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Theorem imval 12892
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 6284 . . 3  |-  ( x  =  A  ->  (
x  /  _i )  =  ( A  /  _i ) )
21fveq2d 5863 . 2  |-  ( x  =  A  ->  (
Re `  ( x  /  _i ) )  =  ( Re `  ( A  /  _i ) ) )
3 df-im 12886 . 2  |-  Im  =  ( x  e.  CC  |->  ( Re `  ( x  /  _i ) ) )
4 fvex 5869 . 2  |-  ( Re
`  ( A  /  _i ) )  e.  _V
52, 3, 4fvmpt 5943 1  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( A  /  _i ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   CCcc 9481   _ici 9485    / cdiv 10197   Recre 12882   Imcim 12883
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 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-im 12886
This theorem is referenced by:  imre  12893  reim  12894  imf  12898  crim  12900  iblcnlem1  21924  itgcnlem  21926  tanregt0  22654  cxpsqrlem  22806  ang180lem2  22865  cnre2csqima  27517  ftc1anclem6  29661
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