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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 4719 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | |
2 | eldifsn 4719 | . 2 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) | |
3 | oveq12 7165 | . . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝑦 / 𝑥) = (𝐵 / 𝐴)) | |
4 | 3 | ancoms 461 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 / 𝑥) = (𝐵 / 𝐴)) |
5 | 4 | fveq2d 6674 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘(𝑦 / 𝑥)) = (log‘(𝐵 / 𝐴))) |
6 | 5 | fveq2d 6674 | . . 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 6683 | . . 3 ⊢ (ℑ‘(log‘(𝐵 / 𝐴))) ∈ V | |
9 | 6, 7, 8 | ovmpoa 7305 | . 2 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴)))) |
10 | 1, 2, 9 | syl2anbr 600 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 {csn 4567 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℂcc 10535 0cc0 10537 / cdiv 11297 ℑcim 14457 logclog 25138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: angcan 25380 angvald 25382 |
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