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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 10619 | . . . . . . . 8 ⊢ i ∈ ℂ | |
3 | recn 10650 | . . . . . . . 8 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ) | |
4 | mulcl 10644 | . . . . . . . 8 ⊢ ((i ∈ ℂ ∧ 𝑧 ∈ ℂ) → (i · 𝑧) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 591 | . . . . . . 7 ⊢ (𝑧 ∈ ℝ → (i · 𝑧) ∈ ℂ) |
6 | efcl 15469 | . . . . . . 7 ⊢ ((i · 𝑧) ∈ ℂ → (exp‘(i · 𝑧)) ∈ ℂ) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ ℂ) |
8 | absefi 15582 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (abs‘(exp‘(i · 𝑧))) = 1) | |
9 | absf 14730 | . . . . . . 7 ⊢ abs:ℂ⟶ℝ | |
10 | ffn 6491 | . . . . . . 7 ⊢ (abs:ℂ⟶ℝ → abs Fn ℂ) | |
11 | fniniseg 6814 | . . . . . . 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 587 | . . . . 5 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ (◡abs “ {1})) |
14 | efifo.2 | . . . . 5 ⊢ 𝐶 = (◡abs “ {1}) | |
15 | 13, 14 | eleqtrrdi 2862 | . . . 4 ⊢ (𝑧 ∈ ℝ → (exp‘(i · 𝑧)) ∈ 𝐶) |
16 | 1, 15 | fmpti 6860 | . . 3 ⊢ 𝐹:ℝ⟶𝐶 |
17 | ffn 6491 | . . 3 ⊢ (𝐹:ℝ⟶𝐶 → 𝐹 Fn ℝ) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ 𝐹 Fn ℝ |
19 | frn 6497 | . . . 4 ⊢ (𝐹:ℝ⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
20 | 16, 19 | ax-mp 5 | . . 3 ⊢ ran 𝐹 ⊆ 𝐶 |
21 | df-ima 5530 | . . . . 5 ⊢ (𝐹 “ (0(,](2 · π))) = ran (𝐹 ↾ (0(,](2 · π))) | |
22 | 1 | reseq1i 5812 | . . . . . . . 8 ⊢ (𝐹 ↾ (0(,](2 · π))) = ((𝑧 ∈ ℝ ↦ (exp‘(i · 𝑧))) ↾ (0(,](2 · π))) |
23 | 0xr 10711 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ* | |
24 | 2re 11733 | . . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ | |
25 | pire 25135 | . . . . . . . . . . . . 13 ⊢ π ∈ ℝ | |
26 | 24, 25 | remulcli 10680 | . . . . . . . . . . . 12 ⊢ (2 · π) ∈ ℝ |
27 | elioc2 12827 | . . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ) → (𝑧 ∈ (0(,](2 · π)) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ (2 · π)))) | |
28 | 23, 26, 27 | mp2an 692 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,](2 · π)) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ (2 · π))) |
29 | 28 | simp1bi 1143 | . . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,](2 · π)) → 𝑧 ∈ ℝ) |
30 | 29 | ssriv 3892 | . . . . . . . . 9 ⊢ (0(,](2 · π)) ⊆ ℝ |
31 | resmpt 5870 | . . . . . . . . 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 2782 | . . . . . . 7 ⊢ (𝐹 ↾ (0(,](2 · π))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
34 | 33 | rneqi 5771 | . . . . . 6 ⊢ ran (𝐹 ↾ (0(,](2 · π))) = ran (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) |
35 | 0re 10666 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
36 | eqid 2759 | . . . . . . . . 9 ⊢ (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) = (𝑧 ∈ (0(,](2 · π)) ↦ (exp‘(i · 𝑧))) | |
37 | 26 | recni 10678 | . . . . . . . . . . . 12 ⊢ (2 · π) ∈ ℂ |
38 | 37 | addid2i 10851 | . . . . . . . . . . 11 ⊢ (0 + (2 · π)) = (2 · π) |
39 | 38 | oveq2i 7154 | . . . . . . . . . 10 ⊢ (0(,](0 + (2 · π))) = (0(,](2 · π)) |
40 | 39 | eqcomi 2768 | . . . . . . . . 9 ⊢ (0(,](2 · π)) = (0(,](0 + (2 · π))) |
