dvlog - Metamath Proof Explorer

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

Theorem dvlog 26628

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 24757	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
3	2	toponrestid 22896	. . . 4 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
4		cnelprrecn 11122	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
6		logcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
7	6	logdmopn 26626	. . . . 5 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
8	7	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
9		logf1o 26541	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
10		f1of1 6773	. . . . . . . . 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 26619	. . . . . . . 8 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
13		f1ores 6788	. . . . . . . 8 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
14	11, 12, 13	mp2an 693	. . . . . . 7 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
15		f1ocnv 6786	. . . . . . 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 26533	. . . . . . . . . . 11 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
18	17	reseq1i 5934	. . . . . . . . . 10 ⊢ (log ↾ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
19	18	cnveqi 5823	. . . . . . . . 9 ⊢ ^◡(log ↾ 𝐷) = ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
20		eff 16037	. . . . . . . . . . 11 ⊢ exp:ℂ⟶ℂ
21		cnvimass 6041	. . . . . . . . . . . 12 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ dom ℑ
22		imf 15066	. . . . . . . . . . . . 13 ⊢ ℑ:ℂ⟶ℝ
23	22	fdmi 6673	. . . . . . . . . . . 12 ⊢ dom ℑ = ℂ
24	21, 23	sseqtri 3971	. . . . . . . . . . 11 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ ℂ
25		fssres 6700	. . . . . . . . . . 11 ⊢ ((exp:ℂ⟶ℂ ∧ (^◡ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ)
26	20, 24, 25	mp2an 693	. . . . . . . . . 10 ⊢ (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ
27		ffun 6665	. . . . . . . . . 10 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (^◡ℑ “ (-π(,]π))))
28		funcnvres2 6572	. . . . . . . . . 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 6041	. . . . . . . . . . 11 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (^◡ℑ “ (-π(,]π)))
31	26	fdmi 6673	. . . . . . . . . . 11 ⊢ dom (exp ↾ (^◡ℑ “ (-π(,]π))) = (^◡ℑ “ (-π(,]π))
32	30, 31	sseqtri 3971	. . . . . . . . . 10 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π))
33		resabs1 5965	. . . . . . . . . 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 2764	. . . . . . . 8 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
36	17	imaeq1i 6016	. . . . . . . . 9 ⊢ (log “ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)
37	36	reseq2i 5935	. . . . . . . 8 ⊢ (exp ↾ (log “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
38	35, 37	eqtr4i 2763	. . . . . . 7 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39		f1oeq1 6762	. . . . . . 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 5823	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = ^◡(exp ↾ (log “ 𝐷))
44		relres 5964	. . . . . . 7 ⊢ Rel (log ↾ 𝐷)
45		dfrel2 6147	. . . . . . 7 ⊢ (Rel (log ↾ 𝐷) ↔ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷))
46	44, 45	mpbi 230	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷)
47	43, 46	eqtr3i 2762	. . . . 5 ⊢ ^◡(exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48		f1of 6774	. . . . . . 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 6030	. . . . . . . 8 ⊢ (log “ 𝐷) ⊆ ran log
51		logrncn 26539	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
52	51	ssriv 3926	. . . . . . . 8 ⊢ ran log ⊆ ℂ
53	50, 52	sstri 3932	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ ℂ
54	6	logcn 26624	. . . . . . 7 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
55		cncfcdm 24875	. . . . . . 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 693	. . . . . 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 2841	. . . 4 ⊢ (⊤ → ^◡(exp ↾ (log “ 𝐷)) ∈ (𝐷–cn→(log “ 𝐷)))
59		ssid 3945	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
60	1, 3	dvres 25888	. . . . . . . . 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 694	. . . . . . . 8 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)))
62		dvef 25957	. . . . . . . . 9 ⊢ (ℂ D exp) = exp
63	1	cnfldtop 24758	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
64	6	dvloglem 26625	. . . . . . . . . 10 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)
65		isopn3i 23057	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷))
66	63, 64, 65	mp2an 693	. . . . . . . . 9 ⊢ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷)
67	62, 66	reseq12i 5936	. . . . . . . 8 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
68	61, 67	eqtri 2760	. . . . . . 7 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
69	68	dmeqi 5853	. . . . . 6 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70		dmres 5971	. . . . . 6 ⊢ dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
71	20	fdmi 6673	. . . . . . . 8 ⊢ dom exp = ℂ
72	53, 71	sseqtrri 3972	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ dom exp
73		dfss2 3908	. . . . . . 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 2764	. . . . 5 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
76	75	a1i 11	. . . 4 ⊢ (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77		neirr 2942	. . . . . 6 ⊢ ¬ 0 ≠ 0
78		resss 5960	. . . . . . . . . . . . 13 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
79	61, 78	eqsstri 3969	. . . . . . . . . . . 12 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
80	79, 62	sseqtri 3971	. . . . . . . . . . 11 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
81	80	rnssi 5889	. . . . . . . . . 10 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82		eff2 16057	. . . . . . . . . . 11 ⊢ exp:ℂ⟶(ℂ ∖ {0})
83		frn 6669	. . . . . . . . . . 11 ⊢ (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
84	82, 83	ax-mp 5	. . . . . . . . . 10 ⊢ ran exp ⊆ (ℂ ∖ {0})
85	81, 84	sstri 3932	. . . . . . . . 9 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
86	85	sseli 3918	. . . . . . . 8 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87		eldifsn 4730	. . . . . . . 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 25954	. . 3 ⊢ (⊤ → (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))))
93	92	mptru 1549	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))))
94	47	oveq2i 7371	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
95	68	fveq1i 6835	. . . . 5 ⊢ ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))
96		f1ocnvfv2 7225	. . . . . 6 ⊢ (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷 ∧ 𝑥 ∈ 𝐷) → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
97	41, 96	mpan 691	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
98	95, 97	eqtrid 2784	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
99	98	oveq2d 7376	. . 3 ⊢ (𝑥 ∈ 𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
100	99	mpteq2ia 5181	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
101	93, 94, 100	3eqtr3i 2768	1 ⊢ (ℂ D (log ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 {csn 4568 {cpr 4570 ↦ cmpt 5167 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 Rel wrel 5629 Fun wfun 6486 ⟶wf 6488 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 -∞cmnf 11168 -cneg 11369 / cdiv 11798 (,]cioc 13290 ℑcim 15051 expce 16017 πcpi 16022 TopOpenctopn 17375 ℂ_fldccnfld 21344 Topctop 22868 intcnt 22992 –cn→ccncf 24853 D cdv 25840 logclog 26531

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-sin 16025 df-cos 16026 df-tan 16027 df-pi 16028 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-log 26533

This theorem is referenced by: dvlog2 26630 dvcncxp1 26720 dvatan 26912 lgamgulmlem2 27007 dvasin 38039 readvrec2 42807 readvrec 42808

W3C validator