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Theorem atanlogsub 26981

Description: A variation on atanlogadd 26979, 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 11131	. . . . . 6 ⊢ 1 ∈ ℂ
2		ax-icn 11132	. . . . . . 7 ⊢ i ∈ ℂ
3		atandm2 26942	. . . . . . . 8 ⊢ (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ (1 − (i · 𝐴)) ≠ 0 ∧ (1 + (i · 𝐴)) ≠ 0))
4	3	simp1bi 1158	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → 𝐴 ∈ ℂ)
5		mulcl 11157	. . . . . . 7 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ)
6	2, 4, 5	sylancr 596	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → (i · 𝐴) ∈ ℂ)
7		addcl 11155	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + (i · 𝐴)) ∈ ℂ)
8	1, 6, 7	sylancr 596	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ∈ ℂ)
9	3	simp3bi 1160	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 + (i · 𝐴)) ≠ 0)
10	8, 9	logcld 26635	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · 𝐴))) ∈ ℂ)
11		subcl 11429	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − (i · 𝐴)) ∈ ℂ)
12	1, 6, 11	sylancr 596	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ∈ ℂ)
13	3	simp2bi 1159	. . . . 5 ⊢ (𝐴 ∈ dom arctan → (1 − (i · 𝐴)) ≠ 0)
14	12, 13	logcld 26635	. . . 4 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · 𝐴))) ∈ ℂ)
15	10, 14	subcld 11542	. . 3 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
16	15	adantr 484	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
17	4	recld 15221	. . . . . . 7 ⊢ (𝐴 ∈ dom arctan → (ℜ‘𝐴) ∈ ℝ)
18		0re 11183	. . . . . . 7 ⊢ 0 ∈ ℝ
19		lttri2 11265	. . . . . . 7 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 0 ∈ ℝ) → ((ℜ‘𝐴) ≠ 0 ↔ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))))
20	17, 18, 19	sylancl 595	. . . . . 6 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) ≠ 0 ↔ ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴))))
21	20	biimpa 480	. . . . 5 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → ((ℜ‘𝐴) < 0 ∨ 0 < (ℜ‘𝐴)))
22	15	imnegd 15237	. . . . . . . . . 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 11558	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
24		mulneg2 11624	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · -𝐴) = -(i · 𝐴))
25	2, 4, 24	sylancr 596	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom arctan → (i · -𝐴) = -(i · 𝐴))
26	25	oveq2d 7412	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 + -(i · 𝐴)))
27		negsub 11479	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 + -(i · 𝐴)) = (1 − (i · 𝐴)))
28	1, 6, 27	sylancr 596	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 + -(i · 𝐴)) = (1 − (i · 𝐴)))
29	26, 28	eqtrd 2797	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 + (i · -𝐴)) = (1 − (i · 𝐴)))
30	29	fveq2d 6871	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 + (i · -𝐴))) = (log‘(1 − (i · 𝐴))))
31	25	oveq2d 7412	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 − -(i · 𝐴)))
32		subneg 11480	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ (i · 𝐴) ∈ ℂ) → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
33	1, 6, 32	sylancr 596	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ dom arctan → (1 − -(i · 𝐴)) = (1 + (i · 𝐴)))
34	31, 33	eqtrd 2797	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ dom arctan → (1 − (i · -𝐴)) = (1 + (i · 𝐴)))
35	34	fveq2d 6871	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ dom arctan → (log‘(1 − (i · -𝐴))) = (log‘(1 + (i · 𝐴))))
36	30, 35	oveq12d 7414	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))) = ((log‘(1 − (i · 𝐴))) − (log‘(1 + (i · 𝐴)))))
37	23, 36	eqtr4d 2800	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom arctan → -((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) = ((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴)))))
38	37	fveq2d 6871	. . . . . . . . . 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 2799	. . . . . . . . 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 484	. . . . . . . 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 26971	. . . . . . . . . 10 ⊢ (𝐴 ∈ dom arctan → -𝐴 ∈ dom arctan)
42	17	lt0neg1d 11756	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom arctan → ((ℜ‘𝐴) < 0 ↔ 0 < -(ℜ‘𝐴)))
43	42	biimpa 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < -(ℜ‘𝐴))
44	4	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 𝐴 ∈ ℂ)
45	44	renegd 15236	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℜ‘-𝐴) = -(ℜ‘𝐴))
46	43, 45	breqtrrd 5128	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → 0 < (ℜ‘-𝐴))
47		atanlogsublem 26980	. . . . . . . . . 10 ⊢ ((-𝐴 ∈ dom arctan ∧ 0 < (ℜ‘-𝐴)) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)π))
48	41, 46, 47	syl2an2r 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)π))
49		picn 26521	. . . . . . . . . . 11 ⊢ π ∈ ℂ
50	49	negnegi 11501	. . . . . . . . . 10 ⊢ --π = π
51	50	oveq2i 7407	. . . . . . . . 9 ⊢ (-π(,)--π) = (-π(,)π)
52	48, 51	eleqtrrdi 2873	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · -𝐴))) − (log‘(1 − (i · -𝐴))))) ∈ (-π(,)--π))
53	40, 52	eqeltrd 2862	. . . . . . 7 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → -(ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)--π))
54		pire 26519	. . . . . . . . 9 ⊢ π ∈ ℝ
55	54	renegcli 11492	. . . . . . . 8 ⊢ -π ∈ ℝ
56	15	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → ((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴)))) ∈ ℂ)
57	56	imcld 15222	. . . . . . . 8 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
58		iooneg 13475	. . . . . . . 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 1486	. . . . . . 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 259	. . . . . 6 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) < 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
61		atanlogsublem 26980	. . . . . 6 ⊢ ((𝐴 ∈ dom arctan ∧ 0 < (ℜ‘𝐴)) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
62	60, 61	jaodan 970	. . . . 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 600	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ (-π(,)π))
64		eliooord 13409	. . . 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 498	. 2 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → -π < (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))))
67	65	simprd 499	. . 3 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) < π)
68	16	imcld 15222	. . . 4 ⊢ ((𝐴 ∈ dom arctan ∧ (ℜ‘𝐴) ≠ 0) → (ℑ‘((log‘(1 + (i · 𝐴))) − (log‘(1 − (i · 𝐴))))) ∈ ℝ)
69		ltle 11271	. . . 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 595	. . 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 26624	. 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 1357	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 208 ∧ wa 399 ∨ wo 858 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 class class class wbr 5100 dom cdm 5647 ran crn 5648 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 ici 11075 + caddc 11076 · cmul 11078 < clt 11216 ≤ cle 11217 − cmin 11414 -cneg 11415 (,)cioo 13349 ℜcre 15124 ℑcim 15125 πcpi 16096 logclog 26619 arctancatan 26929

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-fac 14287 df-bc 14316 df-hash 14344 df-shft 15080 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-ef 16097 df-sin 16099 df-cos 16100 df-tan 16101 df-pi 16102 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-lp 23196 df-perf 23197 df-cn 23287 df-cnp 23288 df-haus 23375 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-limc 25928 df-dv 25929 df-log 26621 df-atan 26932

This theorem is referenced by: atantan 26988

W3C validator