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Theorem dvloglem 24294
Description: Lemma for dvlog 24297. (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.
StepHypRef Expression
1 logf1o 24215 . . . . . 6 log:(ℂ ∖ {0})–1-1-onto→ran log
2 f1ofun 6096 . . . . . 6 (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun log)
31, 2ax-mp 5 . . . . 5 Fun log
4 logcn.d . . . . . . 7 𝐷 = (ℂ ∖ (-∞(,]0))
54logdmss 24288 . . . . . 6 𝐷 ⊆ (ℂ ∖ {0})
6 f1odm 6098 . . . . . . 7 (log:(ℂ ∖ {0})–1-1-onto→ran log → dom log = (ℂ ∖ {0}))
71, 6ax-mp 5 . . . . . 6 dom log = (ℂ ∖ {0})
85, 7sseqtr4i 3617 . . . . 5 𝐷 ⊆ dom log
9 funimass4 6204 . . . . 5 ((Fun log ∧ 𝐷 ⊆ dom log) → ((log “ 𝐷) ⊆ (ℑ “ (-π(,)π)) ↔ ∀𝑥𝐷 (log‘𝑥) ∈ (ℑ “ (-π(,)π))))
103, 8, 9mp2an 707 . . . 4 ((log “ 𝐷) ⊆ (ℑ “ (-π(,)π)) ↔ ∀𝑥𝐷 (log‘𝑥) ∈ (ℑ “ (-π(,)π)))
114ellogdm 24285 . . . . . . 7 (𝑥𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)))
1211simplbi 476 . . . . . 6 (𝑥𝐷𝑥 ∈ ℂ)
134logdmn0 24286 . . . . . 6 (𝑥𝐷𝑥 ≠ 0)
1412, 13logcld 24221 . . . . 5 (𝑥𝐷 → (log‘𝑥) ∈ ℂ)
1514imcld 13869 . . . . . 6 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ∈ ℝ)
1612, 13logimcld 24222 . . . . . . 7 (𝑥𝐷 → (-π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) ≤ π))
1716simpld 475 . . . . . 6 (𝑥𝐷 → -π < (ℑ‘(log‘𝑥)))
184logdmnrp 24287 . . . . . . . . 9 (𝑥𝐷 → ¬ -𝑥 ∈ ℝ+)
19 lognegb 24240 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
2012, 13, 19syl2anc 692 . . . . . . . . . 10 (𝑥𝐷 → (-𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) = π))
2120necon3bbid 2827 . . . . . . . . 9 (𝑥𝐷 → (¬ -𝑥 ∈ ℝ+ ↔ (ℑ‘(log‘𝑥)) ≠ π))
2218, 21mpbid 222 . . . . . . . 8 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ≠ π)
2322necomd 2845 . . . . . . 7 (𝑥𝐷 → π ≠ (ℑ‘(log‘𝑥)))
24 pire 24114 . . . . . . . . 9 π ∈ ℝ
2524a1i 11 . . . . . . . 8 (𝑥𝐷 → π ∈ ℝ)
2616simprd 479 . . . . . . . 8 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ≤ π)
2715, 25, 26leltned 10134 . . . . . . 7 (𝑥𝐷 → ((ℑ‘(log‘𝑥)) < π ↔ π ≠ (ℑ‘(log‘𝑥))))
2823, 27mpbird 247 . . . . . 6 (𝑥𝐷 → (ℑ‘(log‘𝑥)) < π)
2924renegcli 10286 . . . . . . . 8 -π ∈ ℝ
3029rexri 10041 . . . . . . 7 -π ∈ ℝ*
3124rexri 10041 . . . . . . 7 π ∈ ℝ*
32 elioo2 12158 . . . . . . 7 ((-π ∈ ℝ* ∧ π ∈ ℝ*) → ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π)))
3330, 31, 32mp2an 707 . . . . . 6 ((ℑ‘(log‘𝑥)) ∈ (-π(,)π) ↔ ((ℑ‘(log‘𝑥)) ∈ ℝ ∧ -π < (ℑ‘(log‘𝑥)) ∧ (ℑ‘(log‘𝑥)) < π))
3415, 17, 28, 33syl3anbrc 1244 . . . . 5 (𝑥𝐷 → (ℑ‘(log‘𝑥)) ∈ (-π(,)π))
35 imf 13787 . . . . . 6 ℑ:ℂ⟶ℝ
36 ffn 6002 . . . . . 6 (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
37 elpreima 6293 . . . . . 6 (ℑ Fn ℂ → ((log‘𝑥) ∈ (ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π))))
