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Theorem replim 12591
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 9372 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 crre 12589 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( Re `  (
x  +  ( _i  x.  y ) ) )  =  x )
3 crim 12590 . . . . . . 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 2440 . . . 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 22 . . . . 5  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
8 fveq2 5681 . . . . . 6  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  (
Re `  A )  =  ( Re `  ( x  +  (
_i  x.  y )
) ) )
9 fveq2 5681 . . . . . . 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 2449 . . . 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 222 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( A  =  ( x  +  ( _i  x.  y ) )  ->  A  =  ( ( Re `  A
)  +  ( _i  x.  ( Im `  A ) ) ) ) )
1413rexlimivv 2838 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1757   E.wrex 2708   ` cfv 5408  (class class class)co 6082   CCcc 9270   RRcr 9271   _ici 9274    + caddc 9275    x. cmul 9277   Recre 12572   Imcim 12573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-op 3874  df-uni 4082  df-br 4283  df-opab 4341  df-mpt 4342  df-id 4625  df-po 4630  df-so 4631  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-2 10370  df-cj 12574  df-re 12575  df-im 12576
This theorem is referenced by:  remim  12592  reim0b  12594  rereb  12595  mulre  12596  cjreb  12598  reneg  12600  readd  12601  remullem  12603  imneg  12608  imadd  12609  cjcj  12615  imval2  12626  cnrecnv  12640  replimi  12645  replimd  12672  recan  12810  efeul  13431  absef  13466  absefib  13467  efieq1re  13468  cnsubrg  17719  itgconst  21140  tanregt0  21882  tanarg  21955
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