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Theorem dvloglem 26497

Description: Lemma for dvlog 26500. (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 26414	. . . . . 6 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1ofun 6825	. . . . . 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 26491	. . . . . 6 ⊢ 𝐷 ⊆ (ℂ ∖ {0})
6		f1odm 6827	. . . . . . 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 4011	. . . . 5 ⊢ 𝐷 ⊆ dom log
9		funimass4 6946	. . . . 5 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π))))
10	3, 8, 9	mp2an 689	. . . 4 ⊢ ((log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π)) ↔ ∀𝑥 ∈ 𝐷 (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
11	4	ellogdm 26488	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
12	11	simplbi 497	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
13	4	logdmn0 26489	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
14	12, 13	logcld 26420	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ ℂ)
15	14	imcld 15138	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
16	12, 13	logimcld 26421	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
17	16	simpld 494	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → -π < (ℑ‘(log‘𝑥)))
18		pire 26309	. . . . . . . 8 ⊢ π ∈ ℝ
19	18	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → π ∈ ℝ)
20	16	simprd 495	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
21	4	logdmnrp 26490	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → ¬ -𝑥 ∈ ℝ⁺)
22		lognegb 26439	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
23	12, 13, 22	syl2anc 583	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐷 → (-𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) = π))
24	23	necon3bbid 2970	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (¬ -𝑥 ∈ ℝ⁺ ↔ (ℑ‘(log‘𝑥)) ≠ π))
25	21, 24	mpbid 231	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
26	25	necomd 2988	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → π ≠ (ℑ‘(log‘𝑥)))
27	15, 19, 20, 26	leneltd 11364	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) < π)
28	18	renegcli 11517	. . . . . . . 8 ⊢ -π ∈ ℝ
29	28	rexri 11268	. . . . . . 7 ⊢ -π ∈ ℝ^*
30	18	rexri 11268	. . . . . . 7 ⊢ π ∈ ℝ^*
31		elioo2 13361	. . . . . . 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 689	. . . . . 6 ⊢ ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
33	15, 17, 27, 32	syl3anbrc 1340	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
34		imf 15056	. . . . . 6 ⊢ ℑ:ℂ⟶ℝ
35		ffn 6707	. . . . . 6 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
36		elpreima 7049	. . . . . 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 582	. . . 4 ⊢ (𝑥 ∈ 𝐷 → (log‘𝑥) ∈ (^◡ℑ “ (-π(,)π)))
39	10, 38	mprgbir 3060	. . 3 ⊢ (log “ 𝐷) ⊆ (^◡ℑ “ (-π(,)π))
40		df-ioo 13324	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
41		df-ioc 13325	. . . . . . . . . 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 13124	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ℝ^* ∧ π ∈ ℝ^*) → (𝑤 < π → 𝑤 ≤ π))
44	40, 41, 42, 43	ixxssixx 13334	. . . . . . . . 9 ⊢ (-π(,)π) ⊆ (-π(,]π)
45		imass2 6091	. . . . . . . . 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 26408	. . . . . . . 8 ⊢ ran log = (^◡ℑ “ (-π(,]π))
48	46, 47	sseqtrri 4011	. . . . . . 7 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ ran log
49	48	sseli 3970	. . . . . 6 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ ran log)
50		logef 26431	. . . . . 6 ⊢ (𝑥 ∈ ran log → (log‘(exp‘𝑥)) = 𝑥)
51	49, 50	syl 17	. . . . 5 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (log‘(exp‘𝑥)) = 𝑥)
52		elpreima 7049	. . . . . . . . . 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 16022	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
55	54	adantr 480	. . . . . . . . 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 497	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ ℂ)
58	57	imcld 15138	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
59		eliooord 13379	. . . . . . . . . . . 12 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
60	53, 59	simplbiim 504	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
61	60	simprd 495	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) < π)
