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Mirrors > Home > MPE Home > Th. List > efifo | Structured version Visualization version GIF version |
Description: The exponential function of an imaginary number maps the reals onto the unit circle. (Contributed by Mario Carneiro, 13-May-2014.) |
Ref | Expression |
---|---|
efifo.1 | β’ πΉ = (π§ β β β¦ (expβ(i Β· π§))) |
efifo.2 | β’ πΆ = (β‘abs β {1}) |
Ref | Expression |
---|---|
efifo | β’ πΉ:ββontoβπΆ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efifo.1 | . . . 4 β’ πΉ = (π§ β β β¦ (expβ(i Β· π§))) | |
2 | ax-icn 11166 | . . . . . . . 8 β’ i β β | |
3 | recn 11197 | . . . . . . . 8 β’ (π§ β β β π§ β β) | |
4 | mulcl 11191 | . . . . . . . 8 β’ ((i β β β§ π§ β β) β (i Β· π§) β β) | |
5 | 2, 3, 4 | sylancr 586 | . . . . . . 7 β’ (π§ β β β (i Β· π§) β β) |
6 | efcl 16028 | . . . . . . 7 β’ ((i Β· π§) β β β (expβ(i Β· π§)) β β) | |
7 | 5, 6 | syl 17 | . . . . . 6 β’ (π§ β β β (expβ(i Β· π§)) β β) |
8 | absefi 16142 | . . . . . 6 β’ (π§ β β β (absβ(expβ(i Β· π§))) = 1) | |
9 | absf 15286 | . . . . . . 7 β’ abs:ββΆβ | |
10 | ffn 6708 | . . . . . . 7 β’ (abs:ββΆβ β abs Fn β) | |
11 | fniniseg 7052 | . . . . . . 7 β’ (abs Fn β β ((expβ(i Β· π§)) β (β‘abs β {1}) β ((expβ(i Β· π§)) β β β§ (absβ(expβ(i Β· π§))) = 1))) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . 6 β’ ((expβ(i Β· π§)) β (β‘abs β {1}) β ((expβ(i Β· π§)) β β β§ (absβ(expβ(i Β· π§))) = 1)) |
13 | 7, 8, 12 | sylanbrc 582 | . . . . 5 β’ (π§ β β β (expβ(i Β· π§)) β (β‘abs β {1})) |
14 | efifo.2 | . . . . 5 β’ πΆ = (β‘abs β {1}) | |
15 | 13, 14 | eleqtrrdi 2836 | . . . 4 β’ (π§ β β β (expβ(i Β· π§)) β πΆ) |
16 | 1, 15 | fmpti 7104 | . . 3 β’ πΉ:ββΆπΆ |
17 | ffn 6708 | . . 3 β’ (πΉ:ββΆπΆ β πΉ Fn β) | |
18 | 16, 17 | ax-mp 5 | . 2 β’ πΉ Fn β |
19 | frn 6715 | . . . 4 β’ (πΉ:ββΆπΆ β ran πΉ β πΆ) | |
20 | 16, 19 | ax-mp 5 | . . 3 β’ ran πΉ β πΆ |
21 | df-ima 5680 | . . . . 5 β’ (πΉ β (0(,](2 Β· Ο))) = ran (πΉ βΎ (0(,](2 Β· Ο))) | |
22 | 1 | reseq1i 5968 | . . . . . . . 8 β’ (πΉ βΎ (0(,](2 Β· Ο))) = ((π§ β β β¦ (expβ(i Β· π§))) βΎ (0(,](2 Β· Ο))) |
23 | 0xr 11260 | . . . . . . . . . . . 12 β’ 0 β β* | |
24 | 2re 12285 | . . . . . . . . . . . . 13 β’ 2 β β | |
25 | pire 26334 | . . . . . . . . . . . . 13 β’ Ο β β | |
26 | 24, 25 | remulcli 11229 | . . . . . . . . . . . 12 β’ (2 Β· Ο) β β |
27 | elioc2 13388 | . . . . . . . . . . . 12 β’ ((0 β β* β§ (2 Β· Ο) β β) β (π§ β (0(,](2 Β· Ο)) β (π§ β β β§ 0 < π§ β§ π§ β€ (2 Β· Ο)))) | |
28 | 23, 26, 27 | mp2an 689 | . . . . . . . . . . 11 β’ (π§ β (0(,](2 Β· Ο)) β (π§ β β β§ 0 < π§ β§ π§ β€ (2 Β· Ο))) |
29 | 28 | simp1bi 1142 | . . . . . . . . . 10 β’ (π§ β (0(,](2 Β· Ο)) β π§ β β) |
30 | 29 | ssriv 3979 | . . . . . . . . 9 β’ (0(,](2 Β· Ο)) β β |
31 | resmpt 6028 | . . . . . . . . 9 β’ ((0(,](2 Β· Ο)) β β β ((π§ β β β¦ (expβ(i Β· π§))) βΎ (0(,](2 Β· Ο))) = (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§)))) | |
32 | 30, 31 | ax-mp 5 | . . . . . . . 8 β’ ((π§ β β β¦ (expβ(i Β· π§))) βΎ (0(,](2 Β· Ο))) = (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) |
33 | 22, 32 | eqtri 2752 | . . . . . . 7 β’ (πΉ βΎ (0(,](2 Β· Ο))) = (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) |
