![]() |
Mathbox for Saveliy Skresanov |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sigarval | Structured version Visualization version GIF version |
Description: Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.) |
Ref | Expression |
---|---|
sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
Ref | Expression |
---|---|
sigarval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 = 𝐴) | |
2 | 1 | fveq2d 6842 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∗‘𝑥) = (∗‘𝐴)) |
3 | simpr 486 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
4 | 2, 3 | oveq12d 7368 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((∗‘𝑥) · 𝑦) = ((∗‘𝐴) · 𝐵)) |
5 | 4 | fveq2d 6842 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 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 | |
8 | 5, 6, 7 | ovmpoa 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 ℂcc 10983 · cmul 10990 ∗ccj 14916 ℑcim 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 |
Copyright terms: Public domain | W3C validator |