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Theorem dvlog 24397

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 2622	. . . 4 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtop 22587	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) ∈ Top
3	1	cnfldtopon 22586	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
4	3	toponunii 20721	. . . . . . 7 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
5	4	restid 16094	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld))
6	2, 5	ax-mp 5	. . . . 5 ⊢ ((TopOpen‘ℂ_fld) ↾_t ℂ) = (TopOpen‘ℂ_fld)
7	6	eqcomi 2631	. . . 4 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
8		cnelprrecn 10029	. . . . 5 ⊢ ℂ ∈ {ℝ, ℂ}
9	8	a1i 11	. . . 4 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
10		logcn.d	. . . . . 6 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
11	10	logdmopn 24395	. . . . 5 ⊢ 𝐷 ∈ (TopOpen‘ℂ_fld)
12	11	a1i 11	. . . 4 ⊢ (⊤ → 𝐷 ∈ (TopOpen‘ℂ_fld))
13		logf1o 24311	. . . . . . . . 9 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
14		f1of1 6136	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → log:(ℂ ∖ {0})–1-1→ran log)
15	13, 14	ax-mp 5	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1→ran log
16	10	logdmss 24388	. . . . . . . 8 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
17		f1ores 6151	. . . . . . . 8 ⊢ ((log:(ℂ ∖ {0})–1-1→ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷))
18	15, 16, 17	mp2an 708	. . . . . . 7 ⊢ (log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷)
19		f1ocnv 6149	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) → ^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷)
20	18, 19	ax-mp 5	. . . . . 6 ⊢ ^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷
21		df-log 24303	. . . . . . . . . . 11 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
22	21	reseq1i 5392	. . . . . . . . . 10 ⊢ (log ↾ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
23	22	cnveqi 5297	. . . . . . . . 9 ⊢ ^◡(log ↾ 𝐷) = ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷)
24		eff 14812	. . . . . . . . . . 11 ⊢ exp:ℂ⟶ℂ
25		cnvimass 5485	. . . . . . . . . . . 12 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ dom ℑ
26		imf 13853	. . . . . . . . . . . . 13 ⊢ ℑ:ℂ⟶ℝ
27	26	fdmi 6052	. . . . . . . . . . . 12 ⊢ dom ℑ = ℂ
28	25, 27	sseqtri 3637	. . . . . . . . . . 11 ⊢ (^◡ℑ “ (-π(,]π)) ⊆ ℂ
29		fssres 6070	. . . . . . . . . . 11 ⊢ ((exp:ℂ⟶ℂ ∧ (^◡ℑ “ (-π(,]π)) ⊆ ℂ) → (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ)
30	24, 28, 29	mp2an 708	. . . . . . . . . 10 ⊢ (exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ
31		ffun 6048	. . . . . . . . . 10 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))):(^◡ℑ “ (-π(,]π))⟶ℂ → Fun (exp ↾ (^◡ℑ “ (-π(,]π))))
32		funcnvres2 5969	. . . . . . . . . 10 ⊢ (Fun (exp ↾ (^◡ℑ “ (-π(,]π))) → ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)))
33	30, 31, 32	mp2b 10	. . . . . . . . 9 ⊢ ^◡(^◡(exp ↾ (^◡ℑ “ (-π(,]π))) ↾ 𝐷) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
34		cnvimass 5485	. . . . . . . . . . 11 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ dom (exp ↾ (^◡ℑ “ (-π(,]π)))
35	30	fdmi 6052	. . . . . . . . . . 11 ⊢ dom (exp ↾ (^◡ℑ “ (-π(,]π))) = (^◡ℑ “ (-π(,]π))
36	34, 35	sseqtri 3637	. . . . . . . . . 10 ⊢ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π))
37		resabs1 5427	. . . . . . . . . 10 ⊢ ((^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷) ⊆ (^◡ℑ “ (-π(,]π)) → ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)))
