Theorem angvald 26865

Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26862. (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 26862	. 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
7	1, 2, 3, 4, 6	syl22anc 838	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973  {csn 4648  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ℂcc 11182  0cc0 11184   / cdiv 11947  ℑcim 15147  logclog 26614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453

This theorem is referenced by:  angcld  26866  angrteqvd  26867  cosangneg2d  26868  ang180lem4  26873  lawcos  26877  isosctrlem3  26881  angpieqvdlem2  26890