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Theorem absef 13481
Description: The absolute value of the exponential function is the exponential function of the real part. (Contributed by Paul Chapman, 13-Sep-2007.)
Assertion
Ref Expression
absef  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( exp `  ( Re
`  A ) ) )

Proof of Theorem absef
StepHypRef Expression
1 replim 12605 . . . . . 6  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
21fveq2d 5695 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( exp `  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) ) ) )
3 recl 12599 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
43recnd 9412 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
5 ax-icn 9341 . . . . . . 7  |-  _i  e.  CC
6 imcl 12600 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Im `  A )  e.  RR )
76recnd 9412 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  e.  CC )
8 mulcl 9366 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( Im `  A )  e.  CC )  -> 
( _i  x.  (
Im `  A )
)  e.  CC )
95, 7, 8sylancr 663 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  e.  CC )
10 efadd 13379 . . . . . 6  |-  ( ( ( Re `  A
)  e.  CC  /\  ( _i  x.  (
Im `  A )
)  e.  CC )  ->  ( exp `  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) ) )  =  ( ( exp `  ( Re
`  A ) )  x.  ( exp `  (
_i  x.  ( Im `  A ) ) ) ) )
114, 9, 10syl2anc 661 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )  =  ( ( exp `  (
Re `  A )
)  x.  ( exp `  ( _i  x.  (
Im `  A )
) ) ) )
122, 11eqtrd 2475 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( exp `  (
Re `  A )
)  x.  ( exp `  ( _i  x.  (
Im `  A )
) ) ) )
1312fveq2d 5695 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( abs `  ( ( exp `  ( Re
`  A ) )  x.  ( exp `  (
_i  x.  ( Im `  A ) ) ) ) ) )
143reefcld 13373 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( Re `  A ) )  e.  RR )
1514recnd 9412 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( Re `  A ) )  e.  CC )
16 efcl 13368 . . . . 5  |-  ( ( _i  x.  ( Im
`  A ) )  e.  CC  ->  ( exp `  ( _i  x.  ( Im `  A ) ) )  e.  CC )
179, 16syl 16 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( Im `  A ) ) )  e.  CC )
1815, 17absmuld 12940 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( ( exp `  ( Re `  A
) )  x.  ( exp `  ( _i  x.  ( Im `  A ) ) ) ) )  =  ( ( abs `  ( exp `  (
Re `  A )
) )  x.  ( abs `  ( exp `  (
_i  x.  ( Im `  A ) ) ) ) ) )
19 absefi 13480 . . . . 5  |-  ( ( Im `  A )  e.  RR  ->  ( abs `  ( exp `  (
_i  x.  ( Im `  A ) ) ) )  =  1 )
206, 19syl 16 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
_i  x.  ( Im `  A ) ) ) )  =  1 )
2120oveq2d 6107 . . 3  |-  ( A  e.  CC  ->  (
( abs `  ( exp `  ( Re `  A ) ) )  x.  ( abs `  ( exp `  ( _i  x.  ( Im `  A ) ) ) ) )  =  ( ( abs `  ( exp `  (
Re `  A )
) )  x.  1 ) )
2213, 18, 213eqtrd 2479 . 2  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( ( abs `  ( exp `  ( Re `  A ) ) )  x.  1 ) )
2315abscld 12922 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
Re `  A )
) )  e.  RR )
2423recnd 9412 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
Re `  A )
) )  e.  CC )
2524mulid1d 9403 . 2  |-  ( A  e.  CC  ->  (
( abs `  ( exp `  ( Re `  A ) ) )  x.  1 )  =  ( abs `  ( exp `  ( Re `  A ) ) ) )
26 efgt0 13387 . . . . 5  |-  ( ( Re `  A )  e.  RR  ->  0  <  ( exp `  (
Re `  A )
) )
273, 26syl 16 . . . 4  |-  ( A  e.  CC  ->  0  <  ( exp `  (
Re `  A )
) )
28 0re 9386 . . . . 5  |-  0  e.  RR
29 ltle 9463 . . . . 5  |-  ( ( 0  e.  RR  /\  ( exp `  ( Re
`  A ) )  e.  RR )  -> 
( 0  <  ( exp `  ( Re `  A ) )  -> 
0  <_  ( exp `  ( Re `  A
) ) ) )
3028, 14, 29sylancr 663 . . . 4  |-  ( A  e.  CC  ->  (
0  <  ( exp `  ( Re `  A
) )  ->  0  <_  ( exp `  (
Re `  A )
) ) )
3127, 30mpd 15 . . 3  |-  ( A  e.  CC  ->  0  <_  ( exp `  (
Re `  A )
) )
3214, 31absidd 12909 . 2  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
Re `  A )
) )  =  ( exp `  ( Re
`  A ) ) )
3322, 25, 323eqtrd 2479 1  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( exp `  ( Re
`  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   _ici 9284    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419   Recre 12586   Imcim 12587   abscabs 12723   expce 13347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-ico 11306  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356
This theorem is referenced by:  absefib  13482  eff1olem  22004  relog  22045  abscxp  22137  abscxp2  22138  abscxpbnd  22191  zetacvg  27001
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