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Theorem dvlog 27009

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 2763	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtopon 25131	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
3	2	toponrestid 23269	. . . 4 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
4		cnelprrecn 11320	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
6		logcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
7	6	logdmopn 27007	. . . . 5 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
8	7	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
9		logf1o 26922	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})⟶1-1ontoran log
10		f1of1 6845	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})⟶1-1ontoran log → log:(ℂ ∖ {0})⟶1-1ran log)
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})⟶1-1ran log
12	6	logdmss 27000	. . . . . . . 8 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
13		f1ores 6861	. . . . . . . 8 ⊢ ((log:(ℂ ∖ {0})⟶1-1ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷⟶1-1onto(log “ 𝐷))
14	11, 12, 13	mp2an 705	. . . . . . 7 ⊢ (log ↾ 𝐷):𝐷⟶1-1onto(log “ 𝐷)
15		f1ocnv 6859	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷⟶1-1onto(log “ 𝐷) → ^◡(log ↾ 𝐷):(log “ 𝐷)⟶1-1onto𝐷)
16	14, 15	ax-mp 5	. . . . . 6 ⊢ ^◡(log ↾ 𝐷):(log “ 𝐷)⟶1-1onto𝐷
17		df-log 26914	. . . . . . . . . . 11 ⊢ log = ^◡(exp ↾ (^◡ℑ “ ( -π(,]π)))
18	17	reseq1i 5977	. . . . . . . . . 10 ⊢ (log ↾ 𝐷) = (^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) ↾ 𝐷)
19	18	cnveqi 5859	. . . . . . . . 9 ⊢ ^◡(log ↾ 𝐷) = ^◡(^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) ↾ 𝐷)
20		eff 16276	. . . . . . . . . . 11 ⊢ exp:ℂ⟶ℂ
21		cnvimass 6211	. . . . . . . . . . . 12 ⊢ (^◡ℑ “ ( -π(,]π)) ⊆ dom ℑ
22		imf 15309	. . . . . . . . . . . . 13 ⊢ ℑ:ℂ⟶ℝ
23	22	fdmi 6743	. . . . . . . . . . . 12 ⊢ dom ℑ = ℂ
24	21, 23	sseqtri 3961	. . . . . . . . . . 11 ⊢ (^◡ℑ “ ( -π(,]π)) ⊆ ℂ
25		fssres 6770	. . . . . . . . . . 11 ⊢ ((exp:ℂ⟶ℂ ∧ (^◡ℑ “ ( -π(,]π)) ⊆ ℂ) → (exp ↾ (^◡ℑ “ ( -π(,]π))):(^◡ℑ “ ( -π(,]π))⟶ℂ)
26	20, 24, 25	mp2an 705	. . . . . . . . . 10 ⊢ (exp ↾ (^◡ℑ “ ( -π(,]π))):(^◡ℑ “ ( -π(,]π))⟶ℂ
27		ffun 6734	. . . . . . . . . 10 ⊢ ((exp ↾ (^◡ℑ “ ( -π(,]π))):(^◡ℑ “ ( -π(,]π))⟶ℂ → Fun (exp ↾ (^◡ℑ “ ( -π(,]π))))
28		funcnvres2 6642	. . . . . . . . . 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 6211	. . . . . . . . . . 11 ⊢ (^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (^◡ℑ “ ( -π(,]π)))
31	26	fdmi 6743	. . . . . . . . . . 11 ⊢ dom (exp ↾ (^◡ℑ “ ( -π(,]π))) = (^◡ℑ “ ( -π(,]π))
32	30, 31	sseqtri 3961	. . . . . . . . . 10 ⊢ (^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ ( -π(,]π))
33		resabs1 6009	. . . . . . . . . 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 2790	. . . . . . . 8 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) “ 𝐷))
36	17	imaeq1i 6061	. . . . . . . . 9 ⊢ (log “ 𝐷) = (^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) “ 𝐷)
37	36	reseq2i 5978	. . . . . . . 8 ⊢ (exp ↾ (log “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ ( -π(,]π))) “ 𝐷))
38	35, 37	eqtr4i 2789	. . . . . . 7 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39		f1oeq1 6834	. . . . . . 7 ⊢ (^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷)) → (^◡(log ↾ 𝐷):(log “ 𝐷)⟶1-1onto𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)⟶1-1onto𝐷))
40	38, 39	ax-mp 5	. . . . . 6 ⊢ (^◡(log ↾ 𝐷):(log “ 𝐷)⟶1-1onto𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)⟶1-1onto𝐷)
41	16, 40	mpbi 233	. . . . 5 ⊢ (exp ↾ (log “ 𝐷)):(log “ 𝐷)⟶1-1onto𝐷
42	41	a1i 11	. . . 4 ⊢ (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)⟶1-1onto𝐷)
43	38	cnveqi 5859	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = ^◡(exp ↾ (log “ 𝐷))
44		relres 6007	. . . . . . 7 ⊢ Rel (log ↾ 𝐷)
45		dfrel2 6193	. . . . . . 7 ⊢ (Rel (log ↾ 𝐷) ↔ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷))
46	44, 45	mpbi 233	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷)
47	43, 46	eqtr3i 2788	. . . . 5 ⊢ ^◡(exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
48		f1of 6846	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷⟶1-1onto(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
49	14, 48	mp1i 14	. . . . . 6 ⊢ (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
50		imassrn 6210	. . . . . . . 8 ⊢ (log “ 𝐷) ⊆ ran log
51		logrncn 26920	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
52	51	ssriv 3917	. . . . . . . 8 ⊢ ran log ⊆ ℂ
53	50, 52	sstri 3922	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ ℂ
54	6	logcn 27005	. . . . . . 7 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
55		cncfcdm 25249	. . . . . . 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 705	. . . . . 6 ⊢ ((log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
57	49, 56	sylibr 237	. . . . 5 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)))
58	47, 57	eqeltrid 2868	. . . 4 ⊢ (⊤ → ^◡(exp ↾ (log “ 𝐷)) ∈ (𝐷–cn→(log “ 𝐷)))
59		ssid 3935	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
