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Theorem dvlog 26693

Description: The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)

**Hypothesis**
Ref	Expression
logcn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))

**Assertion**
Ref	Expression
dvlog	⊢ (ℂ D (log ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Distinct variable group: 𝑥,𝐷

**Proof of Theorem dvlog**
Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtopon 24803	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
3	2	toponrestid 22927	. . . 4 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
4		cnelprrecn 11248	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
6		logcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
7	6	logdmopn 26691	. . . . 5 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
8	7	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
9		logf1o 26606	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
10		f1of1 6847	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})–1-1→ran log)
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1→ran log
12	6	logdmss 26684	. . . . . . . 8 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
13		f1ores 6862	. . . . . . . 8 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
14	11, 12, 13	mp2an 692	. . . . . . 7 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
15		f1ocnv 6860	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) → ^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ ^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷
17		df-log 26598	. . . . . . . . . . 11 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
18	17	reseq1i 5993	. . . . . . . . . 10 ⊢ (log ↾ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
19	18	cnveqi 5885	. . . . . . . . 9 ⊢ ^◡(log ↾ 𝐷) = ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
20		eff 16117	. . . . . . . . . . 11 ⊢ exp:ℂ⟶ℂ
21		cnvimass 6100	. . . . . . . . . . . 12 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ dom ℑ
22		imf 15152	. . . . . . . . . . . . 13 ⊢ ℑ:ℂ⟶ℝ
23	22	fdmi 6747	. . . . . . . . . . . 12 ⊢ dom ℑ = ℂ
24	21, 23	sseqtri 4032	. . . . . . . . . . 11 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ ℂ
25		fssres 6774	. . . . . . . . . . 11 ⊢ ((exp:ℂ⟶ℂ ∧ (^◡ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ)
26	20, 24, 25	mp2an 692	. . . . . . . . . 10 ⊢ (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ
27		ffun 6739	. . . . . . . . . 10 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (^◡ℑ “ (-π(,]π))))
28		funcnvres2 6646	. . . . . . . . . 10 ⊢ (Fun (exp ↾ (^◡ℑ “ (-π(,]π))) → ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)))
29	26, 27, 28	mp2b 10	. . . . . . . . 9 ⊢ ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
30		cnvimass 6100	. . . . . . . . . . 11 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (^◡ℑ “ (-π(,]π)))
31	26	fdmi 6747	. . . . . . . . . . 11 ⊢ dom (exp ↾ (^◡ℑ “ (-π(,]π))) = (^◡ℑ “ (-π(,]π))
32	30, 31	sseqtri 4032	. . . . . . . . . 10 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π))
33		resabs1 6024	. . . . . . . . . 10 ⊢ ((^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π)) → ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)))
34	32, 33	ax-mp 5	. . . . . . . . 9 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
35	19, 29, 34	3eqtri 2769	. . . . . . . 8 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
36	17	imaeq1i 6075	. . . . . . . . 9 ⊢ (log “ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)
37	36	reseq2i 5994	. . . . . . . 8 ⊢ (exp ↾ (log “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
38	35, 37	eqtr4i 2768	. . . . . . 7 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39		f1oeq1 6836	. . . . . . 7 ⊢ (^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷)) → (^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷))
40	38, 39	ax-mp 5	. . . . . 6 ⊢ (^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷)
41	16, 40	mpbi 230	. . . . 5 ⊢ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷
42	41	a1i 11	. . . 4 ⊢ (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷)
43	38	cnveqi 5885	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = ^◡(exp ↾ (log “ 𝐷))
44		relres 6023	. . . . . . 7 ⊢ Rel (log ↾ 𝐷)
45		dfrel2 6209	. . . . . . 7 ⊢ (Rel (log ↾ 𝐷) ↔ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷))
46	44, 45	mpbi 230	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷)
47	43, 46	eqtr3i 2767	. . . . 5 ⊢ ^◡(exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48		f1of 6848	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
49	14, 48	mp1i 13	. . . . . 6 ⊢ (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
50		imassrn 6089	. . . . . . . 8 ⊢ (log “ 𝐷) ⊆ ran log
51		logrncn 26604	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
52	51	ssriv 3987	. . . . . . . 8 ⊢ ran log ⊆ ℂ
