Theorem angval 25387

Description: Define the angle function, which takes two complex numbers, treated as vectors from the origin, and returns the angle between them, in the range ( − π, π]. To convert from the geometry notation, 𝑚𝐴𝐵𝐶, the measure of the angle with legs 𝐴𝐵, 𝐶𝐵 where 𝐶 is more counterclockwise for positive angles, is represented by ((𝐶 − 𝐵)𝐹(𝐴 − 𝐵)). (Contributed by Mario Carneiro, 23-Sep-2014.)

Hypothesis

Ref	Expression
ang.1	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))

Assertion

Ref	Expression
angval	⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴))))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem angval

Step	Hyp	Ref	Expression
1		eldifsn 4680	. 2 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
2		eldifsn 4680	. 2 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
3		oveq12 7144	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐴) → (𝑦 / 𝑥) = (𝐵 / 𝐴))
4	3	ancoms 462	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑦 / 𝑥) = (𝐵 / 𝐴))
5	4	fveq2d 6649	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (log‘(𝑦 / 𝑥)) = (log‘(𝐵 / 𝐴)))
6	5	fveq2d 6649	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (ℑ‘(log‘(𝑦 / 𝑥))) = (ℑ‘(log‘(𝐵 / 𝐴))))
7		ang.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
8		fvex 6658	. . 3 ⊢ (ℑ‘(log‘(𝐵 / 𝐴))) ∈ V
9	6, 7, 8	ovmpoa 7284	. 2 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴))))
10	1, 2, 9	syl2anbr 601	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴𝐹𝐵) = (ℑ‘(log‘(𝐵 / 𝐴))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878  {csn 4525  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ℂcc 10524  0cc0 10526   / cdiv 11286  ℑcim 14449  logclog 25146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  angcan  25388  angvald  25390