Theorem sigarval 46821

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

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴)
2	1	fveq2d 6869	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∗‘𝑥) = (∗‘𝐴))
3		simpr 484	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	2, 3	oveq12d 7412	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((∗‘𝑥) · 𝑦) = ((∗‘𝐴) · 𝐵))
5	4	fveq2d 6869	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (ℑ‘((∗‘𝑥) · 𝑦)) = (ℑ‘((∗‘𝐴) · 𝐵)))
6		sigar	. 2 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
7		fvex 6878	. 2 ⊢ (ℑ‘((∗‘𝐴) · 𝐵)) ∈ V
8	5, 6, 7	ovmpoa 7551	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6519  (class class class)co 7394   ∈ cmpo 7396  ℂcc 11084   · cmul 11091  ∗ccj 15072  ℑcim 15074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399

This theorem is referenced by:  sigarim  46822  sigarac  46823  sigaraf  46824  sigarmf  46825  sigarls  46828  sigarid  46829  sigardiv  46832  sharhght  46836