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Mirrors > Home > MPE Home > Th. List > eff1o | Structured version Visualization version GIF version |
Description: The exponential function maps the set 𝑆, of complex numbers with imaginary part in the closed-above, open-below interval from -π to π one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 16-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.) |
Ref | Expression |
---|---|
eff1o.1 | ⊢ 𝑆 = (◡ℑ “ (-π(,]π)) |
Ref | Expression |
---|---|
eff1o | ⊢ (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pire 24552 | . . 3 ⊢ π ∈ ℝ | |
2 | 1 | renegcli 10634 | . 2 ⊢ -π ∈ ℝ |
3 | eqid 2799 | . . 3 ⊢ (𝑤 ∈ (-π(,]π) ↦ (exp‘(i · 𝑤))) = (𝑤 ∈ (-π(,]π) ↦ (exp‘(i · 𝑤))) | |
4 | eff1o.1 | . . 3 ⊢ 𝑆 = (◡ℑ “ (-π(,]π)) | |
5 | rexr 10374 | . . . 4 ⊢ (-π ∈ ℝ → -π ∈ ℝ*) | |
6 | iocssre 12502 | . . . 4 ⊢ ((-π ∈ ℝ* ∧ π ∈ ℝ) → (-π(,]π) ⊆ ℝ) | |
7 | 5, 1, 6 | sylancl 581 | . . 3 ⊢ (-π ∈ ℝ → (-π(,]π) ⊆ ℝ) |
8 | picn 24553 | . . . . . . . 8 ⊢ π ∈ ℂ | |
9 | 8 | 2timesi 11458 | . . . . . . 7 ⊢ (2 · π) = (π + π) |
10 | 9 | oveq2i 6889 | . . . . . 6 ⊢ (-π + (2 · π)) = (-π + (π + π)) |
11 | negpicn 24556 | . . . . . . 7 ⊢ -π ∈ ℂ | |
12 | 8, 8 | addcli 10335 | . . . . . . 7 ⊢ (π + π) ∈ ℂ |
13 | 11, 12 | addcomi 10517 | . . . . . 6 ⊢ (-π + (π + π)) = ((π + π) + -π) |
14 | 12, 8 | negsubi 10651 | . . . . . . 7 ⊢ ((π + π) + -π) = ((π + π) − π) |
15 | 8, 8 | pncan3oi 10589 | . . . . . . 7 ⊢ ((π + π) − π) = π |
16 | 14, 15 | eqtri 2821 | . . . . . 6 ⊢ ((π + π) + -π) = π |
17 | 10, 13, 16 | 3eqtrri 2826 | . . . . 5 ⊢ π = (-π + (2 · π)) |
18 | 17 | oveq2i 6889 | . . . 4 ⊢ (-π(,]π) = (-π(,](-π + (2 · π))) |
19 | 18 | efif1olem1 24630 | . . 3 ⊢ ((-π ∈ ℝ ∧ (𝑥 ∈ (-π(,]π) ∧ 𝑦 ∈ (-π(,]π))) → (abs‘(𝑥 − 𝑦)) < (2 · π)) |
20 | 18 | efif1olem2 24631 | . . 3 ⊢ ((-π ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ (-π(,]π)((𝑧 − 𝑦) / (2 · π)) ∈ ℤ) |
21 | 3, 4, 7, 19, 20 | eff1olem 24636 | . 2 ⊢ (-π ∈ ℝ → (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0})) |
22 | 2, 21 | ax-mp 5 | 1 ⊢ (exp ↾ 𝑆):𝑆–1-1-onto→(ℂ ∖ {0}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ∖ cdif 3766 ⊆ wss 3769 {csn 4368 ↦ cmpt 4922 ◡ccnv 5311 ↾ cres 5314 “ cima 5315 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 ici 10226 + caddc 10227 · cmul 10229 ℝ*cxr 10362 − cmin 10556 -cneg 10557 2c2 11368 (,]cioc 12425 ℑcim 14179 expce 15128 πcpi 15133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-fi 8559 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-q 12034 df-rp 12075 df-xneg 12193 df-xadd 12194 df-xmul 12195 df-ioo 12428 df-ioc 12429 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-mod 12924 df-seq 13056 df-exp 13115 df-fac 13314 df-bc 13343 df-hash 13371 df-shft 14148 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-limsup 14543 df-clim 14560 df-rlim 14561 df-sum 14758 df-ef 15134 df-sin 15136 df-cos 15137 df-pi 15139 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-hom 16291 df-cco 16292 df-rest 16398 df-topn 16399 df-0g 16417 df-gsum 16418 df-topgen 16419 df-pt 16420 df-prds 16423 df-xrs 16477 df-qtop 16482 df-imas 16483 df-xps 16485 df-mre 16561 df-mrc 16562 df-acs 16564 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-mulg 17857 df-cntz 18062 df-cmn 18510 df-psmet 20060 df-xmet 20061 df-met 20062 df-bl 20063 df-mopn 20064 df-fbas 20065 df-fg 20066 df-cnfld 20069 df-top 21027 df-topon 21044 df-topsp 21066 df-bases 21079 df-cld 21152 df-ntr 21153 df-cls 21154 df-nei 21231 df-lp 21269 df-perf 21270 df-cn 21360 df-cnp 21361 df-haus 21448 df-tx 21694 df-hmeo 21887 df-fil 21978 df-fm 22070 df-flim 22071 df-flf 22072 df-xms 22453 df-ms 22454 df-tms 22455 df-cncf 23009 df-limc 23971 df-dv 23972 |
This theorem is referenced by: logrn 24646 eff1o2 24651 |
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