ellogrn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ellogrn		Structured version Visualization version GIF version

Theorem ellogrn 26704

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 15167	. . . 4 ⊢ ℑ:ℂ⟶ℝ
2		ffn 6709	. . . 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 26599	. . . . . . . . 9 ⊢ π ∈ ℝ
6	5	renegcli 11522	. . . . . . . 8 ⊢ -π ∈ ℝ
7	6	rexri 11270	. . . . . . 7 ⊢ -π ∈ ℝ^*
8		elioc2 13439	. . . . . . 7 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ) → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
9	7, 5, 8	mp2an 704	. . . . . 6 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))
10		3anass 1109	. . . . . 6 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
11	9, 10	bitri 278	. . . . 5 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
12		imcl 15165	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ)
13	12	biantrurd 541	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))))
14	11, 13	bitr4id 293	. . . 4 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
15	14	pm5.32i 584	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ∈ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
16	4, 15	bitri 278	. 2 ⊢ (𝐴 ∈ (^◡ℑ “ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
17		logrn 26703	. . 3 ⊢ ran log = (^◡ℑ “ (-π(,]π))
18	17	eleq2i 2862	. 2 ⊢ (𝐴 ∈ ran log ↔ 𝐴 ∈ (^◡ℑ “ (-π(,]π)))
19		3anass 1109	. 2 ⊢ ((𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))
20	16, 18, 19	3bitr4i 306	1 ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 ∧ w3a 1101 ∈ wcel 2150 class class class wbr 5114 ^◡ccnv 5664 ran crn 5666 “ cima 5668 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7414 ℂcc 11101 ℝcr 11102 ℝ^*cxr 11245 < clt 11246 ≤ cle 11247 -cneg 11445 (,]cioc 13376 ℑcim 15152 πcpi 16123 logclog 26699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-inf2 9613 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 ax-pre-sup 11181 ax-addf 11182

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-se 5619 df-we 5620 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-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7678 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8899 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9928 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-div 11875 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12508 df-z 12595 df-dec 12715 df-uz 12866 df-q 12976 df-rp 13020 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13379 df-ioc 13380 df-ico 13381 df-icc 13382 df-fz 13539 df-fzo 13686 df-fl 13828 df-mod 13906 df-seq 14041 df-exp 14101 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15107 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-limsup 15525 df-clim 15542 df-rlim 15543 df-sum 15741 df-ef 16124 df-sin 16126 df-cos 16127 df-pi 16129 df-struct 17210 df-sets 17227 df-slot 17245 df-ndx 17257 df-base 17273 df-ress 17294 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17478 df-topn 17479 df-0g 17497 df-gsum 17498 df-topgen 17499 df-pt 17500 df-prds 17503 df-xrs 17559 df-qtop 17564 df-imas 17565 df-xps 17567 df-mre 17641 df-mrc 17642 df-acs 17644 df-mgm 18701 df-sgrp 18780 df-mnd 18796 df-submnd 18845 df-mulg 19137 df-cntz 19390 df-cmn 19855 df-psmet 21497 df-xmet 21498 df-met 21499 df-bl 21500 df-mopn 21501 df-fbas 21502 df-fg 21503 df-cnfld 21506 df-top 23034 df-topon 23051 df-topsp 23073 df-bases 23086 df-cld 23159 df-ntr 23160 df-cls 23161 df-nei 23238 df-lp 23276 df-perf 23277 df-cn 23367 df-cnp 23368 df-haus 23455 df-tx 23702 df-hmeo 23895 df-fil 23986 df-fm 24078 df-flim 24079 df-flf 24080 df-xms 24460 df-ms 24461 df-tms 24462 df-cncf 25020 df-limc 26008 df-dv 26009 df-log 26701

This theorem is referenced by: relogrn 26706 logrncn 26707 logimcl 26714 logrnaddcl 26719 logi 26732 logneg 26733 logcj 26751 logimul 26759 logneg2 26760 logcnlem4 26790 logf1o2 26795 logreclem 26907 asinsin 27037 asin1 27039 atanlogaddlem 27058 atanlogsub 27061 atantan 27068

W3C validator