Theorem imval 15156

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.

Step	Hyp	Ref	Expression
1		fvoveq1 7471	. 2 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i)))
2		df-im 15150	. 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
3		fvex 6933	. 2 ⊢ (ℜ‘(𝐴 / i)) ∈ V
4	1, 2, 3	fvmpt 7029	1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  ici 11186   / cdiv 11947  ℜcre 15146  ℑcim 15147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-im 15150

This theorem is referenced by:  imre  15157  reim  15158  imf  15162  crim  15164  iblcnlem1  25843  itgcnlem  25845  tanregt0  26599  cxpsqrtlem  26762  ang180lem2  26871  cnre2csqima  33857  ftc1anclem6  37658