Users' Mathboxes Mathbox for Saveliy Skresanov < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigarval Structured version   Visualization version   GIF version

Theorem sigarval 39688
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 472 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21fveq2d 6107 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∗‘𝑥) = (∗‘𝐴))
3 simpr 476 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
42, 3oveq12d 6567 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((∗‘𝑥) · 𝑦) = ((∗‘𝐴) · 𝐵))
54fveq2d 6107 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (ℑ‘((∗‘𝑥) · 𝑦)) = (ℑ‘((∗‘𝐴) · 𝐵)))
6 sigar . 2 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
7 fvex 6113 . 2 (ℑ‘((∗‘𝐴) · 𝐵)) ∈ V
85, 6, 7ovmpt2a 6689 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  cmpt2 6551  cc 9813   · cmul 9820  ccj 13684  cim 13686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554
This theorem is referenced by:  sigarim  39689  sigarac  39690  sigaraf  39691  sigarmf  39692  sigarls  39695  sigarid  39696  sigardiv  39699  sharhght  39703
  Copyright terms: Public domain W3C validator