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Theorem angvald 26862
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26859. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
ang.1 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
angvald.1 (𝜑𝑋 ∈ ℂ)
angvald.2 (𝜑𝑋 ≠ 0)
angvald.3 (𝜑𝑌 ∈ ℂ)
angvald.4 (𝜑𝑌 ≠ 0)
Assertion
Ref Expression
angvald (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
Distinct variable groups:   𝑥,𝑦,𝑋   𝑥,𝑌,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem angvald
StepHypRef Expression
1 angvald.1 . 2 (𝜑𝑋 ∈ ℂ)
2 angvald.2 . 2 (𝜑𝑋 ≠ 0)
3 angvald.3 . 2 (𝜑𝑌 ∈ ℂ)
4 angvald.4 . 2 (𝜑𝑌 ≠ 0)
5 ang.1 . . 3 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
65angval 26859 . 2 (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
71, 2, 3, 4, 6syl22anc 839 1 (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = 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:  angcld  26863  angrteqvd  26864  cosangneg2d  26865  ang180lem4  26870  lawcos  26874  isosctrlem3  26878  angpieqvdlem2  26887
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