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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 | ⊢ 𝐹 = (𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) |
efifo.2 | ⊢ 𝐶 = (◡abs “ {1}) |
Ref | Expression |
---|---|
efifo | ⊢ 𝐹:ℝ–onto→𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efifo.1 | . . . 4 ⊢ 𝐹 = (𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) | |
2 | ax-icn 10447 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 10478 | . . . . . . . 8 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ) | |
4 | mulcl 10472 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑧 ∈ ℂ) → (i · 𝑧) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 587 | . . . . . . 7 ⊢ (𝑧 ∈ ℝ → (i · 𝑧) ∈ ℂ) |
6 | efcl 15274 | . . . . . . 7 ⊢ ((i · 𝑧) ∈ ℂ → (exp‘(i · 𝑧)) ∈ ℂ) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ ℂ) |
8 | absefi 15387 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (abs‘(exp‘(i · 𝑧))) = 1) | |
9 | absf 14536 | . . . . . . 7 ⊢ abs:ℂ⟶ℝ | |
10 | ffn 6387 | . . . . . . 7 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
11 | fniniseg 6700 | . . . . . . 7 ⊢ (abs Fn ℂ → ((exp‘(i · 𝑧)) ∈ (◡abs “ {1}) ↔ ((exp‘(i · 𝑧)) ∈ ℂ ∧ (abs‘(exp‘(i · 𝑧))) = 1))) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . 6 ⊢ ((exp‘(i · 𝑧)) ∈ (◡abs “ {1}) ↔ ((exp‘(i · 𝑧)) ∈ ℂ ∧ (abs‘(exp‘(i · 𝑧))) = 1)) |
13 | 7, 8, 12 | sylanbrc 583 | . . . . 5 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ (◡abs “ {1})) |
14 | efifo.2 | . . . . 5 ⊢ 𝐶 = (◡abs “ {1}) | |
15 | 13, 14 | syl6eleqr 2894 | . . . 4 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ 𝐶) |
16 | 1, 15 | fmpti 6744 | . . 3 ⊢ 𝐹:ℝ⟶𝐶 |
17 | ffn 6387 | . . 3 ⊢ (𝐹:ℝ⟶𝐶 → 𝐹 Fn ℝ) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ 𝐹 Fn ℝ |
19 | frn 6393 | . . . 4 ⊢ (𝐹:ℝ⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
20 | 16, 19 | ax-mp 5 | . . 3 ⊢ ran 𝐹 ⊆ 𝐶 |
21 | df-ima 5461 | . . . . 5 ⊢ (𝐹 “ (0(,](2 · π))) = ran (𝐹 ↾ (0(,](2 · π))) | |
22 | 1 | reseq1i 5735 | . . . . . . . 8 ⊢ (𝐹 ↾ (0(,](2 · π))) = ((𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) ↾ (0(,](2 · π))) |
23 | 0xr 10539 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ* | |
24 | 2re 11564 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
25 | pire 24732 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ | |
26 | 24, 25 | remulcli 10508 | . . . . . . . . . . . 12 ⊢ (2 · π) ∈ ℝ |
27 | elioc2 12654 | . . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ) → (𝑧 ∈ (0(,](2 · π)) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ (2 · π)))) | |
28 | 23, 26, 27 | mp2an 688 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,](2 · π)) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ (2 · π))) |
29 | 28 | simp1bi 1138 | . . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,](2 · π)) → 𝑧 ∈ ℝ) |
30 | 29 | ssriv 3897 | . . . . . . . . 9 ⊢ (0(,](2 · π)) ⊆ ℝ |
31 | resmpt 5791 | . . . . . . . . 9 ⊢ ((0(,](2 · π)) ⊆ ℝ → ((𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) ↾ (0(,](2 · π))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧)))) | |
32 | 30, 31 | ax-mp 5 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) ↾ (0(,](2 · π))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
33 | 22, 32 | eqtri 2819 | . . . . . . 7 ⊢ (𝐹 ↾ (0(,](2 · π))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
34 | 33 | rneqi 5694 | . . . . . 6 ⊢ ran (𝐹 ↾ (0(,](2 · π))) = ran (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
35 | 0re 10494 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
36 | eqid 2795 | . . . . . . . . 9 ⊢ (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) | |
