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Theorem absef 12351
Description: The absolute value of the exponential is the exponential 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 11439 . . . . . 6 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
21fveq2d 5643 . . . . 5 |- ( A e. CC -> ( exp ` A ) = ( exp ` ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) )
3 recl 11433 . . . . . . 7 |- ( A e. CC -> ( Re ` A ) e. RR )
43recnd 8210 . . . . . 6 |- ( A e. CC -> ( Re ` A ) e. CC )
5 ax-icn 8129 . . . . . . 7 |- _i e. CC
6 imcl 11434 . . . . . . . 8 |- ( A e. CC -> ( Im ` A ) e. RR )
76recnd 8210 . . . . . . 7 |- ( A e. CC -> ( Im ` A ) e. CC )
8 mulcl 8161 . . . . . . 7 |- ( ( _i e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. ( Im ` A ) ) e. CC )
95, 7, 8sylancr 414 . . . . . 6 |- ( A e. CC -> ( _i x. ( Im ` A ) ) e. CC )
10 efadd 12256 . . . . . 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 411 . . . . 5 |- ( A e. CC -> ( exp ` ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) = ( ( exp ` ( Re ` A ) ) x. ( exp ` ( _i x. ( Im ` A ) ) ) ) )
122, 11eqtrd 2263 . . . 4 |- ( A e. CC -> ( exp ` A ) = ( ( exp ` ( Re ` A ) ) x. ( exp ` ( _i x. ( Im ` A ) ) ) ) )
1312fveq2d 5643 . . 3 |- ( A e. CC -> ( abs ` ( exp ` A ) ) = ( abs ` ( ( exp ` ( Re ` A ) ) x. ( exp ` ( _i x. ( Im ` A ) ) ) ) ) )
143reefcld 12250 . . . . 5 |- ( A e. CC -> ( exp ` ( Re ` A ) ) e. RR )
1514recnd 8210 . . . 4 |- ( A e. CC -> ( exp ` ( Re ` A ) ) e. CC )
16 efcl 12245 . . . . 5 |- ( ( _i x. ( Im ` A ) ) e. CC -> ( exp ` ( _i x. ( Im ` A ) ) ) e. CC )
179, 16syl 14 . . . 4 |- ( A e. CC -> ( exp ` ( _i x. ( Im ` A ) ) ) e. CC )
1815, 17absmuld 11774 . . 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 12350 . . . . 5 |- ( ( Im ` A ) e. RR -> ( abs ` ( exp ` ( _i x. ( Im ` A ) ) ) ) = 1 )
206, 19syl 14 . . . 4 |- ( A e. CC -> ( abs ` ( exp ` ( _i x. ( Im ` A ) ) ) ) = 1 )
2120oveq2d 6036 . . 3 |- ( A e. CC -> ( ( abs ` ( exp ` ( Re ` A ) ) ) x. ( abs ` ( exp ` ( _i x. ( Im ` A ) ) ) ) ) = ( ( abs ` ( exp ` ( Re ` A ) ) ) x. 1 ) )
2213, 18, 213eqtrd 2267 . 2 |- ( A e. CC -> ( abs ` ( exp ` A ) ) = ( ( abs ` ( exp ` ( Re ` A ) ) ) x. 1 ) )
2315abscld 11761 . . . 4 |- ( A e. CC -> ( abs ` ( exp ` ( Re ` A ) ) ) e. RR )
2423recnd 8210 . . 3 |- ( A e. CC -> ( abs ` ( exp ` ( Re ` A ) ) ) e. CC )
2524mulridd 8198 . 2 |- ( A e. CC -> ( ( abs ` ( exp ` ( Re ` A ) ) ) x. 1 ) = ( abs ` ( exp ` ( Re ` A ) ) ) )
26 efgt0 12265 . . . . 5 |- ( ( Re ` A ) e. RR -> 0 < ( exp ` ( Re ` A ) ) )
273, 26syl 14 . . . 4 |- ( A e. CC -> 0 < ( exp ` ( Re ` A ) ) )
28 0re 8181 . . . . 5 |- 0 e. RR
29 ltle 8269 . . . . 5 |- ( ( 0 e. RR /\ ( exp ` ( Re ` A ) ) e. RR ) -> ( 0 < ( exp ` ( Re ` A ) ) -> 0 <_ ( exp ` ( Re ` A ) ) ) )
3028, 14, 29sylancr 414 . . . 4 |- ( A e. CC -> ( 0 < ( exp ` ( Re ` A ) ) -> 0 <_ ( exp ` ( Re ` A ) ) ) )
3127, 30mpd 13 . . 3 |- ( A e. CC -> 0 <_ ( exp ` ( Re ` A ) ) )
3214, 31absidd 11747 . 2 |- ( A e. CC -> ( abs ` ( exp ` ( Re ` A ) ) ) = ( exp ` ( Re ` A ) ) )
3322, 25, 323eqtrd 2267 1 |- ( A e. CC -> ( abs ` ( exp ` A ) ) = ( exp ` ( Re ` A ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2201   class class class wbr 4087   ` cfv 5325  (class class class)co 6020   CCcc 8032   RRcr 8033   0cc0 8034   1c1 8035   _ici 8036   + caddc 8037   x. cmul 8039   < clt 8216   <_ cle 8217   Recre 11420   Imcim 11421   abscabs 11577   expce 12223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685  ax-cnex 8125  ax-resscn 8126  ax-1cn 8127  ax-1re 8128  ax-icn 8129  ax-addcl 8130  ax-addrcl 8131  ax-mulcl 8132  ax-mulrcl 8133  ax-addcom 8134  ax-mulcom 8135  ax-addass 8136  ax-mulass 8137  ax-distr 8138  ax-i2m1 8139  ax-0lt1 8140  ax-1rid 8141  ax-0id 8142  ax-rnegex 8143  ax-precex 8144  ax-cnre 8145  ax-pre-ltirr 8146  ax-pre-ltwlin 8147  ax-pre-lttrn 8148  ax-pre-apti 8149  ax-pre-ltadd 8150  ax-pre-mulgt0 8151  ax-pre-mulext 8152  ax-arch 8153  ax-caucvg 8154
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-disj 4064  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-id 4389  df-po 4392  df-iso 4393  df-iord 4462  df-on 4464  df-ilim 4465  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-isom 5334  df-riota 5973  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-irdg 6538  df-frec 6559  df-1o 6584  df-oadd 6588  df-er 6704  df-en 6912  df-dom 6913  df-fin 6914  df-sup 7185  df-pnf 8218  df-mnf 8219  df-xr 8220  df-ltxr 8221  df-le 8222  df-sub 8354  df-neg 8355  df-reap 8757  df-ap 8764  df-div 8855  df-inn 9146  df-2 9204  df-3 9205  df-4 9206  df-n0 9405  df-z 9482  df-uz 9758  df-q 9856  df-rp 9891  df-ico 10131  df-fz 10246  df-fzo 10380  df-seqfrec 10713  df-exp 10804  df-fac 10991  df-bc 11013  df-ihash 11041  df-cj 11422  df-re 11423  df-im 11424  df-rsqrt 11578  df-abs 11579  df-clim 11859  df-sumdc 11934  df-ef 12229  df-sin 12231  df-cos 12232
This theorem is referenced by:  absefib  12352  abscxp  15665  rpabscxpbnd  15690
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