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Theorem angvald 25366
 Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 25363. (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 25363 . 2 (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
71, 2, 3, 4, 6syl22anc 836 1 (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ≠ wne 3006   ∖ cdif 3909  {csn 4541  ‘cfv 6329  (class class class)co 7131   ∈ cmpo 7133  ℂcc 10511  0cc0 10513   / cdiv 11273  ℑcim 14435  logclog 25122 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 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304 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 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-iota 6288  df-fun 6331  df-fv 6337  df-ov 7134  df-oprab 7135  df-mpo 7136 This theorem is referenced by:  angcld  25367  angrteqvd  25368  cosangneg2d  25369  ang180lem4  25374  lawcos  25378  isosctrlem3  25382  angpieqvdlem2  25391
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