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Theorem atanlogsub 26883

Description: A variation on atanlogadd 26881, to show that √(1 + i𝑧) / √(1 − i𝑧) = √((1 + i𝑧) / (1 − i𝑧)) under more limited conditions. (Contributed by Mario Carneiro, 4-Apr-2015.)

**Assertion**
Ref	Expression
atanlogsub	⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ran log)

**Proof of Theorem atanlogsub**
Step	Hyp	Ref	Expression
1		ax-1cn 11192	. . . . . 6 ⊢ 1 ∈ ℂ
2		ax-icn 11193	. . . . . . 7 ⊢ i ∈ ℂ
3		atandm2 26844	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
4	3	simp1bi 1145	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
5		mulcl 11218	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
6	2, 4, 5	sylancr 587	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
7		addcl 11216	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
8	1, 6, 7	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
9	3	simp3bi 1147	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
10	8, 9	logcld 26536	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
11		subcl 11486	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
12	1, 6, 11	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
13	3	simp2bi 1146	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
14	12, 13	logcld 26536	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
15	10, 14	subcld 11599	. . 3 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
16	15	adantr 480	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
17	4	recld 15218	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (ℜ‘𝐴) ∈ ℝ)
18		0re 11242	. . . . . . 7 ⊢ 0 ∈ ℝ
19		lttri2 11322	. . . . . . 7 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝐴) ≠ 0 ↔ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))))
20	17, 18, 19	sylancl 586	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) ≠ 0 ↔ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))))
21	20	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴)))
22	15	imnegd 15234	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (ℑ‘-((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
23	10, 14	negsubdi2d 11615	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
24		mulneg2 11679	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴))
25	2, 4, 24	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom arctan → (i · -𝐴) = -(i · 𝐴))
26	25	oveq2d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 + -(i · 𝐴)))
27		negsub 11536	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + -(i · 𝐴)) = (1 − (i · 𝐴)))
28	1, 6, 27	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + -(i · 𝐴)) = (1 − (i · 𝐴)))
29	26, 28	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 − (i · 𝐴)))
30	29	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · -𝐴))) = (log‘(1 − (i · 𝐴))))
31	25	oveq2d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 − -(i · 𝐴)))
32		subneg 11537	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
33	1, 6, 32	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
34	31, 33	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 + (i · 𝐴)))
35	34	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · -𝐴))) = (log‘(1 + (i · 𝐴))))
36	30, 35	oveq12d 7428	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
37	23, 36	eqtr4d 2774	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))))
38	37	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → (ℑ‘-((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))))
39	22, 38	eqtr3d 2773	. . . . . . . . 9 ⊢ (𝐴 ∈ dom arctan → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))))
40	39	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) = (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))))
41		atandmneg 26873	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)
42	17	lt0neg1d 11811	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) < 0 ↔ 0 < -(ℜ‘𝐴)))
43	42	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < -(ℜ‘𝐴))
44	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 𝐴 ∈ ℂ)
45	44	renegd 15233	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℜ‘-𝐴) = -(ℜ‘𝐴))
46	43, 45	breqtrrd 5152	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < (ℜ‘-𝐴))
47		atanlogsublem 26882	. . . . . . . . . 10 ⊢ ((-𝐴 ∈ dom arctan ∧ 0 < (ℜ‘-𝐴)) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)π))
48	41, 46, 47	syl2an2r 685	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)π))
49		picn 26424	. . . . . . . . . . 11 ⊢ π ∈ ℂ
50	49	negnegi 11558	. . . . . . . . . 10 ⊢ --π = π
51	50	oveq2i 7421	. . . . . . . . 9 ⊢ (-π(,)--π) = (-π(,)π)
52	48, 51	eleqtrrdi 2846	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)--π))
53	40, 52	eqeltrd 2835	. . . . . . 7 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)--π))
54		pire 26423	. . . . . . . . 9 ⊢ π ∈ ℝ
55	54	renegcli 11549	. . . . . . . 8 ⊢ -π ∈ ℝ
56	15	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
57	56	imcld 15219	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
58		iooneg 13493	. . . . . . . 8 ⊢ ((-π ∈ ℝ ∧ π ∈ ℝ ∧ (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ) → ((ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π) ↔ -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)--π)))
59	55, 54, 57, 58	mp3an12i 1467	. . . . . . 7 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → ((ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π) ↔ -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)--π)))
60	53, 59	mpbird 257	. . . . . 6 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
61		atanlogsublem 26882	. . . . . 6 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
62	60, 61	jaodan 959	. . . . 5 ⊢ ((𝐴 ∈ dom arctan ∧ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
63	21, 62	syldan 591	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
64		eliooord 13427	. . . 4 ⊢ ((ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π) → (-π < (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∧ (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π))
65	63, 64	syl 17	. . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (-π < (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∧ (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π))
66	65	simpld 494	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → -π < (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
67	65	simprd 495	. . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π)
68	16	imcld 15219	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
69		ltle 11328	. . . 4 ⊢ (((ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ≤ π))
70	68, 54, 69	sylancl 586	. . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ≤ π))
71	67, 70	mpd 15	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ≤ π)
72		ellogrn 26525	. 2 ⊢ (((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ran log ↔ (((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ ∧ -π < (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∧ (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ≤ π))
73	16, 66, 71, 72	syl3anbrc 1344	1 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ran log)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 class class class wbr 5124 dom cdm 5659 ran crn 5660 ‘cfv 6536 (class class class)co 7410 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 ici 11136 + caddc 11137 · cmul 11139 < clt 11274 ≤ cle 11275 − cmin 11471 -cneg 11472 (,)cioo 13367 ℜcre 15121 ℑcim 15122 πcpi 16087 logclog 26520 arctancatan 26831

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-tan 16092 df-pi 16093 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-mulg 19056 df-cntz 19305 df-cmn 19768 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825 df-log 26522 df-atan 26834

This theorem is referenced by: atantan 26890

W3C validator