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Mirrors > Home > MPE Home > Th. List > ellogdm | Structured version Visualization version GIF version |
Description: Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.) |
Ref | Expression |
---|---|
logcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
ellogdm | ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logcn.d | . . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
2 | 1 | eleq2i 2898 | . 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (ℂ ∖ (-∞(,]0))) |
3 | eldif 3808 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (-∞(,]0))) | |
4 | mnfxr 10414 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
5 | 0re 10358 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
6 | elioc2 12524 | . . . . . . 7 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0))) | |
7 | 4, 5, 6 | mp2an 685 | . . . . . 6 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0)) |
8 | df-3an 1115 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0) ↔ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0)) | |
9 | mnflt 12243 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
10 | 9 | pm4.71i 557 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴)) |
11 | 10 | anbi1i 619 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) ↔ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0)) |
12 | lenlt 10435 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴)) | |
13 | 5, 12 | mpan2 684 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴)) |
14 | elrp 12114 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
15 | 14 | baib 533 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℝ+ ↔ 0 < 𝐴)) |
16 | 15 | notbid 310 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℝ+ ↔ ¬ 0 < 𝐴)) |
17 | 13, 16 | bitr4d 274 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ ¬ 𝐴 ∈ ℝ+)) |
18 | 17 | pm5.32i 572 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) ↔ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ+)) |
19 | 11, 18 | bitr3i 269 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0) ↔ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ+)) |
20 | 7, 8, 19 | 3bitri 289 | . . . . 5 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ+)) |
21 | 20 | notbii 312 | . . . 4 ⊢ (¬ 𝐴 ∈ (-∞(,]0) ↔ ¬ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ+)) |
22 | iman 392 | . . . 4 ⊢ ((𝐴 ∈ ℝ → 𝐴 ∈ ℝ+) ↔ ¬ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ+)) | |
23 | 21, 22 | bitr4i 270 | . . 3 ⊢ (¬ 𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
24 | 23 | anbi2i 618 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (-∞(,]0)) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
25 | 2, 3, 24 | 3bitri 289 | 1 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∖ cdif 3795 class class class wbr 4873 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 -∞cmnf 10389 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 ℝ+crp 12112 (,]cioc 12464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-addrcl 10313 ax-rnegex 10323 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-rp 12113 df-ioc 12468 |
This theorem is referenced by: logdmn0 24785 logdmnrp 24786 logdmss 24787 logcnlem2 24788 logcnlem3 24789 logcnlem4 24790 logcnlem5 24791 logcn 24792 dvloglem 24793 logf1o2 24795 cxpcn 24888 cxpcn2 24889 dmlogdmgm 25163 rpdmgm 25164 lgamgulmlem2 25169 lgamcvg2 25194 logdivsqrle 31277 binomcxplemdvbinom 39392 |
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