34 | 33 | rneqi 5927 | . . . . . 6 β’ ran (πΉ βΎ (0(,](2 Β· Ο))) = ran (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) |
35 | 0re 11215 | . . . . . . . 8 β’ 0 β β | |
36 | eqid 2724 | . . . . . . . . 9 β’ (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) = (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) | |
37 | 26 | recni 11227 | . . . . . . . . . . . 12 β’ (2 Β· Ο) β β |
38 | 37 | addlidi 11401 | . . . . . . . . . . 11 β’ (0 + (2 Β· Ο)) = (2 Β· Ο) |
39 | 38 | oveq2i 7413 | . . . . . . . . . 10 β’ (0(,](0 + (2 Β· Ο))) = (0(,](2 Β· Ο)) |
40 | 39 | eqcomi 2733 | . . . . . . . . 9 β’ (0(,](2 Β· Ο)) = (0(,](0 + (2 Β· Ο))) |
41 | 36, 14, 40 | efif1o 26421 | . . . . . . . 8 β’ (0 β β β (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))):(0(,](2 Β· Ο))β1-1-ontoβπΆ) |
42 | 35, 41 | ax-mp 5 | . . . . . . 7 β’ (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))):(0(,](2 Β· Ο))β1-1-ontoβπΆ |
43 | f1ofo 6831 | . . . . . . 7 β’ ((π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))):(0(,](2 Β· Ο))β1-1-ontoβπΆ β (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))):(0(,](2 Β· Ο))βontoβπΆ) | |
44 | forn 6799 | . . . . . . 7 β’ ((π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))):(0(,](2 Β· Ο))βontoβπΆ β ran (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) = πΆ) | |
45 | 42, 43, 44 | mp2b 10 | . . . . . 6 β’ ran (π§ β (0(,](2 Β· Ο)) β¦ (expβ(i Β· π§))) = πΆ |
46 | 34, 45 | eqtri 2752 | . . . . 5 β’ ran (πΉ βΎ (0(,](2 Β· Ο))) = πΆ |
47 | 21, 46 | eqtri 2752 | . . . 4 β’ (πΉ β (0(,](2 Β· Ο))) = πΆ |
48 | imassrn 6061 | . . . 4 β’ (πΉ β (0(,](2 Β· Ο))) β ran πΉ | |
49 | 47, 48 | eqsstrri 4010 | . . 3 β’ πΆ β ran πΉ |
50 | 20, 49 | eqssi 3991 | . 2 β’ ran πΉ = πΆ |
51 | df-fo 6540 | . 2 β’ (πΉ:ββontoβπΆ β (πΉ Fn β β§ ran πΉ = πΆ)) | |
52 | 18, 50, 51 | mpbir2an 708 | 1 β’ πΉ:ββontoβπΆ |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3941 {csn 4621 class class class wbr 5139 β¦ cmpt 5222 β‘ccnv 5666 ran crn 5668 βΎ cres 5669 β cima 5670 Fn wfn 6529 βΆwf 6530 βontoβwfo 6532 β1-1-ontoβwf1o 6533 βcfv 6534 (class class class)co 7402 βcc 11105 βcr 11106 0cc0 11107 1c1 11108 ici 11109 + caddc 11110 Β· cmul 11112 β*cxr 11246 < clt 11247 β€ cle 11248 2c2 12266 (,]cioc 13326 abscabs 15183 expce 16007 Οcpi 16012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 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 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-ioc 13330 df-ico 13331 df-icc 13332 df-fz 13486 df-fzo 13629 df-fl 13758 df-mod 13836 df-seq 13968 df-exp 14029 df-fac 14235 df-bc 14264 df-hash 14292 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-pi 16018 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-mulg 18992 df-cntz 19229 df-cmn 19698 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-fbas 21231 df-fg 21232 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-ntr 22868 df-cls 22869 df-nei 22946 df-lp 22984 df-perf 22985 df-cn 23075 df-cnp 23076 df-haus 23163 df-tx 23410 df-hmeo 23603 df-fil 23694 df-fm 23786 df-flim 23787 df-flf 23788 df-xms 24170 df-ms 24171 df-tms 24172 df-cncf 24742 df-limc 25739 df-dv 25740 |
This theorem is referenced by: circgrp 26427 circsubm 26428 circtopn 33337 circcn 33338 |
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