 0)
90	37, 79, 79, 89	divsubdird 12000	. . . 4 ⊢ (𝐴 ∈ dom arctan → (((2 · i) − (𝐴 + i)) / (𝐴 + i)) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
91	74, 77, 90	3eqtr3d 2804	. . 3 ⊢ (𝐴 ∈ dom arctan → ((1 + (i · 𝐴)) / (1 − (i · 𝐴))) = (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))))
92	79, 89	dividd 11959	. . . 4 ⊢ (𝐴 ∈ dom arctan → ((𝐴 + i) / (𝐴 + i)) = 1)
93	92	oveq2d 7407	. . 3 ⊢ (𝐴 ∈ dom arctan → (((2 · i) / (𝐴 + i)) − ((𝐴 + i) / (𝐴 + i))) = (((2 · i) / (𝐴 + i)) − 1))
94	50, 91, 93	3eqtrd 2800	. 2 ⊢ (𝐴 ∈ dom arctan → ((exp‘(log‘(1 + (i · 𝐴)))) / (exp‘(log‘(1 − (i · 𝐴))))) = (((2 · i) / (𝐴 + i)) − 1))
95	43, 45, 94	3eqtrd 2800	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 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 dom cdm 5643 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 0cc0 11067 1c1 11068 ici 11069 + caddc 11070 · cmul 11072 − cmin 11408 -cneg 11409 / cdiv 11838 2c2 12266 expce 16082 logclog 26607 arctancatan 26917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ioc 13348 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-fl 13796 df-mod 13874 df-seq 14009 df-exp 14069 df-fac 14281 df-bc 14310 df-hash 14338 df-shft 15074 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-limsup 15489 df-clim 15506 df-rlim 15507 df-sum 15705 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-lp 23184 df-perf 23185 df-cn 23275 df-cnp 23276 df-haus 23363 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-xms 24368 df-ms 24369 df-tms 24370 df-cncf 24928 df-limc 25916 df-dv 25917 df-log 26609 df-atan 26920

This theorem is referenced by: tanatan 26972

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