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Mirrors > Home > MPE Home > Th. List > angval | Structured version Visualization version GIF version |
Description: Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range ( − π, π]. To convert from the geometry notation, 𝑚𝐴𝐵𝐶, the measure of the angle with legs 𝐴𝐵, 𝐶𝐵 where 𝐶 is more counterclockwise for positive angles, is represented by ((𝐶 − 𝐵)𝐹(𝐴 − 𝐵)). (Contributed by Mario Carneiro, 23-Sep-2014.) |
Ref | Expression |
---|---|
ang.1 | ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) |
Ref | Expression |
---|---|
angval | ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4791 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | |
2 | eldifsn 4791 | . 2 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) | |
3 | oveq12 7429 | . . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝑦 / 𝑥) = (𝐵 / 𝐴)) | |
4 | 3 | ancoms 458 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 / 𝑥) = (𝐵 / 𝐴)) |
5 | 4 | fveq2d 6901 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘(𝑦 / 𝑥)) = (log‘(𝐵 / 𝐴))) |
6 | 5 | fveq2d 6901 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (ℑ‘(log‘(𝑦 / 𝑥))) = (ℑ‘(log‘(𝐵 / 𝐴)))) |
7 | ang.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥)))) | |
8 | fvex 6910 | . . 3 ⊢ (ℑ‘(log‘(𝐵 / 𝐴))) ∈ V | |
9 | 6, 7, 8 | ovmpoa 7576 | . 2 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴)))) |
10 | 1, 2, 9 | syl2anbr 598 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 {csn 4629 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ℂcc 11137 0cc0 11139 / cdiv 11902 ℑcim 15078 logclog 26501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 |
This theorem is referenced by: angcan 26747 angvald 26749 |
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