efif1o - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > efif1o		Structured version Visualization version GIF version

Theorem efif1o 25130

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 10687	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
5		2re 11712	. . . . . . . 8 ⊢ 2 ∈ ℝ
6		pire 25044	. . . . . . . 8 ⊢ π ∈ ℝ
7	5, 6	remulcli 10657	. . . . . . 7 ⊢ (2 · π) ∈ ℝ
8		readdcl 10620	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (2 · π) ∈ ℝ) → (𝐴 + (2 · π)) ∈ ℝ)
9	7, 8	mpan2 689	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 + (2 · π)) ∈ ℝ)
10		elioc2 12800	. . . . . 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 25126	. 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (abs‘(𝑥 − 𝑦)) < (2 · π))
17	3	efif1olem2 25127	. 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 25129	1 ⊢ (𝐴 ∈ ℝ → 𝐹:𝐷–1-1-onto→𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 ^◡ccnv 5554 ↾ cres 5557 “ cima 5558 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 -cneg 10871 / cdiv 11297 2c2 11693 (,]cioc 12740 [,]cicc 12742 abscabs 14593 expce 15415 sincsin 15417 πcpi 15420

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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

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 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465

This theorem is referenced by: efifo 25131

W3C validator