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Theorem imval 11904
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 6080 . . 3  |-  ( x  =  A  ->  (
x  /  _i )  =  ( A  /  _i ) )
21fveq2d 5724 . 2  |-  ( x  =  A  ->  (
Re `  ( x  /  _i ) )  =  ( Re `  ( A  /  _i ) ) )
3 df-im 11898 . 2  |-  Im  =  ( x  e.  CC  |->  ( Re `  ( x  /  _i ) ) )
4 fvex 5734 . 2  |-  ( Re
`  ( A  /  _i ) )  e.  _V
52, 3, 4fvmpt 5798 1  |-  ( A  e.  CC  ->  (
Im `  A )  =  ( Re `  ( A  /  _i ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   _ici 8984    / cdiv 9669   Recre 11894   Imcim 11895
This theorem is referenced by:  imre  11905  reim  11906  imf  11910  crim  11912  iblcnlem1  19671  itgcnlem  19673  tanregt0  20433  cxpsqrlem  20585  ang180lem2  20644  cnre2csqima  24301  ftc1anclem2  26271  ftc1anclem6  26275  ftc1anclem8  26277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-im 11898
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