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Theorem sigarval 39597
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 471 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21fveq2d 5990 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (∗‘𝑥) = (∗‘𝐴))
3 simpr 475 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
42, 3oveq12d 6443 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((∗‘𝑥) · 𝑦) = ((∗‘𝐴) · 𝐵))
54fveq2d 5990 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (ℑ‘((∗‘𝑥) · 𝑦)) = (ℑ‘((∗‘𝐴) · 𝐵)))
6 sigar . 2 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
7 fvex 5996 . 2 (ℑ‘((∗‘𝐴) · 𝐵)) ∈ V
85, 6, 7ovmpt2a 6565 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  cfv 5689  (class class class)co 6425  cmpt2 6427  cc 9688   · cmul 9695  ccj 13541  cim 13543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430
This theorem is referenced by:  sigarim  39598  sigarac  39599  sigaraf  39600  sigarmf  39601  sigarls  39604  sigarid  39605  sigardiv  39608  sharhght  39612
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