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Mirrors > Home > MPE Home > Th. List > ellogrn | Structured version Visualization version GIF version |
Description: Write out the property 𝐴 ∈ ran log explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.) |
Ref | Expression |
---|---|
ellogrn | ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imf 14332 | . . . 4 ⊢ ℑ:ℂ⟶ℝ | |
2 | ffn 6342 | . . . 4 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ) | |
3 | elpreima 6652 | . . . 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 | imcl 14330 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
6 | 5 | biantrurd 525 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)))) |
7 | pire 24763 | . . . . . . . . 9 ⊢ π ∈ ℝ | |
8 | 7 | renegcli 10747 | . . . . . . . 8 ⊢ -π ∈ ℝ |
9 | 8 | rexri 10498 | . . . . . . 7 ⊢ -π ∈ ℝ* |
10 | elioc2 12614 | . . . . . . 7 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) | |
11 | 9, 7, 10 | mp2an 680 | . . . . . 6 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)) |
12 | 3anass 1077 | . . . . . 6 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) | |
13 | 11, 12 | bitri 267 | . . . . 5 ⊢ ((ℑ‘𝐴) ∈ (-π(,]π) ↔ ((ℑ‘𝐴) ∈ ℝ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) |
14 | 6, 13 | syl6rbbr 282 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) ∈ (-π(,]π) ↔ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) |
15 | 14 | pm5.32i 567 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ∈ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) |
16 | 4, 15 | bitri 267 | . 2 ⊢ (𝐴 ∈ (◡ℑ “ (-π(,]π)) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) |
17 | logrn 24859 | . . 3 ⊢ ran log = (◡ℑ “ (-π(,]π)) | |
18 | 17 | eleq2i 2852 | . 2 ⊢ (𝐴 ∈ ran log ↔ 𝐴 ∈ (◡ℑ “ (-π(,]π))) |
19 | 3anass 1077 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π) ↔ (𝐴 ∈ ℂ ∧ (-π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))) | |
20 | 16, 18, 19 | 3bitr4i 295 | 1 ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ∧ w3a 1069 ∈ wcel 2051 class class class wbr 4926 ◡ccnv 5403 ran crn 5405 “ cima 5407 Fn wfn 6181 ⟶wf 6182 ‘cfv 6186 (class class class)co 6975 ℂcc 10332 ℝcr 10333 ℝ*cxr 10472 < clt 10473 ≤ cle 10474 -cneg 10670 (,]cioc 12554 ℑcim 14317 πcpi 15279 logclog 24855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 ax-addf 10413 ax-mulf 10414 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-iin 4792 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-om 7396 df-1st 7500 df-2nd 7501 df-supp 7633 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-ixp 8259 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-fsupp 8628 df-fi 8669 df-sup 8700 df-inf 8701 df-oi 8768 df-card 9161 df-cda 9387 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-dec 11911 df-uz 12058 df-q 12162 df-rp 12204 df-xneg 12323 df-xadd 12324 df-xmul 12325 df-ioo 12557 df-ioc 12558 df-ico 12559 df-icc 12560 df-fz 12708 df-fzo 12849 df-fl 12976 df-mod 13052 df-seq 13184 df-exp 13244 df-fac 13448 df-bc 13477 df-hash 13505 df-shft 14286 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-limsup 14688 df-clim 14705 df-rlim 14706 df-sum 14903 df-ef 15280 df-sin 15282 df-cos 15283 df-pi 15285 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-starv 16435 df-sca 16436 df-vsca 16437 df-ip 16438 df-tset 16439 df-ple 16440 df-ds 16442 df-unif 16443 df-hom 16444 df-cco 16445 df-rest 16551 df-topn 16552 df-0g 16570 df-gsum 16571 df-topgen 16572 df-pt 16573 df-prds 16576 df-xrs 16630 df-qtop 16635 df-imas 16636 df-xps 16638 df-mre 16728 df-mrc 16729 df-acs 16731 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-submnd 17817 df-mulg 18025 df-cntz 18231 df-cmn 18681 df-psmet 20255 df-xmet 20256 df-met 20257 df-bl 20258 df-mopn 20259 df-fbas 20260 df-fg 20261 df-cnfld 20264 df-top 21222 df-topon 21239 df-topsp 21261 df-bases 21274 df-cld 21347 df-ntr 21348 df-cls 21349 df-nei 21426 df-lp 21464 df-perf 21465 df-cn 21555 df-cnp 21556 df-haus 21643 df-tx 21890 df-hmeo 22083 df-fil 22174 df-fm 22266 df-flim 22267 df-flf 22268 df-xms 22649 df-ms 22650 df-tms 22651 df-cncf 23205 df-limc 24183 df-dv 24184 df-log 24857 |
This theorem is referenced by: relogrn 24862 logrncn 24863 logimcl 24870 logrnaddcl 24875 logneg 24888 logcj 24906 logimul 24914 logneg2 24915 logcnlem4 24945 logf1o2 24950 logreclem 25057 asinsin 25187 asin1 25189 atanlogaddlem 25208 atanlogsub 25211 atantan 25218 logi 32519 |
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