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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 10598 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 10629 | . . . . . . . 8 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ) | |
4 | mulcl 10623 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑧 ∈ ℂ) → (i · 𝑧) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 589 | . . . . . . 7 ⊢ (𝑧 ∈ ℝ → (i · 𝑧) ∈ ℂ) |
6 | efcl 15438 | . . . . . . 7 ⊢ ((i · 𝑧) ∈ ℂ → (exp‘(i · 𝑧)) ∈ ℂ) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ ℂ) |
8 | absefi 15551 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (abs‘(exp‘(i · 𝑧))) = 1) | |
9 | absf 14699 | . . . . . . 7 ⊢ abs:ℂ⟶ℝ | |
10 | ffn 6516 | . . . . . . 7 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
11 | fniniseg 6832 | . . . . . . 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 585 | . . . . 5 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ (◡abs “ {1})) |
14 | efifo.2 | . . . . 5 ⊢ 𝐶 = (◡abs “ {1}) | |
15 | 13, 14 | eleqtrrdi 2926 | . . . 4 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ 𝐶) |
16 | 1, 15 | fmpti 6878 | . . 3 ⊢ 𝐹:ℝ⟶𝐶 |
17 | ffn 6516 | . . 3 ⊢ (𝐹:ℝ⟶𝐶 → 𝐹 Fn ℝ) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ 𝐹 Fn ℝ |
19 | frn 6522 | . . . 4 ⊢ (𝐹:ℝ⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
20 | 16, 19 | ax-mp 5 | . . 3 ⊢ ran 𝐹 ⊆ 𝐶 |
21 | df-ima 5570 | . . . . 5 ⊢ (𝐹 “ (0(,](2 · π))) = ran (𝐹 ↾ (0(,](2 · π))) | |
22 | 1 | reseq1i 5851 | . . . . . . . 8 ⊢ (𝐹 ↾ (0(,](2 · π))) = ((𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) ↾ (0(,](2 · π))) |
23 | 0xr 10690 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ* | |
24 | 2re 11714 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
25 | pire 25046 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ | |
26 | 24, 25 | remulcli 10659 | . . . . . . . . . . . 12 ⊢ (2 · π) ∈ ℝ |
27 | elioc2 12802 | . . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ) → (𝑧 ∈ (0(,](2 · π)) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ (2 · π)))) | |
28 | 23, 26, 27 | mp2an 690 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,](2 · π)) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ (2 · π))) |
29 | 28 | simp1bi 1141 | . . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,](2 · π)) → 𝑧 ∈ ℝ) |
30 | 29 | ssriv 3973 | . . . . . . . . 9 ⊢ (0(,](2 · π)) ⊆ ℝ |
31 | resmpt 5907 | . . . . . . . . 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 2846 | . . . . . . 7 ⊢ (𝐹 ↾ (0(,](2 · π))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
34 | 33 | rneqi 5809 | . . . . . 6 ⊢ ran (𝐹 ↾ (0(,](2 · π))) = ran (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
35 | 0re 10645 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
36 | eqid 2823 | . . . . . . . . 9 ⊢ (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) | |
37 | 26 | recni 10657 | . . . . . . . . . . . 12 ⊢ (2 · π) ∈ ℂ |
38 | 37 | addid2i 10830 | . . . . . . . . . . 11 ⊢ (0 + (2 · π)) = (2 · π) |
39 | 38 | oveq2i 7169 | . . . . . . . . . 10 ⊢ (0(,](0 + (2 · π))) = (0(,](2 · π)) |
40 | 39 | eqcomi 2832 | . . . . . . . . 9 ⊢ (0(,](2 · π)) = (0(,](0 + (2 · π))) |
41 | 36, 14, 40 | efif1o 25132 | . . . . . . . 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 6624 | . . . . . . 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 6595 | . . . . . . 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 2846 | . . . . 5 ⊢ ran (𝐹 ↾ (0(,](2 · π))) = 𝐶 |
47 | 21, 46 | eqtri 2846 | . . . 4 ⊢ (𝐹 “ (0(,](2 · π))) = 𝐶 |
48 | imassrn 5942 | . . . 4 ⊢ (𝐹 “ (0(,](2 · π))) ⊆ ran 𝐹 | |
49 | 47, 48 | eqsstrri 4004 | . . 3 ⊢ 𝐶 ⊆ ran 𝐹 |
50 | 20, 49 | eqssi 3985 | . 2 ⊢ ran 𝐹 = 𝐶 |
51 | df-fo 6363 | . 2 ⊢ (𝐹:ℝ–onto→𝐶 ↔ (𝐹 Fn ℝ ∧ ran 𝐹 = 𝐶)) | |
52 | 18, 50, 51 | mpbir2an 709 | 1 ⊢ 𝐹:ℝ–onto→𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 ◡ccnv 5556 ran crn 5558 ↾ cres 5559 “ cima 5560 Fn wfn 6352 ⟶wf 6353 –onto→wfo 6355 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 ici 10541 + caddc 10542 · cmul 10544 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 2c2 11695 (,]cioc 12742 abscabs 14595 expce 15417 πcpi 15422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 |
This theorem is referenced by: circgrp 25138 circsubm 25139 circtopn 31103 circcn 31104 |
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