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 27301

Description: A variation on atanlogadd 27299, 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 11297	. . . . . 6 ⊢ 1 ∈ ℂ
2		ax-icn 11298	. . . . . . 7 ⊢ i ∈ ℂ
3		atandm2 27262	. . . . . . . 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 11323	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
6	2, 4, 5	sylancr 599	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
7		addcl 11321	. . . . . 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 26955	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
11		subcl 11595	. . . . . 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 26955	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
15	10, 14	subcld 11708	. . 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 15404	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (ℜ‘𝐴) ∈ ℝ)
18		0re 11349	. . . . . . 7 ⊢ 0 ∈ ℝ
19		lttri2 11431	. . . . . . 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 15420	. . . . . . . . . 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 11724	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
24		mulneg2 11792	. . . . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 + -(i · 𝐴)))
27		negsub 11645	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 − (i · 𝐴)))
30	29	fveq2d 6920	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · -𝐴))) = (log‘(1 − (i · 𝐴))))
31	25	oveq2d 7467	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 − -(i · 𝐴)))
32		subneg 11646	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 + (i · 𝐴)))
35	34	fveq2d 6920	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · -𝐴))) = (log‘(1 + (i · 𝐴))))
36	30, 35	oveq12d 7469	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
37	23, 36	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))))
38	37	fveq2d 6920	. . . . . . . . . 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 2800	. . . . . . . . 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 27291	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)
42	17	lt0neg1d 11925	. . . . . . . . . . . 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 15419	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℜ‘ -𝐴) = -(ℜ‘𝐴))
46	43, 45	breqtrrd 5139	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < (ℜ‘ -𝐴))
47		atanlogsublem 27300	. . . . . . . . . 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 26842	. . . . . . . . . . 11 ⊢ π ∈ ℂ
50	49	negnegi 11667	. . . . . . . . . 10 ⊢ - -π = π
51	50	oveq2i 7462	. . . . . . . . 9 ⊢ ( -π(,) - -π) = ( -π(,)π)
52	48, 51	eleqtrrdi 2875	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ ( -π(,) - -π))
53	40, 52	eqeltrd 2864	. . . . . . 7 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ( -π(,) - -π))
54		pire 26840	. . . . . . . . 9 ⊢ π ∈ ℝ
55	54	renegcli 11658	. . . . . . . 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 15405	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
58		iooneg 13642	. . . . . . . 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 1495	. . . . . . 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 27300	. . . . . 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 13576	. . . 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 15405	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
69		ltle 11437	. . . 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 26944	. 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 2147 ≠ wne 2959 class class class wbr 5108 dom cdm 5658 ran crn 5659 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 ici 11241 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 − cmin 11580 -cneg 11581 (,)cioo 13516 ℜcre 15307 ℑcim 15308 πcpi 16276 logclog 26939 arctancatan 27249

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-fac 14459 df-bc 14488 df-hash 14516 df-shft 15263 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-limsup 15682 df-clim 15699 df-rlim 15700 df-sum 15898 df-ef 16277 df-sin 16279 df-cos 16280 df-tan 16281 df-pi 16282 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-limc 26241 df-dv 26242 df-log 26941 df-atan 27252

This theorem is used by: atantan 27308

W3C validator