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Theorem atanlogsub 27244

Description: A variation on atanlogadd 27242, 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 11258	. . . . . 6 ⊢ 1 ∈ ℂ
2		ax-icn 11259	. . . . . . 7 ⊢ i ∈ ℂ
3		atandm2 27205	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
4	3	simp1bi 1163	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
5		mulcl 11284	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
6	2, 4, 5	sylancr 599	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
7		addcl 11282	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
8	1, 6, 7	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
9	3	simp3bi 1165	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
10	8, 9	logcld 26898	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
11		subcl 11556	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
12	1, 6, 11	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
13	3	simp2bi 1164	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
14	12, 13	logcld 26898	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
15	10, 14	subcld 11669	. . 3 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
16	15	adantr 486	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
17	4	recld 15361	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (ℜ‘𝐴) ∈ ℝ)
18		0re 11310	. . . . . . 7 ⊢ 0 ∈ ℝ
19		lttri2 11392	. . . . . . 7 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝐴) ≠ 0 ↔ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))))
20	17, 18, 19	sylancl 598	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) ≠ 0 ↔ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))))
21	20	biimpa 482	. . . . 5 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴)))
22	15	imnegd 15377	. . . . . . . . . 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 11685	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
24		mulneg2 11753	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴))
25	2, 4, 24	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom arctan → (i · -𝐴) = -(i · 𝐴))
26	25	oveq2d 7436	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 + -(i · 𝐴)))
27		negsub 11606	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + -(i · 𝐴)) = (1 − (i · 𝐴)))
28	1, 6, 27	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + -(i · 𝐴)) = (1 − (i · 𝐴)))
29	26, 28	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 − (i · 𝐴)))
30	29	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · -𝐴))) = (log‘(1 − (i · 𝐴))))
31	25	oveq2d 7436	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 − -(i · 𝐴)))
32		subneg 11607	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
33	1, 6, 32	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
34	31, 33	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 + (i · 𝐴)))
35	34	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · -𝐴))) = (log‘(1 + (i · 𝐴))))
36	30, 35	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
37	23, 36	eqtr4d 2799	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))))
38	37	fveq2d 6889	. . . . . . . . . 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 2798	. . . . . . . . 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 486	. . . . . . . 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 27234	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)
42	17	lt0neg1d 11885	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) < 0 ↔ 0 < -(ℜ‘𝐴)))
43	42	biimpa 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < -(ℜ‘𝐴))
44	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 𝐴 ∈ ℂ)
45	44	renegd 15376	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℜ‘-𝐴) = -(ℜ‘𝐴))
46	43, 45	breqtrrd 5133	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < (ℜ‘-𝐴))
47		atanlogsublem 27243	. . . . . . . . . 10 ⊢ ((-𝐴 ∈ dom arctan ∧ 0 < (ℜ‘-𝐴)) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)π))
48	41, 46, 47	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)π))
49		picn 26785	. . . . . . . . . . 11 ⊢ π ∈ ℂ
50	49	negnegi 11628	. . . . . . . . . 10 ⊢ --π = π
51	50	oveq2i 7431	. . . . . . . . 9 ⊢ (-π(,)--π) = (-π(,)π)
52	48, 51	eleqtrrdi 2872	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)--π))
53	40, 52	eqeltrd 2861	. . . . . . 7 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)--π))
54		pire 26783	. . . . . . . . 9 ⊢ π ∈ ℝ
55	54	renegcli 11619	. . . . . . . 8 ⊢ -π ∈ ℝ
56	15	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
57	56	imcld 15362	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
58		iooneg 13602	. . . . . . . 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 1494	. . . . . . 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 260	. . . . . 6 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
61		atanlogsublem 27243	. . . . . 6 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
62	60, 61	jaodan 972	. . . . 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 603	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
64		eliooord 13536	. . . 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 18	. . 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 500	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → -π < (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
67	65	simprd 501	. . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π)
68	16	imcld 15362	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
69		ltle 11398	. . . 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 598	. . 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 16	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ≤ π)
72		ellogrn 26887	. 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 1362	1 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ran log)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 dom cdm 5651 ran crn 5652 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 ici 11202 + caddc 11203 · cmul 11205 < clt 11343 ≤ cle 11344 − cmin 11541 -cneg 11542 (,)cioo 13476 ℜcre 15264 ℑcim 15265 πcpi 16232 logclog 26882 arctancatan 27192

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-tan 16237 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884 df-atan 27195

This theorem is used by: atantan 27251

W3C validator