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Theorem dvlog 26825

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 24948	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
3	2	toponrestid 23087	. . . 4 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
4		cnelprrecn 11197	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
5	4	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
6		logcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
7	6	logdmopn 26823	. . . . 5 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
8	7	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
9		logf1o 26738	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
10		f1of1 6819	. . . . . . . . 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 26816	. . . . . . . 8 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
13		f1ores 6835	. . . . . . . 8 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
14	11, 12, 13	mp2an 704	. . . . . . 7 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
15		f1ocnv 6833	. . . . . . 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 26730	. . . . . . . . . . 11 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
18	17	reseq1i 5974	. . . . . . . . . 10 ⊢ (log ↾ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
19	18	cnveqi 5860	. . . . . . . . 9 ⊢ ^◡(log ↾ 𝐷) = ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
20		eff 16139	. . . . . . . . . . 11 ⊢ exp:ℂ⟶ℂ
21		cnvimass 6084	. . . . . . . . . . . 12 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ dom ℑ
22		imf 15169	. . . . . . . . . . . . 13 ⊢ ℑ:ℂ⟶ℝ
23	22	fdmi 6717	. . . . . . . . . . . 12 ⊢ dom ℑ = ℂ
24	21, 23	sseqtri 3985	. . . . . . . . . . 11 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ ℂ
25		fssres 6744	. . . . . . . . . . 11 ⊢ ((exp:ℂ⟶ℂ ∧ (^◡ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ)
26	20, 24, 25	mp2an 704	. . . . . . . . . 10 ⊢ (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ
27		ffun 6708	. . . . . . . . . 10 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (^◡ℑ “ (-π(,]π))))
28		funcnvres2 6616	. . . . . . . . . 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 6084	. . . . . . . . . . 11 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (^◡ℑ “ (-π(,]π)))
31	26	fdmi 6717	. . . . . . . . . . 11 ⊢ dom (exp ↾ (^◡ℑ “ (-π(,]π))) = (^◡ℑ “ (-π(,]π))
32	30, 31	sseqtri 3985	. . . . . . . . . 10 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π))
33		resabs1 6005	. . . . . . . . . 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 6059	. . . . . . . . 9 ⊢ (log “ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)
37	36	reseq2i 5975	. . . . . . . 8 ⊢ (exp ↾ (log “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
38	35, 37	eqtr4i 2789	. . . . . . 7 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
39		f1oeq1 6808	. . . . . . 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 233	. . . . 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 5860	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = ^◡(exp ↾ (log “ 𝐷))
44		relres 6004	. . . . . . 7 ⊢ Rel (log ↾ 𝐷)
45		dfrel2 6187	. . . . . . 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 6820	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
49	14, 48	mp1i 14	. . . . . 6 ⊢ (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
50		imassrn 6073	. . . . . . . 8 ⊢ (log “ 𝐷) ⊆ ran log
51		logrncn 26736	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
52	51	ssriv 3941	. . . . . . . 8 ⊢ ran log ⊆ ℂ
53	50, 52	sstri 3946	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ ℂ
54	6	logcn 26821	. . . . . . 7 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
55		cncfcdm 25066	. . . . . . 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 704	. . . . . 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 2867	. . . 4 ⊢ (⊤ → ^◡(exp ↾ (log “ 𝐷)) ∈ (𝐷–cn→(log “ 𝐷)))
59		ssid 3959	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
60	1, 3	dvres 26079	. . . . . . . . 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 705	. . . . . . . 8 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)))
62		dvef 26148	. . . . . . . . 9 ⊢ (ℂ D exp) = exp
63	1	cnfldtop 24949	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
64	6	dvloglem 26822	. . . . . . . . . 10 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)
65		isopn3i 23248	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷))
66	63, 64, 65	mp2an 704	. . . . . . . . 9 ⊢ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷)
67	62, 66	reseq12i 5976	. . . . . . . 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 5894	. . . . . 6 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
70		dmres 6011	. . . . . 6 ⊢ dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
71	20	fdmi 6717	. . . . . . . 8 ⊢ dom exp = ℂ
72	53, 71	sseqtrri 3986	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ dom exp
73		dfss2 3923	. . . . . . 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 2967	. . . . . 6 ⊢ ¬ 0 ≠ 0
78		resss 6000	. . . . . . . . . . . . 13 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
79	61, 78	eqsstri 3983	. . . . . . . . . . . 12 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
80	79, 62	sseqtri 3985	. . . . . . . . . . 11 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
81	80	rnssi 5930	. . . . . . . . . 10 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
82		eff2 16159	. . . . . . . . . . 11 ⊢ exp:ℂ⟶(ℂ ∖ {0})
83		frn 6713	. . . . . . . . . . 11 ⊢ (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
84	82, 83	ax-mp 5	. . . . . . . . . 10 ⊢ ran exp ⊆ (ℂ ∖ {0})
85	81, 84	sstri 3946	. . . . . . . . 9 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
86	85	sseli 3933	. . . . . . . 8 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
87		eldifsn 4753	. . . . . . . 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 500	. . . . . 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 26145	. . 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 7421	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
95	68	fveq1i 6882	. . . . 5 ⊢ ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))
96		f1ocnvfv2 7275	. . . . . 6 ⊢ (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷 ∧ 𝑥 ∈ 𝐷) → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
97	41, 96	mpan 702	. . . . 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 7426	. . 3 ⊢ (𝑥 ∈ 𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
100	99	mpteq2ia 5206	. 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 400 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3902 ∩ cin 3904 ⊆ wss 3905 {csn 4589 {cpr 4591 ↦ cmpt 5192 ^◡ccnv 5660 dom cdm 5661 ran crn 5662 ↾ cres 5663 “ cima 5664 Rel wrel 5666 Fun wfun 6530 ⟶wf 6532 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ℂcc 11102 ℝcr 11103 0cc0 11104 1c1 11105 -∞cmnf 11245 -cneg 11446 / cdiv 11875 (,]cioc 13377 ℑcim 15154 expce 16119 πcpi 16124 TopOpenctopn 17478 ℂ_fldccnfld 21531 Topctop 23059 intcnt 23183 –cn→ccncf 25044 D cdv 26031 logclog 26728

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13380 df-ioc 13381 df-ico 13382 df-icc 13383 df-fz 13540 df-fzo 13688 df-fl 13830 df-mod 13908 df-seq 14043 df-exp 14103 df-fac 14315 df-bc 14344 df-hash 14372 df-shft 15109 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-limsup 15527 df-clim 15544 df-rlim 15545 df-sum 15743 df-ef 16125 df-sin 16127 df-cos 16128 df-tan 16129 df-pi 16130 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-xrs 17560 df-qtop 17565 df-imas 17566 df-xps 17568 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-mulg 19138 df-cntz 19391 df-cmn 19856 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-fbas 21528 df-fg 21529 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cld 23185 df-ntr 23186 df-cls 23187 df-nei 23264 df-lp 23302 df-perf 23303 df-cn 23393 df-cnp 23394 df-haus 23481 df-cmp 23553 df-tx 23728 df-hmeo 23921 df-fil 24012 df-fm 24104 df-flim 24105 df-flf 24106 df-xms 24486 df-ms 24487 df-tms 24488 df-cncf 25046 df-limc 26034 df-dv 26035 df-log 26730

This theorem is used by: dvlog2 26827 dvcncxp1 26917 dvatan 27109 lgamgulmlem2 27203 dvasin 38383 readvrec2 43150 readvrec 43151

W3C validator