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Theorem angvald 24754
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 24751. (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 24751 . 2 (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
71, 2, 3, 4, 6syl22anc 1476 1 (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wne 2942  cdif 3718  {csn 4314  cfv 6031  (class class class)co 6792  cmpt2 6794  cc 10135  0cc0 10137   / cdiv 10885  cim 14045  logclog 24521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797
This theorem is referenced by:  angcld  24755  angrteqvd  24756  cosangneg2d  24757  ang180lem4  24762  lawcos  24766  isosctrlem3  24770  angpieqvdlem2  24776
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