ellogdm - Metamath Proof Explorer

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Theorem ellogdm 26618

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 2825	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐴 ∈ (ℂ ∖ (-∞(,]0)))
3		eldif 3941	. 2 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝐴 ∈ ℂ ∧ ¬ 𝐴 ∈ (-∞(,]0)))
4		mnfxr 11300	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
5		0re 11245	. . . . . . 7 ⊢ 0 ∈ ℝ
6		elioc2 13432	. . . . . . 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 1088	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0) ↔ ((𝐴 ∈ ℝ ∧ -∞ < 𝐴) ∧ 𝐴 ≤ 0))
9		mnflt 13147	. . . . . . . . 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 11321	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴))
13	5, 12	mpan2 691	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ ¬ 0 < 𝐴))
14		elrp 13018	. . . . . . . . . . 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 1086 = wceq 1539 ∈ wcel 2107 ∖ cdif 3928 class class class wbr 5123 (class class class)co 7413 ℂcc 11135 ℝcr 11136 0cc0 11137 -∞cmnf 11275 ℝ^*cxr 11276 < clt 11277 ≤ cle 11278 ℝ⁺crp 13016 (,]cioc 13370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-addrcl 11198 ax-rnegex 11208 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-po 5572 df-so 5573 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8727 df-en 8968 df-dom 8969 df-sdom 8970 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-rp 13017 df-ioc 13374

This theorem is referenced by: logdmn0 26619 logdmnrp 26620 logdmss 26621 logcnlem2 26622 logcnlem3 26623 logcnlem4 26624 logcnlem5 26625 logcn 26626 dvloglem 26627 logf1o2 26629 cxpcn 26724 cxpcnOLD 26725 cxpcn2 26726 dmlogdmgm 27004 rpdmgm 27005 lgamgulmlem2 27010 lgamcvg2 27035 logdivsqrle 34640 binomcxplemdvbinom 44344

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