Theorem angvald 26741

Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26738. (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 26738	. 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 1541   ∈ wcel 2111   ≠ wne 2928   ∖ cdif 3894  {csn 4573  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  ℂcc 11004  0cc0 11006   / cdiv 11774  ℑcim 15005  logclog 26490

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  angcld  26742  angrteqvd  26743  cosangneg2d  26744  ang180lem4  26749  lawcos  26753  isosctrlem3  26757  angpieqvdlem2  26766