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Theorem imval 11592
Description: The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.)
Assertion
Ref Expression
imval  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( A  /  _i ) ) )

Proof of Theorem imval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq1 5865 . . 3  |-  ( x  =  A  ->  (
x  /  _i )  =  ( A  /  _i ) )
21fveq2d 5529 . 2  |-  ( x  =  A  ->  (
Re `  ( x  /  _i ) )  =  ( Re `  ( A  /  _i ) ) )
3 df-im 11586 . 2  |-  Im  =  ( x  e.  CC  |->  ( Re `  ( x  /  _i ) ) )
4 fvex 5539 . 2  |-  ( Re
`  ( A  /  _i ) )  e.  _V
52, 3, 4fvmpt 5602 1  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( A  /  _i ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   _ici 8739    / cdiv 9423   Recre 11582   Imcim 11583
This theorem is referenced by:  imre  11593  reim  11594  imf  11598  crim  11600  iblcnlem1  19142  itgcnlem  19144  tanregt0  19901  cxpsqrlem  20049  ang180lem2  20108  cnre2csqima  23295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-im 11586
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