60	1, 3	dvres 26261	. . . . . . . . 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 706	. . . . . . . 8 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)))
62		dvef 26330	. . . . . . . . 9 ⊢ (ℂ D exp) = exp
63	1	cnfldtop 25132	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
64	6	dvloglem 27006	. . . . . . . . . 10 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)
65		isopn3i 23430	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷))
66	63, 64, 65	mp2an 705	. . . . . . . . 9 ⊢ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷)
67	62, 66	reseq12i 5979	. . . . . . . 8 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
68	61, 67	eqtri 2786	. . . . . . 7 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
69	68	dmeqi 5893	. . . . . 6 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70		dmres 6015	. . . . . 6 ⊢ dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
71	20	fdmi 6743	. . . . . . . 8 ⊢ dom exp = ℂ
72	53, 71	sseqtrri 3962	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ dom exp
73		dfss2 4097	. . . . . . 7 ⊢ ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
74	72, 73	mpbi 233	. . . . . 6 ⊢ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
75	69, 70, 74	3eqtri 2790	. . . . 5 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
76	75	a1i 11	. . . 4 ⊢ (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
77		neirr 2968	. . . . . 6 ⊢ ¬ 0 ≠ 0
78		resss 6003	. . . . . . . . . . . . 13 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
79	61, 78	eqsstri 3959	. . . . . . . . . . . 12 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
80	79, 62	sseqtri 3961	. . . . . . . . . . 11 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
81	80	rnssi 5929	. . . . . . . . . 10 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82		eff2 16296	. . . . . . . . . . 11 ⊢ exp:ℂ⟶(ℂ ∖ {0})
83		frn 6739	. . . . . . . . . . 11 ⊢ (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
84	82, 83	ax-mp 5	. . . . . . . . . 10 ⊢ ran exp ⊆ (ℂ ∖ {0})
85	81, 84	sstri 3922	. . . . . . . . 9 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
86	85	sseli 3909	. . . . . . . 8 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87		eldifsn 4748	. . . . . . . 8 ⊢ (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
88	86, 87	sylib 221	. . . . . . 7 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
89	88	simprd 501	. . . . . 6 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
90	77, 89	mto 200	. . . . 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 26327	. . 3 ⊢ (⊤ → (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))))
93	92	mptru 1577	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))))
94	47	oveq2i 7454	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
95	68	fveq1i 6909	. . . . 5 ⊢ ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))
96		f1ocnvfv2 7308	. . . . . 6 ⊢ (((exp ↾ (log “ 𝐷)):(log “ 𝐷)⟶1-1onto𝐷 ∧ 𝑥 ∈ 𝐷) → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
97	41, 96	mpan 703	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
98	95, 97	eqtrid 2810	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
99	98	oveq2d 7459	. . 3 ⊢ (𝑥 ∈ 𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
100	99	mpteq2ia 5203	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
101	93, 94, 100	3eqtr3i 2794	1 ⊢ (ℂ D (log ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2147 ≠ wne 2959 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 {csn 4588 {cpr 4590 ↦ cmpt 5189 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 Rel wrel 5660 Fun wfun 6552 ⟶wf 6554 ⟶1-1wf1 6555 ⟶1-1ontowf1o 6557 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 -∞cmnf 11368 -cneg 11569 / cdiv 12000 (,]cioc 13504 ℑcim 15294 expce 16256 πcpi 16261 TopOpenctopn 17621 ℂ_fldccnfld 21708 Topctop 23241 intcnt 23365 –cn→ccncf 25227 D cdv 26213 logclog 26912

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 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306

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 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ioc 13508 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-mod 14037 df-seq 14172 df-exp 14232 df-fac 14445 df-bc 14474 df-hash 14502 df-shft 15249 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-limsup 15667 df-clim 15684 df-rlim 15685 df-sum 15883 df-ef 16262 df-sin 16264 df-cos 16265 df-tan 16266 df-pi 16267 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-submnd 19008 df-mulg 19307 df-cntz 19561 df-cmn 20026 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-fbas 21705 df-fg 21706 df-cnfld 21709 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cld 23367 df-ntr 23368 df-cls 23369 df-nei 23446 df-lp 23484 df-perf 23485 df-cn 23575 df-cnp 23576 df-haus 23663 df-cmp 23735 df-tx 23911 df-hmeo 24104 df-fil 24195 df-fm 24287 df-flim 24288 df-flf 24289 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229 df-limc 26216 df-dv 26217 df-log 26914

This theorem is used by: dvlog2 27011 dvcncxp1 27101 dvatan 27293 lgamgulmlem2 27387 dvasin 38814 readvrec2 43605 readvrec 43606

W3C validator