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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 10663 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
2 | 1 | fveq2d 5433 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴))))) |
3 | recl 10657 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
4 | 3 | recnd 7818 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
5 | ax-icn 7739 | . . . . . . 7 ⊢ i ∈ ℂ | |
6 | imcl 10658 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
7 | 6 | recnd 7818 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
8 | mulcl 7771 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
9 | 5, 7, 8 | sylancr 411 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
10 | efadd 11418 | . . . . . 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 409 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
12 | 2, 11 | eqtrd 2173 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
13 | 12 | fveq2d 5433 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴)))))) |
14 | 3 | reefcld 11412 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℝ) |
15 | 14 | recnd 7818 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℂ) |
16 | efcl 11407 | . . . . 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 10998 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴)))))) |
19 | absefi 11511 | . . . . 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 5798 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
22 | 13, 18, 21 | 3eqtrd 2177 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
23 | 15 | abscld 10985 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℝ) |
24 | 23 | recnd 7818 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℂ) |
25 | 24 | mulid1d 7807 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · 1) = (abs‘(exp‘(ℜ‘𝐴)))) |
26 | efgt0 11427 | . . . . 5 ⊢ ((ℜ‘𝐴) ∈ ℝ → 0 < (exp‘(ℜ‘𝐴))) | |
27 | 3, 26 | syl 14 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 < (exp‘(ℜ‘𝐴))) |
28 | 0re 7790 | . . . . 5 ⊢ 0 ∈ ℝ | |
29 | ltle 7875 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ (exp‘(ℜ‘𝐴)) ∈ ℝ) → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) | |
30 | 28, 14, 29 | sylancr 411 | . . . 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 10971 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) = (exp‘(ℜ‘𝐴))) |
33 | 22, 25, 32 | 3eqtrd 2177 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 ici 7646 + caddc 7647 · cmul 7649 < clt 7824 ≤ cle 7825 ℜcre 10644 ℑcim 10645 abscabs 10801 expce 11385 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-ico 9707 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 |
This theorem is referenced by: absefib 11513 abscxp 13043 rpabscxpbnd 13067 |
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