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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 14197 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
2 | 1 | fveq2d 6415 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴))))) |
3 | recl 14191 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
4 | 3 | recnd 10357 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
5 | ax-icn 10283 | . . . . . . 7 ⊢ i ∈ ℂ | |
6 | imcl 14192 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
7 | 6 | recnd 10357 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
8 | mulcl 10308 | . . . . . . 7 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
9 | 5, 7, 8 | sylancr 582 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
10 | efadd 15160 | . . . . . 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 580 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
12 | 2, 11 | eqtrd 2833 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) |
13 | 12 | fveq2d 6415 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴)))))) |
14 | 3 | reefcld 15154 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℝ) |
15 | 14 | recnd 10357 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘(ℜ‘𝐴)) ∈ ℂ) |
16 | efcl 15149 | . . . . 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 14534 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘((exp‘(ℜ‘𝐴)) · (exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴)))))) |
19 | absefi 15262 | . . . . 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 6894 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · (abs‘(exp‘(i · (ℑ‘𝐴))))) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
22 | 13, 18, 21 | 3eqtrd 2837 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = ((abs‘(exp‘(ℜ‘𝐴))) · 1)) |
23 | 15 | abscld 14516 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℝ) |
24 | 23 | recnd 10357 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) ∈ ℂ) |
25 | 24 | mulid1d 10346 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘(exp‘(ℜ‘𝐴))) · 1) = (abs‘(exp‘(ℜ‘𝐴)))) |
26 | efgt0 15169 | . . . . 5 ⊢ ((ℜ‘𝐴) ∈ ℝ → 0 < (exp‘(ℜ‘𝐴))) | |
27 | 3, 26 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℂ → 0 < (exp‘(ℜ‘𝐴))) |
28 | 0re 10330 | . . . . 5 ⊢ 0 ∈ ℝ | |
29 | ltle 10416 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ (exp‘(ℜ‘𝐴)) ∈ ℝ) → (0 < (exp‘(ℜ‘𝐴)) → 0 ≤ (exp‘(ℜ‘𝐴)))) | |
30 | 28, 14, 29 | sylancr 582 | . . . 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 14502 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘(ℜ‘𝐴))) = (exp‘(ℜ‘𝐴))) |
33 | 22, 25, 32 | 3eqtrd 2837 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(exp‘𝐴)) = (exp‘(ℜ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 ℝcr 10223 0cc0 10224 1c1 10225 ici 10226 + caddc 10227 · cmul 10229 < clt 10363 ≤ cle 10364 ℜcre 14178 ℑcim 14179 abscabs 14315 expce 15128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-addf 10303 ax-mulf 10304 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-ico 12430 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-fac 13314 df-bc 13343 df-hash 13371 df-shft 14148 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-limsup 14543 df-clim 14560 df-rlim 14561 df-sum 14758 df-ef 15134 df-sin 15136 df-cos 15137 |
This theorem is referenced by: absefib 15264 eff1olem 24636 relog 24684 abscxp 24779 abscxp2 24780 abscxpbnd 24838 zetacvg 25093 |
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