Theorem dvloglem 25949

Description: Lemma for dvlog 25952. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypothesis

Ref	Expression
logcn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))

Assertion

Ref	Expression
dvloglem	⊢ (log “ 𝐷) ∈ (TopOpen‘ℂfld)

Proof of Theorem dvloglem

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		logf1o 25866	. . . . . 6 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1ofun 6783	. . . . . 6 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
3	1, 2	ax-mp 5	. . . . 5 ⊢ Fun log
4		logcn.d	. . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
5	4	logdmss 25943	. . . . . 6 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1odm 6785	. . . . . . 7 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → dom log = (ℂ ∖ {0}))
7	1, 6	ax-mp 5	. . . . . 6 ⊢ dom log = (ℂ ∖ {0})
8	5, 7	sseqtrri 3979	. . . . 5 ⊢ 𝐷 ⊆ dom log
9		funimass4 6904	. . . . 5 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (◡ℑ “ (-π(,)π))))
10	3, 8, 9	mp2an 690	. . . 4 ⊢ ((log “ 𝐷) ⊆ (◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (◡ℑ “ (-π(,)π)))
11	4	ellogdm 25940	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)))
12	11	simplbi 498	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
13	4	logdmn0 25941	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
14	12, 13	logcld 25872	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
15	14	imcld 15034	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
16	12, 13	logimcld 25873	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
17	16	simpld 495	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
18		pire 25761	. . . . . . . 8 ⊢ π ∈ ℝ
19	18	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → π ∈ ℝ)
20	16	simprd 496	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
21	4	logdmnrp 25942	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ+)
22		lognegb 25891	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
23	12, 13, 22	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
24	23	necon3bbid 2979	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (¬ -𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) ≠ π))
25	21, 24	mpbid 231	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
26	25	necomd 2997	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
27	15, 19, 20, 26	leneltd 11267	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
28	18	renegcli 11420	. . . . . . . 8 ⊢ -π ∈ ℝ
29	28	rexri 11171	. . . . . . 7 ⊢ -π ∈ ℝ*
30	18	rexri 11171	. . . . . . 7 ⊢ π ∈ ℝ*
31		elioo2 13259	. . . . . . 7 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ*) → ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π)))
32	29, 30, 31	mp2an 690	. . . . . 6 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
33	15, 17, 27, 32	syl3anbrc 1343	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
34		imf 14952	. . . . . 6 ⊢ ℑ:ℂ⟶ℝ
35		ffn 6665	. . . . . 6 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
36		elpreima 7005	. . . . . 6 ⊢ (ℑ Fn ℂ → ((log‘𝑥) ∈ (◡ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π))))
37	34, 35, 36	mp2b 10	. . . . 5 ⊢ ((log‘𝑥) ∈ (◡ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π)))
38	14, 33, 37	sylanbrc 583	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (◡ℑ “ (-π(,)π)))
39	10, 38	mprgbir 3069	. . 3 ⊢ (log “ 𝐷) ⊆ (◡ℑ “ (-π(,)π))
40		df-ioo 13222	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
41		df-ioc 13223	. . . . . . . . . 10 ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)})
42		idd 24	. . . . . . . . . 10 ⊢ ((-π ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → (-π < 𝑤 → -π < 𝑤))
43		xrltle 13022	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ* ∧ π ∈ ℝ*) → (𝑤 < π → 𝑤 ≤ π))
44	40, 41, 42, 43	ixxssixx 13232	. . . . . . . . 9 ⊢ (-π(,)π) ⊆ (-π(,]π)
45		imass2 6052	. . . . . . . . 9 ⊢ ((-π(,)π) ⊆ (-π(,]π) → (◡ℑ “ (-π(,)π)) ⊆ (◡ℑ “ (-π(,]π)))
46	44, 45	ax-mp 5	. . . . . . . 8 ⊢ (◡ℑ “ (-π(,)π)) ⊆ (◡ℑ “ (-π(,]π))
47		logrn 25860	. . . . . . . 8 ⊢ ran log = (◡ℑ “ (-π(,]π))
48	46, 47	sseqtrri 3979	. . . . . . 7 ⊢ (◡ℑ “ (-π(,)π)) ⊆ ran log
49	48	sseli 3938	. . . . . 6 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → 𝑥 ∈ ran log)
50		logef 25883	. . . . . 6 ⊢ (𝑥 ∈ ran log → (log‘(exp‘𝑥)) = 𝑥)
51	49, 50	syl 17	. . . . 5 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (log‘(exp‘𝑥)) = 𝑥)
52		elpreima 7005	. . . . . . . . . 10 ⊢ (ℑ Fn ℂ → (𝑥 ∈ (◡ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π))))
53	34, 35, 52	mp2b 10	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)))
54		efcl 15919	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
55	54	adantr 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
56	53, 55	sylbi 216	. . . . . . . 8 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
57	53	simplbi 498	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → 𝑥 ∈ ℂ)
58	57	imcld 15034	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
59		eliooord 13277	. . . . . . . . . . . 12 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
