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Theorem dvlog 26585

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 2731	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtopon 24695	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
3	2	toponrestid 22834	. . . 4 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
4		cnelprrecn 11096	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
6		logcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
7	6	logdmopn 26583	. . . . 5 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
8	7	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
9		logf1o 26498	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
10		f1of1 6762	. . . . . . . . 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 26576	. . . . . . . 8 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
13		f1ores 6777	. . . . . . . 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 6775	. . . . . . 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 26490	. . . . . . . . . . 11 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
18	17	reseq1i 5924	. . . . . . . . . 10 ⊢ (log ↾ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
19	18	cnveqi 5814	. . . . . . . . 9 ⊢ ^◡(log ↾ 𝐷) = ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
20		eff 15985	. . . . . . . . . . 11 ⊢ exp:ℂ⟶ℂ
21		cnvimass 6031	. . . . . . . . . . . 12 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ dom ℑ
22		imf 15017	. . . . . . . . . . . . 13 ⊢ ℑ:ℂ⟶ℝ
23	22	fdmi 6662	. . . . . . . . . . . 12 ⊢ dom ℑ = ℂ
24	21, 23	sseqtri 3983	. . . . . . . . . . 11 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ ℂ
25		fssres 6689	. . . . . . . . . . 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 6654	. . . . . . . . . 10 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (^◡ℑ “ (-π(,]π))))
28		funcnvres2 6561	. . . . . . . . . 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 6031	. . . . . . . . . . 11 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (^◡ℑ “ (-π(,]π)))
31	26	fdmi 6662	. . . . . . . . . . 11 ⊢ dom (exp ↾ (^◡ℑ “ (-π(,]π))) = (^◡ℑ “ (-π(,]π))
32	30, 31	sseqtri 3983	. . . . . . . . . 10 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π))
33		resabs1 5955	. . . . . . . . . 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 2758	. . . . . . . 8 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
36	17	imaeq1i 6006	. . . . . . . . 9 ⊢ (log “ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)
37	36	reseq2i 5925	. . . . . . . 8 ⊢ (exp ↾ (log “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
38	35, 37	eqtr4i 2757	. . . . . . 7 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39		f1oeq1 6751	. . . . . . 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 5814	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = ^◡(exp ↾ (log “ 𝐷))
44		relres 5954	. . . . . . 7 ⊢ Rel (log ↾ 𝐷)
45		dfrel2 6136	. . . . . . 7 ⊢ (Rel (log ↾ 𝐷) ↔ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷))
46	44, 45	mpbi 230	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷)
47	43, 46	eqtr3i 2756	. . . . 5 ⊢ ^◡(exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48		f1of 6763	. . . . . . 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 6020	. . . . . . . 8 ⊢ (log “ 𝐷) ⊆ ran log
51		logrncn 26496	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
52	51	ssriv 3938	. . . . . . . 8 ⊢ ran log ⊆ ℂ
53	50, 52	sstri 3944	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ ℂ
54	6	logcn 26581	. . . . . . 7 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
55		cncfcdm 24816	. . . . . . 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 2835	. . . 4 ⊢ (⊤ → ^◡(exp ↾ (log “ 𝐷)) ∈ (𝐷–cn→(log “ 𝐷)))
59		ssid 3957	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
60	1, 3	dvres 25837	. . . . . . . . 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 25909	. . . . . . . . 9 ⊢ (ℂ D exp) = exp
63	1	cnfldtop 24696	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
64	6	dvloglem 26582	. . . . . . . . . 10 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)
65		isopn3i 22995	. . . . . . . . . 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 5926	. . . . . . . 8 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
68	61, 67	eqtri 2754	. . . . . . 7 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
69	68	dmeqi 5844	. . . . . 6 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70		dmres 5961	. . . . . 6 ⊢ dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
71	20	fdmi 6662	. . . . . . . 8 ⊢ dom exp = ℂ
72	53, 71	sseqtrri 3984	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ dom exp
73		dfss2 3920	. . . . . . 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 2758	. . . . 5 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
76	75	a1i 11	. . . 4 ⊢ (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77		neirr 2937	. . . . . 6 ⊢ ¬ 0 ≠ 0
78		resss 5950	. . . . . . . . . . . . 13 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
79	61, 78	eqsstri 3981	. . . . . . . . . . . 12 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
80	79, 62	sseqtri 3983	. . . . . . . . . . 11 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
81	80	rnssi 5880	. . . . . . . . . 10 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82		eff2 16005	. . . . . . . . . . 11 ⊢ exp:ℂ⟶(ℂ ∖ {0})
83		frn 6658	. . . . . . . . . . 11 ⊢ (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
84	82, 83	ax-mp 5	. . . . . . . . . 10 ⊢ ran exp ⊆ (ℂ ∖ {0})
85	81, 84	sstri 3944	. . . . . . . . 9 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
86	85	sseli 3930	. . . . . . . 8 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87		eldifsn 4738	. . . . . . . 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 25906	. . 3 ⊢ (⊤ → (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))))
93	92	mptru 1548	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))))
94	47	oveq2i 7357	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
95	68	fveq1i 6823	. . . . 5 ⊢ ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))
96		f1ocnvfv2 7211	. . . . . 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 2778	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
99	98	oveq2d 7362	. . 3 ⊢ (𝑥 ∈ 𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
100	99	mpteq2ia 5186	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
101	93, 94, 100	3eqtr3i 2762	1 ⊢ (ℂ D (log ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 {csn 4576 {cpr 4578 ↦ cmpt 5172 ^◡ccnv 5615 dom cdm 5616 ran crn 5617 ↾ cres 5618 “ cima 5619 Rel wrel 5621 Fun wfun 6475 ⟶wf 6477 –1-1→wf1 6478 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ℂcc 11001 ℝcr 11002 0cc0 11003 1c1 11004 -∞cmnf 11141 -cneg 11342 / cdiv 11771 (,]cioc 13243 ℑcim 15002 expce 15965 πcpi 15970 TopOpenctopn 17322 ℂ_fldccnfld 21289 Topctop 22806 intcnt 22930 –cn→ccncf 24794 D cdv 25789 logclog 26488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ioc 13247 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-fac 14178 df-bc 14207 df-hash 14235 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-sum 15591 df-ef 15971 df-sin 15973 df-cos 15974 df-tan 15975 df-pi 15976 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-cmp 23300 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793 df-log 26490

This theorem is referenced by: dvlog2 26587 dvcncxp1 26677 dvatan 26870 lgamgulmlem2 26965 dvasin 37743 readvrec2 42393 readvrec 42394

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