MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  absef Unicode version

Theorem absef 12473
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 11597 . . . . . 6  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
21fveq2d 5491 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( exp `  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) ) ) )
3 recl 11591 . . . . . . 7  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
43recnd 8858 . . . . . 6  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
5 ax-icn 8793 . . . . . . 7  |-  _i  e.  CC
6 imcl 11592 . . . . . . . 8  |-  ( A  e.  CC  ->  (
Im `  A )  e.  RR )
76recnd 8858 . . . . . . 7  |-  ( A  e.  CC  ->  (
Im `  A )  e.  CC )
8 mulcl 8818 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( Im `  A )  e.  CC )  -> 
( _i  x.  (
Im `  A )
)  e.  CC )
95, 7, 8sylancr 646 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  ( Im `  A ) )  e.  CC )
10 efadd 12371 . . . . . 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 644 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )  =  ( ( exp `  (
Re `  A )
)  x.  ( exp `  ( _i  x.  (
Im `  A )
) ) ) )
122, 11eqtrd 2318 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  =  ( ( exp `  (
Re `  A )
)  x.  ( exp `  ( _i  x.  (
Im `  A )
) ) ) )
1312fveq2d 5491 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( abs `  ( ( exp `  ( Re
`  A ) )  x.  ( exp `  (
_i  x.  ( Im `  A ) ) ) ) ) )
143reefcld 12365 . . . . 5  |-  ( A  e.  CC  ->  ( exp `  ( Re `  A ) )  e.  RR )
1514recnd 8858 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( Re `  A ) )  e.  CC )
16 efcl 12360 . . . . 5  |-  ( ( _i  x.  ( Im
`  A ) )  e.  CC  ->  ( exp `  ( _i  x.  ( Im `  A ) ) )  e.  CC )
179, 16syl 17 . . . 4  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  ( Im `  A ) ) )  e.  CC )
1815, 17absmuld 11932 . . 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 12472 . . . . 5  |-  ( ( Im `  A )  e.  RR  ->  ( abs `  ( exp `  (
_i  x.  ( Im `  A ) ) ) )  =  1 )
206, 19syl 17 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
_i  x.  ( Im `  A ) ) ) )  =  1 )
2120oveq2d 5837 . . 3  |-  ( A  e.  CC  ->  (
( abs `  ( exp `  ( Re `  A ) ) )  x.  ( abs `  ( exp `  ( _i  x.  ( Im `  A ) ) ) ) )  =  ( ( abs `  ( exp `  (
Re `  A )
) )  x.  1 ) )
2213, 18, 213eqtrd 2322 . 2  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( ( abs `  ( exp `  ( Re `  A ) ) )  x.  1 ) )
2315abscld 11914 . . . 4  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
Re `  A )
) )  e.  RR )
2423recnd 8858 . . 3  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
Re `  A )
) )  e.  CC )
2524mulid1d 8849 . 2  |-  ( A  e.  CC  ->  (
( abs `  ( exp `  ( Re `  A ) ) )  x.  1 )  =  ( abs `  ( exp `  ( Re `  A ) ) ) )
26 efgt0 12379 . . . . 5  |-  ( ( Re `  A )  e.  RR  ->  0  <  ( exp `  (
Re `  A )
) )
273, 26syl 17 . . . 4  |-  ( A  e.  CC  ->  0  <  ( exp `  (
Re `  A )
) )
28 0reALT 9140 . . . . 5  |-  0  e.  RR
29 ltle 8907 . . . . 5  |-  ( ( 0  e.  RR  /\  ( exp `  ( Re
`  A ) )  e.  RR )  -> 
( 0  <  ( exp `  ( Re `  A ) )  -> 
0  <_  ( exp `  ( Re `  A
) ) ) )
3028, 14, 29sylancr 646 . . . 4  |-  ( A  e.  CC  ->  (
0  <  ( exp `  ( Re `  A
) )  ->  0  <_  ( exp `  (
Re `  A )
) ) )
3127, 30mpd 16 . . 3  |-  ( A  e.  CC  ->  0  <_  ( exp `  (
Re `  A )
) )
3214, 31absidd 11901 . 2  |-  ( A  e.  CC  ->  ( abs `  ( exp `  (
Re `  A )
) )  =  ( exp `  ( Re
`  A ) ) )
3322, 25, 323eqtrd 2322 1  |-  ( A  e.  CC  ->  ( abs `  ( exp `  A
) )  =  ( exp `  ( Re
`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1625    e. wcel 1687   class class class wbr 4026   ` cfv 5223  (class class class)co 5821   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735   _ici 8736    + caddc 8737    x. cmul 8739    < clt 8864    <_ cle 8865   Recre 11578   Imcim 11579   abscabs 11715   expce 12339
This theorem is referenced by:  absefib  12474  eff1olem  19906  relog  19946  abscxp  20035  abscxp2  20036  abscxpbnd  20089  zetacvg  23095
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-inf2 7339  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-se 4354  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-isom 5232  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-oadd 6480  df-er 6657  df-pm 6772  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-sup 7191  df-oi 7222  df-card 7569  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-n0 9963  df-z 10022  df-uz 10228  df-rp 10352  df-ico 10658  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348
  Copyright terms: Public domain W3C validator