3835, 36, 37mp2b 10 . . . . 5 ((log‘𝑥) ∈ (ℑ “ (-π(,)π)) ↔ ((log‘𝑥) ∈ ℂ ∧ (ℑ‘(log‘𝑥)) ∈ (-π(,)π)))
3914, 34, 38sylanbrc 697 . . . 4 (𝑥𝐷 → (log‘𝑥) ∈ (ℑ “ (-π(,)π)))
4010, 39mprgbir 2922 . . 3 (log “ 𝐷) ⊆ (ℑ “ (-π(,)π))
41 df-ioo 12121 . . . . . . . . . 10 (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧 < 𝑦)})
42 df-ioc 12122 . . . . . . . . . 10 (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 < 𝑧𝑧𝑦)})
43 idd 24 . . . . . . . . . 10 ((-π ∈ ℝ*𝑤 ∈ ℝ*) → (-π < 𝑤 → -π < 𝑤))
44 xrltle 11926 . . . . . . . . . 10 ((𝑤 ∈ ℝ* ∧ π ∈ ℝ*) → (𝑤 < π → 𝑤 ≤ π))
4541, 42, 43, 44ixxssixx 12131 . . . . . . . . 9 (-π(,)π) ⊆ (-π(,]π)
46 imass2 5460 . . . . . . . . 9 ((-π(,)π) ⊆ (-π(,]π) → (ℑ “ (-π(,)π)) ⊆ (ℑ “ (-π(,]π)))
4745, 46ax-mp 5 . . . . . . . 8 (ℑ “ (-π(,)π)) ⊆ (ℑ “ (-π(,]π))
48 logrn 24209 . . . . . . . 8 ran log = (ℑ “ (-π(,]π))
4947, 48sseqtr4i 3617 . . . . . . 7 (ℑ “ (-π(,)π)) ⊆ ran log
5049sseli 3579 . . . . . 6 (𝑥 ∈ (ℑ “ (-π(,)π)) → 𝑥 ∈ ran log)
51 logef 24232 . . . . . 6 (𝑥 ∈ ran log → (log‘(exp‘𝑥)) = 𝑥)
5250, 51syl 17 . . . . 5 (𝑥 ∈ (ℑ “ (-π(,)π)) → (log‘(exp‘𝑥)) = 𝑥)
53 elpreima 6293 . . . . . . . . . 10 (ℑ Fn ℂ → (𝑥 ∈ (ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π))))
5435, 36, 53mp2b 10 . . . . . . . . 9 (𝑥 ∈ (ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)))
55 efcl 14738 . . . . . . . . . 10 (𝑥 ∈ ℂ → (exp‘𝑥) ∈ ℂ)
5655adantr 481 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
5754, 56sylbi 207 . . . . . . . 8 (𝑥 ∈ (ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ ℂ)
5854simplbi 476 . . . . . . . . . . 11 (𝑥 ∈ (ℑ “ (-π(,)π)) → 𝑥 ∈ ℂ)
5958imcld 13869 . . . . . . . . . 10 (𝑥 ∈ (ℑ “ (-π(,)π)) → (ℑ‘𝑥) ∈ ℝ)
6054simprbi 480 . . . . . . . . . . . 12 (𝑥 ∈ (ℑ “ (-π(,)π)) → (ℑ‘𝑥) ∈ (-π(,)π))
61 eliooord 12175 . . . . . . . . . . . 12 ((ℑ‘𝑥) ∈ (-π(,)π) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
6260, 61syl 17 . . . . . . . . . . 11 (𝑥 ∈ (ℑ “ (-π(,)π)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
6362simprd 479 . . . . . . . . . 10 (𝑥 ∈ (ℑ “ (-π(,)π)) → (ℑ‘𝑥) < π)
6459, 63ltned 10117 . . . . . . . . 9 (𝑥 ∈ (ℑ “ (-π(,)π)) → (ℑ‘𝑥) ≠ π)
6552adantr 481 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (log‘(exp‘𝑥)) = 𝑥)
6665fveq2d 6152 . . . . . . . . . . . 12 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘(log‘(exp‘𝑥))) = (ℑ‘𝑥))
67 simpr 477 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ (-∞(,]0))
68 mnfxr 10040 . . . . . . . . . . . . . . . . . 18 -∞ ∈ ℝ*
69 0re 9984 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
