Theorem angvald 26829

Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 26826. (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 26826	. 2 ⊢ (((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) ∧ (𝑌 ∈ ℂ ∧ 𝑌 ≠ 0)) → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))
7	1, 2, 3, 4, 6	syl22anc 837	1 ⊢ (𝜑 → (𝑋𝐹𝑌) = (ℑ‘(log‘(𝑌 / 𝑋))))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2100   ≠ wne 2933   ∖ cdif 3955  {csn 4634  ‘cfv 6556  (class class class)co 7427   ∈ cmpo 7429  ℂcc 11157  0cc0 11159   / cdiv 11922  ℑcim 15116  logclog 26578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-sbc 3788  df-dif 3961  df-un 3963  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-br 5155  df-opab 5217  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6508  df-fun 6558  df-fv 6564  df-ov 7430  df-oprab 7431  df-mpo 7432

This theorem is referenced by:  angcld  26830  angrteqvd  26831  cosangneg2d  26832  ang180lem4  26837  lawcos  26841  isosctrlem3  26845  angpieqvdlem2  26854