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Theorem angvald 24731
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 24728. (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 24728 . 2 (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
71, 2, 3, 4, 6syl22anc 1478 1 (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2137  wne 2930  cdif 3710  {csn 4319  cfv 6047  (class class class)co 6811  cmpt2 6813  cc 10124  0cc0 10126   / cdiv 10874  cim 14035  logclog 24498
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 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-iota 6010  df-fun 6049  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816
This theorem is referenced by:  angcld  24732  angrteqvd  24733  cosangneg2d  24734  ang180lem4  24739  lawcos  24743  isosctrlem3  24747  angpieqvdlem2  24753
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