62	58, 61	ltned 11346	. . . . . . . . 9 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (ℑ‘𝑥) ≠ π)
63	51	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (log‘(exp‘𝑥)) = 𝑥)
64	63	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘(log‘(exp‘𝑥))) = (ℑ‘𝑥))
65		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ (-∞(,]0))
66		mnfxr 11267	. . . . . . . . . . . . . . . . 17 ⊢ -∞ ∈ ℝ^*
67		0re 11212	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℝ
68		elioc2 13383	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → ((exp‘𝑥) ∈ (-∞(,]0) ↔ ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0)))
69	66, 67, 68	mp2an 689	. . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ ℝ)
72		0red 11213	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ∈ ℝ)
73	70	simp3d 1141	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≤ 0)
74		efne0 16036	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → (exp‘𝑥) ≠ 0)
75	57, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (exp‘𝑥) ≠ 0)
76	75	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≠ 0)
77	76	necomd 2988	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ≠ (exp‘𝑥))
78	71, 72, 73, 77	leneltd 11364	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) < 0)
79	71, 78	negelrpd 13004	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → -(exp‘𝑥) ∈ ℝ⁺)
80		lognegb 26439	. . . . . . . . . . . . . . 15 ⊢ (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (-(exp‘𝑥) ∈ ℝ⁺ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
81	56, 75, 80	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (-(exp‘𝑥) ∈ ℝ⁺ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
82	81	adantr 480	. . . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (^◡ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘𝑥) = π)
85	84	ex 412	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → ((exp‘𝑥) ∈ (-∞(,]0) → (ℑ‘𝑥) = π))
86	85	necon3ad 2945	. . . . . . . . 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 3951	. . . . . . 7 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
89	88, 4	eleqtrrdi 2836	. . . . . 6 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
90		funfvima2 7224	. . . . . . 7 ⊢ ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
91	3, 8, 90	mp2an 689	. . . . . 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 2826	. . . 4 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
94	93	ssriv 3978	. . 3 ⊢ (^◡ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
95	39, 94	eqssi 3990	. 2 ⊢ (log “ 𝐷) = (^◡ℑ “ (-π(,)π))
96		imcncf 24744	. . . 4 ⊢ ℑ ∈ (ℂ–cn→ℝ)
97		ssid 3996	. . . . 5 ⊢ ℂ ⊆ ℂ
98		ax-resscn 11162	. . . . 5 ⊢ ℝ ⊆ ℂ
99		eqid 2724	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
100	99	cnfldtopon 24620	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
101	100	toponrestid 22744	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
102	99	tgioo2 24640	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
103	99, 101, 102	cncfcn 24751	. . . . 5 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,))))
104	97, 98, 103	mp2an 689	. . . 4 ⊢ (ℂ–cn→ℝ) = ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,)))
105	96, 104	eleqtri 2823	. . 3 ⊢ ℑ ∈ ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,)))
106		iooretop 24603	. . 3 ⊢ (-π(,)π) ∈ (topGen‘ran (,))
107		cnima 23090	. . 3 ⊢ ((ℑ ∈ ((TopOpen‘ℂ_fld) Cn (topGen‘ran (,))) ∧ (-π(,)π) ∈ (topGen‘ran (,))) → (^◡ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂ_fld))
108	105, 106, 107	mp2an 689	. 2 ⊢ (^◡ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂ_fld)
109	95, 108	eqeltri 2821	1 ⊢ (log “ 𝐷) ∈ (TopOpen‘ℂ_fld)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∖ cdif 3937 ⊆ wss 3940 {csn 4620 class class class wbr 5138 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 “ cima 5669 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7401 ℂcc 11103 ℝcr 11104 0cc0 11105 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 -cneg 11441 ℝ⁺crp 12970 (,)cioo 13320 (,]cioc 13321 ℑcim 15041 expce 16001 πcpi 16006 TopOpenctopn 17365 topGenctg 17381 ℂ_fldccnfld 21227 Cn ccn 23049 –cn→ccncf 24717 logclog 26404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 df-log 26406

This theorem is referenced by: dvlog 26500

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