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Theorem ellogrn 26545

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 15070	. . . 4 ⊢ ℑ:ℂ⟶ℝ
2		ffn 6659	. . . 4 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
3		elpreima 7003	. . . 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 26443	. . . . . . . . 9 ⊢ π ∈ ℝ
6	5	renegcli 11450	. . . . . . . 8 ⊢ -π ∈ ℝ
7	6	rexri 11198	. . . . . . 7 ⊢ -π ∈ ℝ^*
8		elioc2 13357	. . . . . . 7 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ) → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
9	7, 5, 8	mp2an 699	. . . . . 6 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))
10		3anass 1101	. . . . . 6 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
11	9, 10	bitri 277	. . . . 5 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
12		imcl 15068	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ)
13	12	biantrurd 538	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))))
14	11, 13	bitr4id 292	. . . 4 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
15	14	pm5.32i 580	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ∈ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
16	4, 15	bitri 277	. 2 ⊢ (𝐴 ∈ (^◡ℑ “ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
17		logrn 26544	. . 3 ⊢ ran log = (^◡ℑ “ (-π(,]π))
18	17	eleq2i 2833	. 2 ⊢ (𝐴 ∈ ran log ↔ 𝐴 ∈ (^◡ℑ “ (-π(,]π)))
19		3anass 1101	. 2 ⊢ ((𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
20	16, 18, 19	3bitr4i 305	1 ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 397 ∧ w3a 1093 ∈ wcel 2121 class class class wbr 5075 ^◡ccnv 5620 ran crn 5622 “ cima 5624 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 ℝcr 11032 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 -cneg 11373 (,]cioc 13294 ℑcim 15055 πcpi 16026 logclog 26540

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24307 df-ms 24308 df-tms 24309 df-cncf 24867 df-limc 25855 df-dv 25856 df-log 26542

This theorem is referenced by: relogrn 26547 logrncn 26548 logimcl 26555 logrnaddcl 26560 logi 26573 logneg 26574 logcj 26592 logimul 26600 logneg2 26601 logcnlem4 26631 logf1o2 26636 logreclem 26748 asinsin 26878 asin1 26880 atanlogaddlem 26899 atanlogsub 26902 atantan 26909

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