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
39	23, 33, 38	3eqtri 2648	. . . . . . . 8 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
40	21	imaeq1i 5463	. . . . . . . . 9 ⊢ (log “ 𝐷) = (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷)
41	40	reseq2i 5393	. . . . . . . 8 ⊢ (exp ↾ (log “ 𝐷)) = (exp ↾ (^◡(exp ↾ (^◡ℑ “ (-π(,]π))) “ 𝐷))
42	39, 41	eqtr4i 2647	. . . . . . 7 ⊢ ^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷))
43		f1oeq1 6127	. . . . . . 7 ⊢ (^◡(log ↾ 𝐷) = (exp ↾ (log “ 𝐷)) → (^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷))
44	42, 43	ax-mp 5	. . . . . 6 ⊢ (^◡(log ↾ 𝐷):(log “ 𝐷)–1-1-onto→𝐷 ↔ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷)
45	20, 44	mpbi 220	. . . . 5 ⊢ (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷
46	45	a1i 11	. . . 4 ⊢ (⊤ → (exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷)
47	42	cnveqi 5297	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = ^◡(exp ↾ (log “ 𝐷))
48		relres 5426	. . . . . . 7 ⊢ Rel (log ↾ 𝐷)
49		dfrel2 5583	. . . . . . 7 ⊢ (Rel (log ↾ 𝐷) ↔ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷))
50	48, 49	mpbi 220	. . . . . 6 ⊢ ^◡^◡(log ↾ 𝐷) = (log ↾ 𝐷)
51	47, 50	eqtr3i 2646	. . . . 5 ⊢ ^◡(exp ↾ (log “ 𝐷)) = (log ↾ 𝐷)
52		f1of 6137	. . . . . . 7 ⊢ ((log ↾ 𝐷):𝐷–1-1-onto→(log “ 𝐷) → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
53	18, 52	mp1i 13	. . . . . 6 ⊢ (⊤ → (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
54		imassrn 5477	. . . . . . . 8 ⊢ (log “ 𝐷) ⊆ ran log
55		logrncn 24309	. . . . . . . . 9 ⊢ (𝑥 ∈ ran log → 𝑥 ∈ ℂ)
56	55	ssriv 3607	. . . . . . . 8 ⊢ ran log ⊆ ℂ
57	54, 56	sstri 3612	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ ℂ
58	10	logcn 24393	. . . . . . 7 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
59		cncffvrn 22701	. . . . . . 7 ⊢ (((log “ 𝐷) ⊆ ℂ ∧ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)) → ((log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷)))
60	57, 58, 59	mp2an 708	. . . . . 6 ⊢ ((log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)) ↔ (log ↾ 𝐷):𝐷⟶(log “ 𝐷))
61	53, 60	sylibr 224	. . . . 5 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐷–cn→(log “ 𝐷)))
62	51, 61	syl5eqel 2705	. . . 4 ⊢ (⊤ → ^◡(exp ↾ (log “ 𝐷)) ∈ (𝐷–cn→(log “ 𝐷)))
63		ssid 3624	. . . . . . . . 9 ⊢ ℂ ⊆ ℂ
64	1, 7	dvres 23675	. . . . . . . . 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))))
65	63, 24, 63, 57, 64	mp4an 709	. . . . . . . 8 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)))
66		dvef 23743	. . . . . . . . 9 ⊢ (ℂ D exp) = exp
67	10	dvloglem 24394	. . . . . . . . . 10 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)
68		isopn3i 20886	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)) → ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷))
69	2, 67, 68	mp2an 708	. . . . . . . . 9 ⊢ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷)) = (log “ 𝐷)
70	66, 69	reseq12i 5394	. . . . . . . 8 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) = (exp ↾ (log “ 𝐷))
71	65, 70	eqtri 2644	. . . . . . 7 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) = (exp ↾ (log “ 𝐷))
72	71	dmeqi 5325	. . . . . 6 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = dom (exp ↾ (log “ 𝐷))
73		dmres 5419	. . . . . 6 ⊢ dom (exp ↾ (log “ 𝐷)) = ((log “ 𝐷) ∩ dom exp)
74	24	fdmi 6052	. . . . . . . 8 ⊢ dom exp = ℂ
