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Theorem dvloglem 26148

Description: Lemma for dvlog 26151. (Contributed by Mario Carneiro, 24-Feb-2015.)

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

**Assertion**
Ref	Expression
dvloglem	⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)

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 26065	. . . . . 6 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1ofun 6833	. . . . . 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 26142	. . . . . 6 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1odm 6835	. . . . . . 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 4019	. . . . 5 ⊢ 𝐷 ⊆ dom log
9		funimass4 6954	. . . . 5 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π))))
10	3, 8, 9	mp2an 691	. . . 4 ⊢ ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
11	4	ellogdm 26139	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
12	11	simplbi 499	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
13	4	logdmn0 26140	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
14	12, 13	logcld 26071	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
15	14	imcld 15139	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
16	12, 13	logimcld 26072	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
17	16	simpld 496	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
18		pire 25960	. . . . . . . 8 ⊢ π ∈ ℝ
19	18	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → π ∈ ℝ)
20	16	simprd 497	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
21	4	logdmnrp 26141	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ⁺)
22		lognegb 26090	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
23	12, 13, 22	syl2anc 585	. . . . . . . . . 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 11365	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
28	18	renegcli 11518	. . . . . . . 8 ⊢ -π ∈ ℝ
29	28	rexri 11269	. . . . . . 7 ⊢ -π ∈ ℝ^*
30	18	rexri 11269	. . . . . . 7 ⊢ π ∈ ℝ^*
31		elioo2 13362	. . . . . . 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 691	. . . . . 6 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
33	15, 17, 27, 32	syl3anbrc 1344	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
34		imf 15057	. . . . . 6 ⊢ ℑ:ℂ⟶ℝ
35		ffn 6715	. . . . . 6 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
36		elpreima 7057	. . . . . 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 584	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
39	10, 38	mprgbir 3069	. . 3 ⊢ (log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π))
40		df-ioo 13325	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
41		df-ioc 13326	. . . . . . . . . 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 13125	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ^* ∧ π ∈ ℝ^*) → (𝑤 < π → 𝑤 ≤ π))
44	40, 41, 42, 43	ixxssixx 13335	. . . . . . . . 9 ⊢ (-π(,)π) ⊆ (-π(,]π)
45		imass2 6099	. . . . . . . . 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 26059	. . . . . . . 8 ⊢ ran log = (^◡ℑ “ (-π(,]π))
48	46, 47	sseqtrri 4019	. . . . . . 7 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ ran log
49	48	sseli 3978	. . . . . 6 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ ran log)
50		logef 26082	. . . . . 6 ⊢ (𝑥 ∈ ran log → (log‘(exp‘𝑥)) = 𝑥)
51	49, 50	syl 17	. . . . 5 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (log‘(exp‘𝑥)) = 𝑥)
52		elpreima 7057	. . . . . . . . . 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 16023	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
55	54	adantr 482	. . . . . . . . 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 499	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ ℂ)
58	57	imcld 15139	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
59		eliooord 13380	. . . . . . . . . . . 12 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
60	53, 59	simplbiim 506	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
61	60	simprd 497	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) < π)
62	58, 61	ltned 11347	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) ≠ π)
63	51	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (log‘(exp‘𝑥)) = 𝑥)
64	63	fveq2d 6893	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘(log‘(exp‘𝑥))) = (ℑ‘𝑥))
65		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ (-∞(,]0))
66		mnfxr 11268	. . . . . . . . . . . . . . . . 17 ⊢ -∞ ∈ ℝ^*
67		0re 11213	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
68		elioc2 13384	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → ((exp‘𝑥) ∈ (-∞(,]0) ↔ ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0)))
69	66, 67, 68	mp2an 691	. . . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ ℝ)
72		0red 11214	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ∈ ℝ)
73	70	simp3d 1145	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≤ 0)
74		efne0 16037	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ≠ 0)
75	57, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (exp‘𝑥) ≠ 0)
76	75	adantr 482	. . . . . . . . . . . . . . . 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 11365	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) < 0)
79	71, 78	negelrpd 13005	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → -(exp‘𝑥) ∈ ℝ⁺)
80		lognegb 26090	. . . . . . . . . . . . . . 15 ⊢ (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (-(exp‘𝑥) ∈ ℝ⁺ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
81	56, 75, 80	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (-(exp‘𝑥) ∈ ℝ⁺ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
82	81	adantr 482	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘𝑥) = π)
85	84	ex 414	. . . . . . . . . 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 3959	. . . . . . 7 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
89	88, 4	eleqtrrdi 2845	. . . . . 6 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
90		funfvima2 7230	. . . . . . 7 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
91	3, 8, 90	mp2an 691	. . . . . 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 2835	. . . 4 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
94	93	ssriv 3986	. . 3 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
95	39, 94	eqssi 3998	. 2 ⊢ (log “ 𝐷) = (^◡ℑ “ (-π(,)π))
96		imcncf 24411	. . . 4 ⊢ ℑ ∈ (ℂ–cn→ℝ)
97		ssid 4004	. . . . 5 ⊢ ℂ ⊆ ℂ
98		ax-resscn 11164	. . . . 5 ⊢ ℝ ⊆ ℂ
99		eqid 2733	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
100	99	cnfldtopon 24291	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
101	100	toponrestid 22415	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
102	99	tgioo2 24311	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
103	99, 101, 102	cncfcn 24418	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,))))
104	97, 98, 103	mp2an 691	. . . 4 ⊢ (ℂ–cn→ℝ) = ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,)))
105	96, 104	eleqtri 2832	. . 3 ⊢ ℑ ∈ ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,)))
106		iooretop 24274	. . 3 ⊢ (-π(,)π) ∈ (topGen‘ran (,))
107		cnima 22761	. . 3 ⊢ ((ℑ ∈ ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,))) ∧ (-π(,)π) ∈ (topGen‘ran (,))) → (^◡ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂ_fld))
108	105, 106, 107	mp2an 691	. 2 ⊢ (^◡ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂ_fld)
109	95, 108	eqeltri 2830	1 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 “ cima 5679 Fun wfun 6535 Fn wfn 6536 ⟶wf 6537 –1-1-onto→wf1o 6540 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 -cneg 11442 ℝ⁺crp 12971 (,)cioo 13321 (,]cioc 13322 ℑcim 15042 expce 16002 πcpi 16007 TopOpenctopn 17364 topGenctg 17380 ℂ_fldccnfld 20937 Cn ccn 22720 –cn→ccncf 24384 logclog 26055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cncf 24386 df-limc 25375 df-dv 25376 df-log 26057

This theorem is referenced by: dvlog 26151

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