53	50, 52	sstri 3993	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ ℂ
54	6	logcn 26689	. . . . . . 7 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
55		cncfcdm 24924	. . . . . . 7 ⊢ (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
56	53, 54, 55	mp2an 692	. . . . . 6 ⊢ ((log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
57	49, 56	sylibr 234	. . . . 5 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)))
58	47, 57	eqeltrid 2845	. . . 4 ⊢ (⊤ → ^◡(exp ↾ (log “ 𝐷)) ∈ (𝐷–cn→(log “ 𝐷)))
59		ssid 4006	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
60	1, 3	dvres 25946	. . . . . . . . 9 ⊢ (((ℂ ⊆ ℂ ∧ exp:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ (log “ 𝐷) ⊆ ℂ)) → (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))))
61	59, 20, 59, 53, 60	mp4an 693	. . . . . . . 8 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)))
62		dvef 26018	. . . . . . . . 9 ⊢ (ℂ D exp) = exp
63	1	cnfldtop 24804	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
64	6	dvloglem 26690	. . . . . . . . . 10 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)
65		isopn3i 23090	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷))
66	63, 64, 65	mp2an 692	. . . . . . . . 9 ⊢ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷)
67	62, 66	reseq12i 5995	. . . . . . . 8 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
68	61, 67	eqtri 2765	. . . . . . 7 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
69	68	dmeqi 5915	. . . . . 6 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70		dmres 6030	. . . . . 6 ⊢ dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
71	20	fdmi 6747	. . . . . . . 8 ⊢ dom exp = ℂ
72	53, 71	sseqtrri 4033	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ dom exp
73		dfss2 3969	. . . . . . 7 ⊢ ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
74	72, 73	mpbi 230	. . . . . 6 ⊢ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
75	69, 70, 74	3eqtri 2769	. . . . 5 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
76	75	a1i 11	. . . 4 ⊢ (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77		neirr 2949	. . . . . 6 ⊢ ¬ 0 ≠ 0
78		resss 6019	. . . . . . . . . . . . 13 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
79	61, 78	eqsstri 4030	. . . . . . . . . . . 12 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
80	79, 62	sseqtri 4032	. . . . . . . . . . 11 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
81	80	rnssi 5951	. . . . . . . . . 10 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82		eff2 16135	. . . . . . . . . . 11 ⊢ exp:ℂ⟶(ℂ ∖ {0})
83		frn 6743	. . . . . . . . . . 11 ⊢ (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
84	82, 83	ax-mp 5	. . . . . . . . . 10 ⊢ ran exp ⊆ (ℂ ∖ {0})
85	81, 84	sstri 3993	. . . . . . . . 9 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
86	85	sseli 3979	. . . . . . . 8 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87		eldifsn 4786	. . . . . . . 8 ⊢ (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
88	86, 87	sylib 218	. . . . . . 7 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
89	88	simprd 495	. . . . . 6 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
90	77, 89	mto 197	. . . . 5 ⊢ ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷)))
91	90	a1i 11	. . . 4 ⊢ (⊤ → ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))))
92	1, 3, 5, 8, 42, 58, 76, 91	dvcnv 26015	. . 3 ⊢ (⊤ → (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))))
93	92	mptru 1547	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))))
94	47	oveq2i 7442	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
95	68	fveq1i 6907	. . . . 5 ⊢ ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))
96		f1ocnvfv2 7297	. . . . . 6 ⊢ (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷 ∧ 𝑥 ∈ 𝐷) → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
97	41, 96	mpan 690	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
98	95, 97	eqtrid 2789	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
99	98	oveq2d 7447	. . 3 ⊢ (𝑥 ∈ 𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
100	99	mpteq2ia 5245	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
101	93, 94, 100	3eqtr3i 2773	1 ⊢ (ℂ D (log ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 {csn 4626 {cpr 4628 ↦ cmpt 5225 ^◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Rel wrel 5690 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 -∞cmnf 11293 -cneg 11493 / cdiv 11920 (,]cioc 13388 ℑcim 15137 expce 16097 πcpi 16102 TopOpenctopn 17466 ℂ_fldccnfld 21364 Topctop 22899 intcnt 23025 –cn→ccncf 24902 D cdv 25898 logclog 26596

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598

This theorem is referenced by: dvlog2 26695 dvcncxp1 26785 dvatan 26978 lgamgulmlem2 27073 dvasin 37711 readvrec2 42391 readvrec 42392

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