atanlogsub - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > atanlogsub		Structured version Visualization version GIF version

Theorem atanlogsub 27154

Description: A variation on atanlogadd 27152, 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 11183	. . . . . 6 ⊢ 1 ∈ ℂ
2		ax-icn 11184	. . . . . . 7 ⊢ i ∈ ℂ
3		atandm2 27115	. . . . . . . 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 11209	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
6	2, 4, 5	sylancr 599	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
7		addcl 11207	. . . . . 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 26808	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
11		subcl 11481	. . . . . 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 26808	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
15	10, 14	subcld 11594	. . 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 15282	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (ℜ‘𝐴) ∈ ℝ)
18		0re 11235	. . . . . . 7 ⊢ 0 ∈ ℝ
19		lttri2 11317	. . . . . . 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 15298	. . . . . . . . . 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 11610	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
24		mulneg2 11676	. . . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 + -(i · 𝐴)))
27		negsub 11531	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 − (i · 𝐴)))
30	29	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · -𝐴))) = (log‘(1 − (i · 𝐴))))
31	25	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 − -(i · 𝐴)))
32		subneg 11532	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 + (i · 𝐴)))
35	34	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · -𝐴))) = (log‘(1 + (i · 𝐴))))
36	30, 35	oveq12d 7432	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
37	23, 36	eqtr4d 2798	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))))
38	37	fveq2d 6883	. . . . . . . . . 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 2797	. . . . . . . . 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 27144	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)
42	17	lt0neg1d 11808	. . . . . . . . . . . 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 15297	. . . . . . . . . . 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 27153	. . . . . . . . . 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 26695	. . . . . . . . . . 11 ⊢ π ∈ ℂ
50	49	negnegi 11553	. . . . . . . . . 10 ⊢ --π = π
51	50	oveq2i 7425	. . . . . . . . 9 ⊢ (-π(,)--π) = (-π(,)π)
52	48, 51	eleqtrrdi 2871	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)--π))
53	40, 52	eqeltrd 2860	. . . . . . 7 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)--π))
54		pire 26693	. . . . . . . . 9 ⊢ π ∈ ℝ
55	54	renegcli 11544	. . . . . . . 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 15283	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
58		iooneg 13525	. . . . . . . 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 27153	. . . . . 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 13459	. . . 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 15283	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
69		ltle 11323	. . . 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 26797	. 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 2955 class class class wbr 5103 dom cdm 5655 ran crn 5656 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 ici 11127 + caddc 11128 · cmul 11130 < clt 11268 ≤ cle 11269 − cmin 11466 -cneg 11467 (,)cioo 13399 ℜcre 15185 ℑcim 15186 πcpi 16153 logclog 26792 arctancatan 27102

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9621 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ioc 13404 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-fl 13854 df-mod 13932 df-seq 14067 df-exp 14127 df-fac 14339 df-bc 14368 df-hash 14396 df-shft 15141 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-limsup 15559 df-clim 15576 df-rlim 15577 df-sum 15775 df-ef 16154 df-sin 16156 df-cos 16157 df-tan 16158 df-pi 16159 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-limc 26094 df-dv 26095 df-log 26794 df-atan 27105

This theorem is used by: atantan 27161

W3C validator