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Mirrors > Home > MPE Home > Th. List > absef | Structured version Visualization version 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 14267 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
2 | 1 | fveq2d 6452 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴))))) |
3 | recl 14261 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
4 | 3 | recnd 10407 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
5 | ax-icn 10333 | . . . . . . 7 ⊢ i ∈ ℂ | |
6 | imcl 14262 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
7 | 6 | recnd 10407 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
8 | mulcl 10358 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
9 | 5, 7, 8 | sylancr 581 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
10 | efadd 15230 | . . . . . 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 579 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
12 | 2, 11 | eqtrd 2814 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
13 | 12 | fveq2d 6452 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴)))))) |
14 | 3 | reefcld 15224 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℝ) |
15 | 14 | recnd 10407 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℂ) |
16 | efcl 15219 | . . . . 5 ⊢ ((i · (ℑ‘𝐴)) ∈ ℂ → (exp‘(i · (ℑ‘𝐴))) ∈ ℂ) | |
17 | 9, 16 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(i · (ℑ‘𝐴))) ∈ ℂ) |
18 | 15, 17 | absmuld 14605 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴)))))) |
19 | absefi 15332 | . . . . 5 ⊢ ((ℑ‘𝐴) ∈ ℝ → (abs‘(exp‘(i · (ℑ‘𝐴)))) = 1) | |
20 | 6, 19 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(i · (ℑ‘𝐴)))) = 1) |
21 | 20 | oveq2d 6940 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
22 | 13, 18, 21 | 3eqtrd 2818 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
23 | 15 | abscld 14587 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℝ) |
24 | 23 | recnd 10407 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℂ) |
25 | 24 | mulid1d 10396 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · 1) = (abs‘(exp‘(ℜ‘𝐴)))) |
26 | efgt0 15239 | . . . . 5 ⊢ ((ℜ‘𝐴) ∈ ℝ → 0 < (exp‘(ℜ‘𝐴))) | |
27 | 3, 26 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 < (exp‘(ℜ‘𝐴))) |
28 | 0re 10380 | . . . . 5 ⊢ 0 ∈ ℝ | |
29 | ltle 10467 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ (exp‘(ℜ‘𝐴)) ∈ ℝ) → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) | |
30 | 28, 14, 29 | sylancr 581 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) |
31 | 27, 30 | mpd 15 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (exp‘(ℜ‘𝐴))) |
32 | 14, 31 | absidd 14573 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) = (exp‘(ℜ‘𝐴))) |
33 | 22, 25, 32 | 3eqtrd 2818 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 ici 10276 + caddc 10277 · cmul 10279 < clt 10413 ≤ cle 10414 ℜcre 14248 ℑcim 14249 abscabs 14385 expce 15198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-ico 12497 df-fz 12648 df-fzo 12789 df-fl 12916 df-seq 13124 df-exp 13183 df-fac 13383 df-bc 13412 df-hash 13440 df-shft 14218 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-limsup 14614 df-clim 14631 df-rlim 14632 df-sum 14829 df-ef 15204 df-sin 15206 df-cos 15207 |
This theorem is referenced by: absefib 15334 eff1olem 24736 relog 24784 abscxp 24879 abscxp2 24880 abscxpbnd 24938 zetacvg 25197 |
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