Theorem imval 15020

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 7375	. 2 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i)))
2		df-im 15014	. 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
3		fvex 6841	. 2 ⊢ (ℜ‘(𝐴 / i)) ∈ V
4	1, 2, 3	fvmpt 6935	1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6487  (class class class)co 7352  ℂcc 11010  ici 11014   / cdiv 11780  ℜcre 15010  ℑcim 15011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6443  df-fun 6489  df-fv 6495  df-ov 7355  df-im 15014

This theorem is referenced by:  imre  15021  reim  15022  imf  15026  crim  15028  iblcnlem1  25722  itgcnlem  25724  tanregt0  26481  cxpsqrtlem  26644  ang180lem2  26753  cnre2csqima  33931  ftc1anclem6  37744