60	53, 59	simplbiim 505	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
61	60	simprd 496	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) < π)
62	58, 61	ltned 11249	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) ≠ π)
63	51	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (log‘(exp‘𝑥)) = 𝑥)
64	63	fveq2d 6843	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘(log‘(exp‘𝑥))) = (ℑ‘𝑥))
65		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ (-∞(,]0))
66		mnfxr 11170	. . . . . . . . . . . . . . . . 17 ⊢ -∞ ∈ ℝ*
67		0re 11115	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
68		elioc2 13281	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((exp‘𝑥) ∈ (-∞(,]0) ↔ ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0)))
69	66, 67, 68	mp2an 690	. . . . . . . . . . . . . . . 16 ⊢ ((exp‘𝑥) ∈ (-∞(,]0) ↔ ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0))
70	65, 69	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0))
71	70	simp1d 1142	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ ℝ)
72		0red 11116	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ∈ ℝ)
73	70	simp3d 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≤ 0)
74		efne0 15933	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ≠ 0)
75	57, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (exp‘𝑥) ≠ 0)
76	75	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≠ 0)
77	76	necomd 2997	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ≠ (exp‘𝑥))
78	71, 72, 73, 77	leneltd 11267	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) < 0)
79	71, 78	negelrpd 12903	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → -(exp‘𝑥) ∈ ℝ+)
80		lognegb 25891	. . . . . . . . . . . . . . 15 ⊢ (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (-(exp‘𝑥) ∈ ℝ+ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
81	56, 75, 80	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (-(exp‘𝑥) ∈ ℝ+ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
82	81	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (-(exp‘𝑥) ∈ ℝ+ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
83	79, 82	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘(log‘(exp‘𝑥))) = π)
84	64, 83	eqtr3d 2778	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘𝑥) = π)
85	84	ex 413	. . . . . . . . . 10 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → ((exp‘𝑥) ∈ (-∞(,]0) → (ℑ‘𝑥) = π))
86	85	necon3ad 2954	. . . . . . . . 9 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → ((ℑ‘𝑥) ≠ π → ¬ (exp‘𝑥) ∈ (-∞(,]0)))
87	62, 86	mpd 15	. . . . . . . 8 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → ¬ (exp‘𝑥) ∈ (-∞(,]0))
88	56, 87	eldifd 3919	. . . . . . 7 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
89	88, 4	eleqtrrdi 2849	. . . . . 6 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
90		funfvima2 7177	. . . . . . 7 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
91	3, 8, 90	mp2an 690	. . . . . 6 ⊢ ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
92	89, 91	syl 17	. . . . 5 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
93	51, 92	eqeltrrd 2839	. . . 4 ⊢ (𝑥 ∈ (◡ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
94	93	ssriv 3946	. . 3 ⊢ (◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
95	39, 94	eqssi 3958	. 2 ⊢ (log “ 𝐷) = (◡ℑ “ (-π(,)π))
96		imcncf 24212	. . . 4 ⊢ ℑ ∈ (ℂ–cn→ℝ)
97		ssid 3964	. . . . 5 ⊢ ℂ ⊆ ℂ
98		ax-resscn 11066	. . . . 5 ⊢ ℝ ⊆ ℂ
99		eqid 2736	. . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
100	99	cnfldtopon 24092	. . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
101	100	toponrestid 22216	. . . . . 6 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
102	99	tgioo2 24112	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
103	99, 101, 102	cncfcn 24219	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = ((TopOpen‘ℂfld) Cn (topGen‘ran (,))))
104	97, 98, 103	mp2an 690	. . . 4 ⊢ (ℂ–cn→ℝ) = ((TopOpen‘ℂfld) Cn (topGen‘ran (,)))
105	96, 104	eleqtri 2836	. . 3 ⊢ ℑ ∈ ((TopOpen‘ℂfld) Cn (topGen‘ran (,)))
106		iooretop 24075	. . 3 ⊢ (-π(,)π) ∈ (topGen‘ran (,))
107		cnima 22562	. . 3 ⊢ ((ℑ ∈ ((TopOpen‘ℂfld) Cn (topGen‘ran (,))) ∧ (-π(,)π) ∈ (topGen‘ran (,))) → (◡ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂfld))
108	105, 106, 107	mp2an 690	. 2 ⊢ (◡ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂfld)
109	95, 108	eqeltri 2834	1 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂfld)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2941  ∀wral 3062   ∖ cdif 3905   ⊆ wss 3908  {csn 4584   class class class wbr 5103  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  Fun wfun 6487   Fn wfn 6488  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7351  ℂcc 11007  ℝcr 11008  0cc0 11009  -∞cmnf 11145  ℝ^*cxr 11146   < clt 11147   ≤ cle 11148  -cneg 11344  ℝ⁺crp 12869  (,)cioo 13218  (,]cioc 13219  ℑcim 14937  expce 15898  πcpi 15903  TopOpenctopn 17257  topGenctg 17273  ℂ_fldccnfld 20743   Cn ccn 22521  –cn→ccncf 24185  logclog 25856

