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Theorem imval 13797
 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 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))

Proof of Theorem imval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq1 6622 . . 3 (𝑥 = 𝐴 → (𝑥 / i) = (𝐴 / i))
21fveq2d 6162 . 2 (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i)))
3 df-im 13791 . 2 ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
4 fvex 6168 . 2 (ℜ‘(𝐴 / i)) ∈ V
52, 3, 4fvmpt 6249 1 (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ‘cfv 5857  (class class class)co 6615  ℂcc 9894  ici 9898   / cdiv 10644  ℜcre 13787  ℑcim 13788 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-im 13791 This theorem is referenced by:  imre  13798  reim  13799  imf  13803  crim  13805  iblcnlem1  23494  itgcnlem  23496  tanregt0  24223  cxpsqrtlem  24382  ang180lem2  24474  cnre2csqima  29781  ftc1anclem6  33161
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