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Theorem sigarval 44882
Description: Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
Hypothesis
Ref Expression
sigar 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
Assertion
Ref Expression
sigarval ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐺(𝑥,𝑦)

Proof of Theorem sigarval
StepHypRef Expression
1 simpl 484 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21fveq2d 6842 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∗‘𝑥) = (∗‘𝐴))
3 simpr 486 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
42, 3oveq12d 7368 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((∗‘𝑥) · 𝑦) = ((∗‘𝐴) · 𝐵))
54fveq2d 6842 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (ℑ‘((∗‘𝑥) · 𝑦)) = (ℑ‘((∗‘𝐴) · 𝐵)))
6 sigar . 2 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
7 fvex 6851 . 2 (ℑ‘((∗‘𝐴) · 𝐵)) ∈ V
85, 6, 7ovmpoa 7503 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cfv 6492  (class class class)co 7350  cmpo 7352  cc 10983   · cmul 10990  ccj 14916  cim 14918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6444  df-fun 6494  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355
This theorem is referenced by:  sigarim  44883  sigarac  44884  sigaraf  44885  sigarmf  44886  sigarls  44889  sigarid  44890  sigardiv  44893  sharhght  44897
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