efif1o - Metamath Proof Explorer

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Theorem efif1o 25124

Description: The exponential function of an imaginary number maps any open-below, closed-above interval of length 2π one-to-one onto the unit circle. (Contributed by Paul Chapman, 16-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.)

**Hypotheses**
Ref	Expression
efif1o.1	⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
efif1o.2	⊢ 𝐶 = (^◡abs “ {1})
efif1o.3	⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π)))

**Assertion**
Ref	Expression
efif1o	⊢ (𝐴 ∈ ℝ → 𝐹:𝐷–1-1-onto→𝐶)

Distinct variable groups: 𝑤,𝐴 𝑤,𝐶 𝑤,𝐷 Allowed substitution hint: 𝐹(𝑤)

Proof of Theorem efif1o Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efif1o.1	. 2 ⊢ 𝐹 = (𝑤 ∈ 𝐷 ↦ (exp‘(i · 𝑤)))
2		efif1o.2	. 2 ⊢ 𝐶 = (^◡abs “ {1})
3		efif1o.3	. . 3 ⊢ 𝐷 = (𝐴(,](𝐴 + (2 · π)))
4		rexr 10681	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
5		2re 11705	. . . . . . . 8 ⊢ 2 ∈ ℝ
6		pire 25038	. . . . . . . 8 ⊢ π ∈ ℝ
7	5, 6	remulcli 10651	. . . . . . 7 ⊢ (2 · π) ∈ ℝ
8		readdcl 10614	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝐴 + (2 · π)) ∈ ℝ)
9	7, 8	mpan2 689	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 + (2 · π)) ∈ ℝ)
10		elioc2 12793	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 + (2 · π)) ∈ ℝ) → (𝑥 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π)))))
11	4, 9, 10	syl2anc 586	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝑥 ∈ (𝐴(,](𝐴 + (2 · π))) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π)))))
12		simp1 1132	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ (𝐴 + (2 · π))) → 𝑥 ∈ ℝ)
13	11, 12	syl6bi 255	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝑥 ∈ (𝐴(,](𝐴 + (2 · π))) → 𝑥 ∈ ℝ))
14	13	ssrdv 3973	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,](𝐴 + (2 · π))) ⊆ ℝ)
15	3, 14	eqsstrid 4015	. 2 ⊢ (𝐴 ∈ ℝ → 𝐷 ⊆ ℝ)
16	3	efif1olem1 25120	. 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
17	3	efif1olem2 25121	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ∃𝑦 ∈ 𝐷 ((𝑧 − 𝑦) / (2 · π)) ∈ ℤ)
18		eqid 2821	. 2 ⊢ (sin ↾ (-(π / 2)[,](π / 2))) = (sin ↾ (-(π / 2)[,](π / 2)))
19	1, 2, 15, 16, 17, 18	efif1olem4 25123	1 ⊢ (𝐴 ∈ ℝ → 𝐹:𝐷–1-1-onto→𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {csn 4561 class class class wbr 5059 ↦ cmpt 5139 ^◡ccnv 5549 ↾ cres 5552 “ cima 5553 –1-1-onto→wf1o 6349 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 1c1 10532 ici 10533 + caddc 10534 · cmul 10536 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 -cneg 10865 / cdiv 11291 2c2 11686 (,]cioc 12733 [,]cicc 12735 abscabs 14587 expce 15409 sincsin 15411 πcpi 15414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459

This theorem is referenced by: efifo 25125

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