Theorem imval 14148

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 6869	. 2 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i)))
2		df-im 14142	. 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i)))
3		fvex 6392	. 2 ⊢ (ℜ‘(𝐴 / i)) ∈ V
4	1, 2, 3	fvmpt 6475	1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i)))

Syntax hints:   → wi 4   = wceq 1652   ∈ wcel 2155  ‘cfv 6070  (class class class)co 6846  ℂcc 10191  ici 10195   / cdiv 10943  ℜcre 14138  ℑcim 14139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6849  df-im 14142

This theorem is referenced by:  imre  14149  reim  14150  imf  14154  crim  14156  iblcnlem1  23861  itgcnlem  23863  tanregt0  24593  cxpsqrtlem  24755  ang180lem2  24847  cnre2csqima  30427  ftc1anclem6  33936