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Theorem imval 12994
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 6239 . . 3  |-  ( x  =  A  ->  (
x  /  _i )  =  ( A  /  _i ) )
21fveq2d 5807 . 2  |-  ( x  =  A  ->  (
Re `  ( x  /  _i ) )  =  ( Re `  ( A  /  _i ) ) )
3 df-im 12988 . 2  |-  Im  =  ( x  e.  CC  |->  ( Re `  ( x  /  _i ) ) )
4 fvex 5813 . 2  |-  ( Re
`  ( A  /  _i ) )  e.  _V
52, 3, 4fvmpt 5886 1  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( A  /  _i ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   ` cfv 5523  (class class class)co 6232   CCcc 9438   _ici 9442    / cdiv 10165   Recre 12984   Imcim 12985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-im 12988
This theorem is referenced by:  imre  12995  reim  12996  imf  13000  crim  13002  iblcnlem1  22376  itgcnlem  22378  tanregt0  23108  cxpsqrtlem  23267  ang180lem2  23359  cnre2csqima  28227  ftc1anclem6  31432
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