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Theorem ellogrn 26060

Description: Write out the property 𝐴 ∈ ran log explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)

**Assertion**
Ref	Expression
ellogrn	⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))

**Proof of Theorem ellogrn**
Step	Hyp	Ref	Expression
1		imf 15057	. . . 4 ⊢ ℑ:ℂ⟶ℝ
2		ffn 6715	. . . 4 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
3		elpreima 7057	. . . 4 ⊢ (ℑ Fn ℂ → (𝐴 ∈ (^◡ℑ “ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ∈ (-π(,]π))))
4	1, 2, 3	mp2b 10	. . 3 ⊢ (𝐴 ∈ (^◡ℑ “ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ∈ (-π(,]π)))
5		pire 25960	. . . . . . . . 9 ⊢ π ∈ ℝ
6	5	renegcli 11518	. . . . . . . 8 ⊢ -π ∈ ℝ
7	6	rexri 11269	. . . . . . 7 ⊢ -π ∈ ℝ^*
8		elioc2 13384	. . . . . . 7 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ) → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
9	7, 5, 8	mp2an 691	. . . . . 6 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))
10		3anass 1096	. . . . . 6 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
11	9, 10	bitri 275	. . . . 5 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
12		imcl 15055	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ)
13	12	biantrurd 534	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))))
14	11, 13	bitr4id 290	. . . 4 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
15	14	pm5.32i 576	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ∈ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
16	4, 15	bitri 275	. 2 ⊢ (𝐴 ∈ (^◡ℑ “ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
17		logrn 26059	. . 3 ⊢ ran log = (^◡ℑ “ (-π(,]π))
18	17	eleq2i 2826	. 2 ⊢ (𝐴 ∈ ran log ↔ 𝐴 ∈ (^◡ℑ “ (-π(,]π)))
19		3anass 1096	. 2 ⊢ ((𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
20	16, 18, 19	3bitr4i 303	1 ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5148 ^◡ccnv 5675 ran crn 5677 “ cima 5679 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 -cneg 11442 (,]cioc 13322 ℑcim 15042 πcpi 16007 logclog 26055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-lp 22632 df-perf 22633 df-cn 22723 df-cnp 22724 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cncf 24386 df-limc 25375 df-dv 25376 df-log 26057

This theorem is referenced by: relogrn 26062 logrncn 26063 logimcl 26070 logrnaddcl 26075 logneg 26088 logcj 26106 logimul 26114 logneg2 26115 logcnlem4 26145 logf1o2 26150 logreclem 26257 asinsin 26387 asin1 26389 atanlogaddlem 26408 atanlogsub 26411 atantan 26418 logi 34693

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