37 | 26 | recni 10506 | . . . . . . . . . . . 12 ⊢ (2 · π) ∈ ℂ |
38 | 37 | addid2i 10680 | . . . . . . . . . . 11 ⊢ (0 + (2 · π)) = (2 · π) |
39 | 38 | oveq2i 7032 | . . . . . . . . . 10 ⊢ (0(,](0 + (2 · π))) = (0(,](2 · π)) |
40 | 39 | eqcomi 2804 | . . . . . . . . 9 ⊢ (0(,](2 · π)) = (0(,](0 + (2 · π))) |
41 | 36, 14, 40 | efif1o 24816 | . . . . . . . 8 ⊢ (0 ∈ ℝ → (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))):(0(,](2 · π))–1-1-onto→𝐶) |
42 | 35, 41 | ax-mp 5 | . . . . . . 7 ⊢ (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))):(0(,](2 · π))–1-1-onto→𝐶 |
43 | f1ofo 6495 | . . . . . . 7 ⊢ ((𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))):(0(,](2 · π))–1-1-onto→𝐶 → (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))):(0(,](2 · π))–onto→𝐶) | |
44 | forn 6466 | . . . . . . 7 ⊢ ((𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))):(0(,](2 · π))–onto→𝐶 → ran (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) = 𝐶) | |
45 | 42, 43, 44 | mp2b 10 | . . . . . 6 ⊢ ran (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) = 𝐶 |
46 | 34, 45 | eqtri 2819 | . . . . 5 ⊢ ran (𝐹 ↾ (0(,](2 · π))) = 𝐶 |
47 | 21, 46 | eqtri 2819 | . . . 4 ⊢ (𝐹 “ (0(,](2 · π))) = 𝐶 |
48 | imassrn 5822 | . . . 4 ⊢ (𝐹 “ (0(,](2 · π))) ⊆ ran 𝐹 | |
49 | 47, 48 | eqsstrri 3927 | . . 3 ⊢ 𝐶 ⊆ ran 𝐹 |
50 | 20, 49 | eqssi 3909 | . 2 ⊢ ran 𝐹 = 𝐶 |
51 | df-fo 6236 | . 2 ⊢ (𝐹:ℝ–onto→𝐶 ↔ (𝐹 Fn ℝ ∧ ran 𝐹 = 𝐶)) | |
52 | 18, 50, 51 | mpbir2an 707 | 1 ⊢ 𝐹:ℝ–onto→𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ⊆ wss 3863 {csn 4476 class class class wbr 4966 ↦ cmpt 5045 ◡ccnv 5447 ran crn 5449 ↾ cres 5450 “ cima 5451 Fn wfn 6225 ⟶wf 6226 –onto→wfo 6228 –1-1-onto→wf1o 6229 ‘cfv 6230 (class class class)co 7021 ℂcc 10386 ℝcr 10387 0cc0 10388 1c1 10389 ici 10390 + caddc 10391 · cmul 10393 ℝ*cxr 10525 < clt 10526 ≤ cle 10527 2c2 11545 (,]cioc 12594 abscabs 14432 expce 15253 πcpi 15258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-pre-sup 10466 ax-addf 10467 ax-mulf 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-oadd 7962 df-er 8144 df-map 8263 df-pm 8264 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-fi 8726 df-sup 8757 df-inf 8758 df-oi 8825 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-div 11151 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-q 12203 df-rp 12245 df-xneg 12362 df-xadd 12363 df-xmul 12364 df-ioo 12597 df-ioc 12598 df-ico 12599 df-icc 12600 df-fz 12748 df-fzo 12889 df-fl 13017 df-mod 13093 df-seq 13225 df-exp 13285 df-fac 13489 df-bc 13518 df-hash 13546 df-shft 14265 df-cj 14297 df-re 14298 df-im 14299 df-sqrt 14433 df-abs 14434 df-limsup 14667 df-clim 14684 df-rlim 14685 df-sum 14882 df-ef 15259 df-sin 15261 df-cos 15262 df-pi 15264 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-hom 16423 df-cco 16424 df-rest 16530 df-topn 16531 df-0g 16549 df-gsum 16550 df-topgen 16551 df-pt 16552 df-prds 16555 df-xrs 16609 df-qtop 16614 df-imas 16615 df-xps 16617 df-mre 16691 df-mrc 16692 df-acs 16694 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-mulg 17987 df-cntz 18193 df-cmn 18640 df-psmet 20224 df-xmet 20225 df-met 20226 df-bl 20227 df-mopn 20228 df-fbas 20229 df-fg 20230 df-cnfld 20233 df-top 21191 df-topon 21208 df-topsp 21230 df-bases 21243 df-cld 21316 df-ntr 21317 df-cls 21318 df-nei 21395 df-lp 21433 df-perf 21434 df-cn 21524 df-cnp 21525 df-haus 21612 df-tx 21859 df-hmeo 22052 df-fil 22143 df-fm 22235 df-flim 22236 df-flf 22237 df-xms 22618 df-ms 22619 df-tms 22620 df-cncf 23174 df-limc 24152 df-dv 24153 |
This theorem is referenced by: circgrp 24822 circsubm 24823 circtopn 30723 circcn 30724 |
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