Theorem sigarval 47278

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 6844	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∗‘𝑥) = (∗‘𝐴))
3		simpr 484	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
4	2, 3	oveq12d 7385	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((∗‘𝑥) · 𝑦) = ((∗‘𝐴) · 𝐵))
5	4	fveq2d 6844	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (ℑ‘((∗‘𝑥) · 𝑦)) = (ℑ‘((∗‘𝐴) · 𝐵)))
6		sigar	. 2 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))
7		fvex 6853	. 2 ⊢ (ℑ‘((∗‘𝐴) · 𝐵)) ∈ V
8	5, 6, 7	ovmpoa 7522	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  ℂcc 11036   · cmul 11043  ∗ccj 15058  ℑcim 15060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372

This theorem is referenced by:  sigarim  47279  sigarac  47280  sigaraf  47281  sigarmf  47282  sigarls  47285  sigarid  47286  sigardiv  47289  sharhght  47293