41 | 36, 14, 40 | efif1o 25222 | . . . . . . . 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 6602 | . . . . . . 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 6572 | . . . . . . 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 2782 | . . . . 5 ⊢ ran (𝐹 ↾ (0(,](2 · π))) = 𝐶 |
47 | 21, 46 | eqtri 2782 | . . . 4 ⊢ (𝐹 “ (0(,](2 · π))) = 𝐶 |
48 | imassrn 5905 | . . . 4 ⊢ (𝐹 “ (0(,](2 · π))) ⊆ ran 𝐹 | |
49 | 47, 48 | eqsstrri 3923 | . . 3 ⊢ 𝐶 ⊆ ran 𝐹 |
50 | 20, 49 | eqssi 3904 | . 2 ⊢ ran 𝐹 = 𝐶 |
51 | df-fo 6334 | . 2 ⊢ (𝐹:ℝ–onto→𝐶 ↔ (𝐹 Fn ℝ ∧ ran 𝐹 = 𝐶)) | |
52 | 18, 50, 51 | mpbir2an 711 | 1 ⊢ 𝐹:ℝ–onto→𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ⊆ wss 3854 {csn 4515 class class class wbr 5025 ↦ cmpt 5105 ◡ccnv 5516 ran crn 5518 ↾ cres 5519 “ cima 5520 Fn wfn 6323 ⟶wf 6324 –onto→wfo 6326 –1-1-onto→wf1o 6327 ‘cfv 6328 (class class class)co 7143 ℂcc 10558 ℝcr 10559 0cc0 10560 1c1 10561 ici 10562 + caddc 10563 · cmul 10565 ℝ*cxr 10697 < clt 10698 ≤ cle 10699 2c2 11714 (,]cioc 12765 abscabs 14626 expce 15448 πcpi 15453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-inf2 9122 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 ax-addf 10639 ax-mulf 10640 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-iin 4879 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-se 5477 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-of 7398 df-om 7573 df-1st 7686 df-2nd 7687 df-supp 7829 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8473 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-fsupp 8852 df-fi 8893 df-sup 8924 df-inf 8925 df-oi 8992 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-q 12374 df-rp 12416 df-xneg 12533 df-xadd 12534 df-xmul 12535 df-ioo 12768 df-ioc 12769 df-ico 12770 df-icc 12771 df-fz 12925 df-fzo 13068 df-fl 13196 df-mod 13272 df-seq 13404 df-exp 13465 df-fac 13669 df-bc 13698 df-hash 13726 df-shft 14459 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 df-limsup 14861 df-clim 14878 df-rlim 14879 df-sum 15076 df-ef 15454 df-sin 15456 df-cos 15457 df-pi 15459 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-ress 16534 df-plusg 16621 df-mulr 16622 df-starv 16623 df-sca 16624 df-vsca 16625 df-ip 16626 df-tset 16627 df-ple 16628 df-ds 16630 df-unif 16631 df-hom 16632 df-cco 16633 df-rest 16739 df-topn 16740 df-0g 16758 df-gsum 16759 df-topgen 16760 df-pt 16761 df-prds 16764 df-xrs 16818 df-qtop 16823 df-imas 16824 df-xps 16826 df-mre 16900 df-mrc 16901 df-acs 16903 df-mgm 17903 df-sgrp 17952 df-mnd 17963 df-submnd 18008 df-mulg 18277 df-cntz 18499 df-cmn 18960 df-psmet 20143 df-xmet 20144 df-met 20145 df-bl 20146 df-mopn 20147 df-fbas 20148 df-fg 20149 df-cnfld 20152 df-top 21579 df-topon 21596 df-topsp 21618 df-bases 21631 df-cld 21704 df-ntr 21705 df-cls 21706 df-nei 21783 df-lp 21821 df-perf 21822 df-cn 21912 df-cnp 21913 df-haus 22000 df-tx 22247 df-hmeo 22440 df-fil 22531 df-fm 22623 df-flim 22624 df-flf 22625 df-xms 23007 df-ms 23008 df-tms 23009 df-cncf 23564 df-limc 24550 df-dv 24551 |
This theorem is referenced by: circgrp 25228 circsubm 25229 circtopn 31293 circcn 31294 |
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