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Theorem imval 14454
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 fvoveq1 7168 . 2 (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i)))
2 df-im 14448 . 2 ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
3 fvex 6676 . 2 (ℜ‘(𝐴 / i)) ∈ V
41, 2, 3fvmpt 6761 1 (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  cc 10523  ici 10527   / cdiv 11285  cre 14444  cim 14445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-im 14448
This theorem is referenced by:  imre  14455  reim  14456  imf  14460  crim  14462  iblcnlem1  24315  itgcnlem  24317  tanregt0  25050  cxpsqrtlem  25212  ang180lem2  25315  cnre2csqima  31053  ftc1anclem6  34853
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