70 elioc2 12178 . . . . . . . . . . . . . . . . . 18 ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → ((exp‘𝑥) ∈ (-∞(,]0) ↔ ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0)))
7168, 69, 70mp2an 707 . . . . . . . . . . . . . . . . 17 ((exp‘𝑥) ∈ (-∞(,]0) ↔ ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0))
7267, 71sylib 208 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → ((exp‘𝑥) ∈ ℝ ∧ -∞ < (exp‘𝑥) ∧ (exp‘𝑥) ≤ 0))
7372simp1d 1071 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ∈ ℝ)
7473renegcld 10401 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → -(exp‘𝑥) ∈ ℝ)
75 efne0 14752 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → (exp‘𝑥) ≠ 0)
7658, 75syl 17 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℑ “ (-π(,)π)) → (exp‘𝑥) ≠ 0)
7776adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≠ 0)
7877necomd 2845 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ≠ (exp‘𝑥))
79 0red 9985 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 ∈ ℝ)
8072simp3d 1073 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) ≤ 0)
8173, 79, 80leltned 10134 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → ((exp‘𝑥) < 0 ↔ 0 ≠ (exp‘𝑥)))
8278, 81mpbird 247 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (exp‘𝑥) < 0)
8373lt0neg1d 10541 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → ((exp‘𝑥) < 0 ↔ 0 < -(exp‘𝑥)))
8482, 83mpbid 222 . . . . . . . . . . . . . 14 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → 0 < -(exp‘𝑥))
8574, 84elrpd 11813 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → -(exp‘𝑥) ∈ ℝ+)
86 lognegb 24240 . . . . . . . . . . . . . . 15 (((exp‘𝑥) ∈ ℂ ∧ (exp‘𝑥) ≠ 0) → (-(exp‘𝑥) ∈ ℝ+ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
8757, 76, 86syl2anc 692 . . . . . . . . . . . . . 14 (𝑥 ∈ (ℑ “ (-π(,)π)) → (-(exp‘𝑥) ∈ ℝ+ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
8887adantr 481 . . . . . . . . . . . . 13 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (-(exp‘𝑥) ∈ ℝ+ ↔ (ℑ‘(log‘(exp‘𝑥))) = π))
8985, 88mpbid 222 . . . . . . . . . . . 12 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘(log‘(exp‘𝑥))) = π)
9066, 89eqtr3d 2657 . . . . . . . . . . 11 ((𝑥 ∈ (ℑ “ (-π(,)π)) ∧ (exp‘𝑥) ∈ (-∞(,]0)) → (ℑ‘𝑥) = π)
9190ex 450 . . . . . . . . . 10 (𝑥 ∈ (ℑ “ (-π(,)π)) → ((exp‘𝑥) ∈ (-∞(,]0) → (ℑ‘𝑥) = π))
9291necon3ad 2803 . . . . . . . . 9 (𝑥 ∈ (ℑ “ (-π(,)π)) → ((ℑ‘𝑥) ≠ π → ¬ (exp‘𝑥) ∈ (-∞(,]0)))
9364, 92mpd 15 . . . . . . . 8 (𝑥 ∈ (ℑ “ (-π(,)π)) → ¬ (exp‘𝑥) ∈ (-∞(,]0))
9457, 93eldifd 3566 . . . . . . 7 (𝑥 ∈ (ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ (ℂ ∖ (-∞(,]0)))
9594, 4syl6eleqr 2709 . . . . . 6 (𝑥 ∈ (ℑ “ (-π(,)π)) → (exp‘𝑥) ∈ 𝐷)
