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Theorem replim 11922
Description: Reconstruct a complex number from its real and imaginary parts. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 7-Nov-2013.)
Assertion
Ref Expression
replim  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )

Proof of Theorem replim
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 9088 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 crre 11920 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  x )
3 crim 11921 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Im `  (
x  +  ( _i  x.  y ) ) )  =  y )
43oveq2d 6098 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( _i  x.  (
Im `  ( x  +  ( _i  x.  y ) ) ) )  =  ( _i  x.  y ) )
52, 4oveq12d 6100 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( Re `  ( x  +  (
_i  x.  y )
) )  +  ( _i  x.  ( Im
`  ( x  +  ( _i  x.  y
) ) ) ) )  =  ( x  +  ( _i  x.  y ) ) )
65eqcomd 2442 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  ( _i  x.  y ) )  =  ( ( Re `  ( x  +  ( _i  x.  y ) ) )  +  ( _i  x.  ( Im `  ( x  +  ( _i  x.  y ) ) ) ) ) )
7 id 21 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
8 fveq2 5729 . . . . . 6  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
Re `  A )  =  ( Re `  ( x  +  (
_i  x.  y )
) ) )
9 fveq2 5729 . . . . . . 7  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
Im `  A )  =  ( Im `  ( x  +  (
_i  x.  y )
) ) )
109oveq2d 6098 . . . . . 6  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
_i  x.  ( Im `  A ) )  =  ( _i  x.  (
Im `  ( x  +  ( _i  x.  y ) ) ) ) )
118, 10oveq12d 6100 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) )  =  ( ( Re
`  ( x  +  ( _i  x.  y
) ) )  +  ( _i  x.  (
Im `  ( x  +  ( _i  x.  y ) ) ) ) ) )
127, 11eqeq12d 2451 . . . 4  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( A  =  ( (
Re `  A )  +  ( _i  x.  ( Im `  A ) ) )  <->  ( x  +  ( _i  x.  y ) )  =  ( ( Re `  ( x  +  (
_i  x.  y )
) )  +  ( _i  x.  ( Im
`  ( x  +  ( _i  x.  y
) ) ) ) ) ) )
136, 12syl5ibrcom 215 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  A  =  ( ( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) ) ) )
1413rexlimivv 2836 . 2  |-  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
151, 14syl 16 1  |-  ( A  e.  CC  ->  A  =  ( ( Re
`  A )  +  ( _i  x.  (
Im `  A )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   _ici 8993    + caddc 8994    x. cmul 8996   Recre 11903   Imcim 11904
This theorem is referenced by:  remim  11923  reim0b  11925  rereb  11926  mulre  11927  cjreb  11929  reneg  11931  readd  11932  remullem  11934  imneg  11939  imadd  11940  cjcj  11946  imval2  11957  cnrecnv  11971  replimi  11976  replimd  12003  recan  12141  efeul  12764  absef  12799  absefib  12800  efieq1re  12801  cnsubrg  16760  itgconst  19711  tanregt0  20442  tanarg  20515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-2 10059  df-cj 11905  df-re 11906  df-im 11907
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