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Theorem imval 14464
 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 7169 . 2 (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i)))
2 df-im 14458 . 2 ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
3 fvex 6672 . 2 (ℜ‘(𝐴 / i)) ∈ V
41, 2, 3fvmpt 6757 1 (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ‘cfv 6344  (class class class)co 7146  ℂcc 10529  ici 10533   / cdiv 11291  ℜcre 14454  ℑcim 14455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-im 14458 This theorem is referenced by:  imre  14465  reim  14466  imf  14470  crim  14472  iblcnlem1  24389  itgcnlem  24391  tanregt0  25129  cxpsqrtlem  25291  ang180lem2  25394  cnre2csqima  31181  ftc1anclem6  35047
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