75	57, 74	sseqtr4i 3638	. . . . . . 7 ⊢ (log “ 𝐷) ⊆ dom exp
76		df-ss 3588	. . . . . . 7 ⊢ ((log “ 𝐷) ⊆ dom exp ↔ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷))
77	75, 76	mpbi 220	. . . . . 6 ⊢ ((log “ 𝐷) ∩ dom exp) = (log “ 𝐷)
78	72, 73, 77	3eqtri 2648	. . . . 5 ⊢ dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷)
79	78	a1i 11	. . . 4 ⊢ (⊤ → dom (ℂ D (exp ↾ (log “ 𝐷))) = (log “ 𝐷))
80		neirr 2803	. . . . . 6 ⊢ ¬ 0 ≠ 0
81		resss 5422	. . . . . . . . . . . . 13 ⊢ ((ℂ D exp) ↾ ((int‘(TopOpen‘ℂ_fld))‘(log “ 𝐷))) ⊆ (ℂ D exp)
82	65, 81	eqsstri 3635	. . . . . . . . . . . 12 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ D exp)
83	82, 66	sseqtri 3637	. . . . . . . . . . 11 ⊢ (ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp
84		rnss 5354	. . . . . . . . . . 11 ⊢ ((ℂ D (exp ↾ (log “ 𝐷))) ⊆ exp → ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp)
85	83, 84	ax-mp 5	. . . . . . . . . 10 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ ran exp
86		eff2 14829	. . . . . . . . . . 11 ⊢ exp:ℂ⟶(ℂ ∖ {0})
87		frn 6053	. . . . . . . . . . 11 ⊢ (exp:ℂ⟶(ℂ ∖ {0}) → ran exp ⊆ (ℂ ∖ {0}))
88	86, 87	ax-mp 5	. . . . . . . . . 10 ⊢ ran exp ⊆ (ℂ ∖ {0})
89	85, 88	sstri 3612	. . . . . . . . 9 ⊢ ran (ℂ D (exp ↾ (log “ 𝐷))) ⊆ (ℂ ∖ {0})
90	89	sseli 3599	. . . . . . . 8 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ∈ (ℂ ∖ {0}))
91		eldifsn 4317	. . . . . . . 8 ⊢ (0 ∈ (ℂ ∖ {0}) ↔ (0 ∈ ℂ ∧ 0 ≠ 0))
92	90, 91	sylib 208	. . . . . . 7 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → (0 ∈ ℂ ∧ 0 ≠ 0))
93	92	simprd 479	. . . . . 6 ⊢ (0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))) → 0 ≠ 0)
94	80, 93	mto 188	. . . . 5 ⊢ ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷)))
95	94	a1i 11	. . . 4 ⊢ (⊤ → ¬ 0 ∈ ran (ℂ D (exp ↾ (log “ 𝐷))))
96	1, 7, 9, 12, 46, 62, 79, 95	dvcnv 23740	. . 3 ⊢ (⊤ → (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))))
97	96	trud 1493	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))))
98	51	oveq2i 6661	. 2 ⊢ (ℂ D ^◡(exp ↾ (log “ 𝐷))) = (ℂ D (log ↾ 𝐷))
99	71	fveq1i 6192	. . . . 5 ⊢ ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))
100		f1ocnvfv2 6533	. . . . . 6 ⊢ (((exp ↾ (log “ 𝐷)):(log “ 𝐷)–1-1-onto→𝐷 ∧ 𝑥 ∈ 𝐷) → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
101	45, 100	mpan 706	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → ((exp ↾ (log “ 𝐷))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
102	99, 101	syl5eq 2668	. . . 4 ⊢ (𝑥 ∈ 𝐷 → ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)) = 𝑥)
103	102	oveq2d 6666	. . 3 ⊢ (𝑥 ∈ 𝐷 → (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥))) = (1 / 𝑥))
104	103	mpteq2ia 4740	. 2 ⊢ (𝑥 ∈ 𝐷 ↦ (1 / ((ℂ D (exp ↾ (log “ 𝐷)))‘(^◡(exp ↾ (log “ 𝐷))‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
105	97, 98, 104	3eqtr3i 2652	1 ⊢ (ℂ D (log ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 {csn 4177 {cpr 4179 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 Rel wrel 5119 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 -∞cmnf 10072 -cneg 10267 / cdiv 10684 (,]cioc 12176 ℑcim 13838 expce 14792 πcpi 14797 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 Topctop 20698 intcnt 20821 –cn→ccncf 22679 D cdv 23627 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: dvlog2 24399 dvcncxp1 24484 dvatan 24662 lgamgulmlem2 24756 dvasin 33496

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