ellogdm - Metamath Proof Explorer

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Theorem ellogdm 26681

Description: Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)

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

**Assertion**
Ref	Expression
ellogdm	⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺)))

**Proof of Theorem ellogdm**
Step	Hyp	Ref	Expression
1		logcn.d	. . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
2	1	eleq2i 2833	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
3		eldif 3961	. 2 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (-∞(,]0)))
4		mnfxr 11318	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
5		0re 11263	. . . . . . 7 ⊢ 0 ∈ ℝ
6		elioc2 13450	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ) → (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0)))
7	4, 5, 6	mp2an 692	. . . . . 6 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0))
8		df-3an 1089	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0) ↔ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0))
9		mnflt 13165	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
10	9	pm4.71i 559	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴))
11	10	anbi1i 624	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) ↔ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0))
12		lenlt 11339	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴))
13	5, 12	mpan2 691	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴))
14		elrp 13036	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴))
15	14	baib 535	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℝ⁺ ↔ 0 < 𝐴))
16	15	notbid 318	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℝ⁺ ↔ ¬ 0 < 𝐴))
17	13, 16	bitr4d 282	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ ¬ 𝐴 ∈ ℝ⁺))
18	17	pm5.32i 574	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) ↔ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ⁺))
19	11, 18	bitr3i 277	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0) ↔ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ⁺))
20	7, 8, 19	3bitri 297	. . . . 5 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ⁺))
21	20	notbii 320	. . . 4 ⊢ (¬ 𝐴 ∈ (-∞(,]0) ↔ ¬ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ⁺))
22		iman 401	. . . 4 ⊢ ((𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺) ↔ ¬ (𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℝ⁺))
23	21, 22	bitr4i 278	. . 3 ⊢ (¬ 𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺))
24	23	anbi2i 623	. 2 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (-∞(,]0)) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺)))
25	2, 3, 24	3bitri 297	1 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ⁺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℝ⁺crp 13034 (,]cioc 13388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-rp 13035 df-ioc 13392

This theorem is referenced by: logdmn0 26682 logdmnrp 26683 logdmss 26684 logcnlem2 26685 logcnlem3 26686 logcnlem4 26687 logcnlem5 26688 logcn 26689 dvloglem 26690 logf1o2 26692 cxpcn 26787 cxpcnOLD 26788 cxpcn2 26789 dmlogdmgm 27067 rpdmgm 27068 lgamgulmlem2 27073 lgamcvg2 27098 logdivsqrle 34665 binomcxplemdvbinom 44372

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