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Mirrors > Home > ILE Home > Th. List > absef | GIF version |
Description: The absolute value of the exponential is the exponential of the real part. (Contributed by Paul Chapman, 13-Sep-2007.) |
Ref | Expression |
---|---|
absef | ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | replim 10631 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
2 | 1 | fveq2d 5425 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴))))) |
3 | recl 10625 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
4 | 3 | recnd 7794 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
5 | ax-icn 7715 | . . . . . . 7 ⊢ i ∈ ℂ | |
6 | imcl 10626 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
7 | 6 | recnd 7794 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
8 | mulcl 7747 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
9 | 5, 7, 8 | sylancr 410 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
10 | efadd 11381 | . . . . . 6 ⊢ (((ℜ‘𝐴) ∈ ℂ ∧ (i · (ℑ‘𝐴)) ∈ ℂ) → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) | |
11 | 4, 9, 10 | syl2anc 408 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
12 | 2, 11 | eqtrd 2172 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
13 | 12 | fveq2d 5425 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴)))))) |
14 | 3 | reefcld 11375 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℝ) |
15 | 14 | recnd 7794 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℂ) |
16 | efcl 11370 | . . . . 5 ⊢ ((i · (ℑ‘𝐴)) ∈ ℂ → (exp‘(i · (ℑ‘𝐴))) ∈ ℂ) | |
17 | 9, 16 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (ℑ‘𝐴))) ∈ ℂ) |
18 | 15, 17 | absmuld 10966 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴)))))) |
19 | absefi 11475 | . . . . 5 ⊢ ((ℑ‘𝐴) ∈ ℝ → (abs‘(exp‘(i · (ℑ‘𝐴)))) = 1) | |
20 | 6, 19 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(i · (ℑ‘𝐴)))) = 1) |
21 | 20 | oveq2d 5790 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
22 | 13, 18, 21 | 3eqtrd 2176 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
23 | 15 | abscld 10953 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℝ) |
24 | 23 | recnd 7794 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℂ) |
25 | 24 | mulid1d 7783 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · 1) = (abs‘(exp‘(ℜ‘𝐴)))) |
26 | efgt0 11390 | . . . . 5 ⊢ ((ℜ‘𝐴) ∈ ℝ → 0 < (exp‘(ℜ‘𝐴))) | |
27 | 3, 26 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 < (exp‘(ℜ‘𝐴))) |
28 | 0re 7766 | . . . . 5 ⊢ 0 ∈ ℝ | |
29 | ltle 7851 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ (exp‘(ℜ‘𝐴)) ∈ ℝ) → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) | |
30 | 28, 14, 29 | sylancr 410 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) |
31 | 27, 30 | mpd 13 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (exp‘(ℜ‘𝐴))) |
32 | 14, 31 | absidd 10939 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) = (exp‘(ℜ‘𝐴))) |
33 | 22, 25, 32 | 3eqtrd 2176 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 ici 7622 + caddc 7623 · cmul 7625 < clt 7800 ≤ cle 7801 ℜcre 10612 ℑcim 10613 abscabs 10769 expce 11348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-ico 9677 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-bc 10494 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 df-sin 11356 df-cos 11357 |
This theorem is referenced by: absefib 11477 |
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