96 funfvima2 6447 . . . . . . 7 ((Fun log ∧ 𝐷 ⊆ dom log) → ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷)))
973, 8, 96mp2an 707 . . . . . 6 ((exp‘𝑥) ∈ 𝐷 → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
9895, 97syl 17 . . . . 5 (𝑥 ∈ (ℑ “ (-π(,)π)) → (log‘(exp‘𝑥)) ∈ (log “ 𝐷))
9952, 98eqeltrrd 2699 . . . 4 (𝑥 ∈ (ℑ “ (-π(,)π)) → 𝑥 ∈ (log “ 𝐷))
10099ssriv 3587 . . 3 (ℑ “ (-π(,)π)) ⊆ (log “ 𝐷)
10140, 100eqssi 3599 . 2 (log “ 𝐷) = (ℑ “ (-π(,)π))
102 imcncf 22614 . . . 4 ℑ ∈ (ℂ–cn→ℝ)
103 ssid 3603 . . . . 5 ℂ ⊆ ℂ
104 ax-resscn 9937 . . . . 5 ℝ ⊆ ℂ
105 eqid 2621 . . . . . 6 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
106105cnfldtop 22497 . . . . . . . 8 (TopOpen‘ℂfld) ∈ Top
107105cnfldtopon 22496 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
108107toponunii 20647 . . . . . . . . 9 ℂ = (TopOpen‘ℂfld)
109108restid 16015 . . . . . . . 8 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
110106, 109ax-mp 5 . . . . . . 7 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
111110eqcomi 2630 . . . . . 6 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
112105tgioo2 22514 . . . . . 6 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
113105, 111, 112cncfcn 22620 . . . . 5 ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = ((TopOpen‘ℂfld) Cn (topGen‘ran (,))))
114103, 104, 113mp2an 707 . . . 4 (ℂ–cn→ℝ) = ((TopOpen‘ℂfld) Cn (topGen‘ran (,)))
115102, 114eleqtri 2696 . . 3 ℑ ∈ ((TopOpen‘ℂfld) Cn (topGen‘ran (,)))
116 iooretop 22479 . . 3 (-π(,)π) ∈ (topGen‘ran (,))
117 cnima 20979 . . 3 ((ℑ ∈ ((TopOpen‘ℂfld) Cn (topGen‘ran (,))) ∧ (-π(,)π) ∈ (topGen‘ran (,))) → (ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂfld))
118115, 116, 117mp2an 707 . 2 (ℑ “ (-π(,)π)) ∈ (TopOpen‘ℂfld)
119101, 118eqeltri 2694 1 (log “ 𝐷) ∈ (TopOpen‘ℂfld)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cdif 3552  wss 3555  {csn 4148   class class class wbr 4613  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cc 9878  cr 9879  0cc0 9880  -∞cmnf 10016  *cxr 10017   < clt 10018  cle 10019  -cneg 10211  +crp 11776  (,)cioo 12117  (,]cioc 12118  cim 13772  expce 14717  πcpi 14722  t crest 16002  TopOpenctopn 16003  topGenctg 16019  fldccnfld 19665  Topctop 20617   Cn ccn 20938  cnccncf 22587  logclog 24205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-shft 13741  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-limsup 14136  df-clim 14153  df-rlim 14154  df-sum 14351  df-ef 14723  df-sin 14725  df-cos 14726  df-pi 14728  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536  df-dv 23537  df-log 24207
This theorem is referenced by:  dvlog  24297
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