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Theorem angval 26859
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 ((𝐶𝐵)𝐹(𝐴𝐵)). (Contributed by Mario Carneiro, 23-Sep-2014.)
Hypothesis
Ref Expression
ang.1 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
Assertion
Ref Expression
angval (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴))))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem angval
StepHypRef 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 7440 . . . . . 6 ((𝑦 = 𝐵𝑥 = 𝐴) → (𝑦 / 𝑥) = (𝐵 / 𝐴))
43ancoms 458 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 / 𝑥) = (𝐵 / 𝐴))
54fveq2d 6911 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (log‘(𝑦 / 𝑥)) = (log‘(𝐵 / 𝐴)))
65fveq2d 6911 . . 3 ((𝑥 = 𝐴𝑦 = 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 6920 . . 3 (ℑ‘(log‘(𝐵 / 𝐴))) ∈ V
96, 7, 8ovmpoa 7588 . 2 ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴))))
101, 2, 9syl2anbr 599 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 1537  wcel 2106  wne 2938  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  cmpo 7433  cc 11151  0cc0 11153   / cdiv 11918  cim 15134  logclog 26611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  angcan  26860  angvald  26862
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