Theorem angvald 24334

Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 24331. (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	⊢ (𝜑 → 𝑋 ∈ ℂ)
angvald.2	⊢ (𝜑 → 𝑋 ≠ 0)
angvald.3	⊢ (𝜑 → 𝑌 ∈ ℂ)
angvald.4	⊢ (𝜑 → 𝑌 ≠ 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

Step	Hyp	Ref	Expression
1		angvald.1	. 2 ⊢ (𝜑 → 𝑋 ∈ ℂ)
2		angvald.2	. 2 ⊢ (𝜑 → 𝑋 ≠ 0)
3		angvald.3	. 2 ⊢ (𝜑 → 𝑌 ∈ ℂ)
4		angvald.4	. 2 ⊢ (𝜑 → 𝑌 ≠ 0)
5		ang.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
6	5	angval 24331	. 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
7	1, 2, 3, 4, 6	syl22anc 1319	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537  {csn 4125  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ℂcc 9813  0cc0 9815   / cdiv 10563  ℑcim 13686  logclog 24105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  angcld  24335  angrteqvd  24336  cosangneg2d  24337  ang180lem4  24342  lawcos  24346  isosctrlem3  24350  angpieqvdlem2  24356