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-inf2 9535  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088  ax-mulf 11089

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609  df-om 7795  df-1st 7913  df-2nd 7914  df-supp 8085  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-2o 8405  df-er 8606  df-map 8725  df-pm 8726  df-ixp 8794  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-fsupp 9264  df-fi 9305  df-sup 9336  df-inf 9337  df-oi 9404  df-card 9833  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-div 11771  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-q 12828  df-rp 12870  df-xneg 12987  df-xadd 12988  df-xmul 12989  df-ioo 13222  df-ioc 13223  df-ico 13224  df-icc 13225  df-fz 13379  df-fzo 13522  df-fl 13651  df-mod 13729  df-seq 13861  df-exp 13922  df-fac 14128  df-bc 14157  df-hash 14185  df-shft 14906  df-cj 14938  df-re 14939  df-im 14940  df-sqrt 15074  df-abs 15075  df-limsup 15307  df-clim 15324  df-rlim 15325  df-sum 15525  df-ef 15904  df-sin 15906  df-cos 15907  df-pi 15909  df-struct 16973  df-sets 16990  df-slot 17008  df-ndx 17020  df-base 17038  df-ress 17067  df-plusg 17100  df-mulr 17101  df-starv 17102  df-sca 17103  df-vsca 17104  df-ip 17105  df-tset 17106  df-ple 17107  df-ds 17109  df-unif 17110  df-hom 17111  df-cco 17112  df-rest 17258  df-topn 17259  df-0g 17277  df-gsum 17278  df-topgen 17279  df-pt 17280  df-prds 17283  df-xrs 17338  df-qtop 17343  df-imas 17344  df-xps 17346  df-mre 17420  df-mrc 17421  df-acs 17423  df-mgm 18451  df-sgrp 18500  df-mnd 18511  df-submnd 18556  df-mulg 18826  df-cntz 19050  df-cmn 19517  df-psmet 20735  df-xmet 20736  df-met 20737  df-bl 20738  df-mopn 20739  df-fbas 20740  df-fg 20741  df-cnfld 20744  df-top 22189  df-topon 22206  df-topsp 22228  df-bases 22242  df-cld 22316  df-ntr 22317  df-cls 22318  df-nei 22395  df-lp 22433  df-perf 22434  df-cn 22524  df-cnp 22525  df-haus 22612  df-tx 22859  df-hmeo 23052  df-fil 23143  df-fm 23235  df-flim 23236  df-flf 23237  df-xms 23619  df-ms 23620  df-tms 23621  df-cncf 24187  df-limc 25176  df-dv 25177  df-log 25858

This theorem is referenced by:  dvlog  25952