Theorem angval 24531

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 ( - pi , pi ]. To convert from the geometry notation, m A B C, the measure of the angle with legs A B, C B 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 e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )

Assertion

Ref	Expression
angval	|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A F B ) = ( Im ` ( 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 4317	. 2 |- ( A e. ( CC \ { 0 } ) <-> ( A e. CC /\ A =/= 0 ) )
2		eldifsn 4317	. 2 |- ( B e. ( CC \ { 0 } ) <-> ( B e. CC /\ B =/= 0 ) )
3		oveq12 6659	. . . . . 6 |- ( ( y = B /\ x = A ) -> ( y / x ) = ( B / A ) )
4	3	ancoms 469	. . . . 5 |- ( ( x = A /\ y = B ) -> ( y / x ) = ( B / A ) )
5	4	fveq2d 6195	. . . 4 |- ( ( x = A /\ y = B ) -> ( log ` ( y / x ) ) = ( log ` ( B / A ) ) )
6	5	fveq2d 6195	. . 3 |- ( ( x = A /\ y = B ) -> ( Im ` ( log ` ( y / x ) ) ) = ( Im ` ( log ` ( B / A ) ) ) )
7		ang.1	. . 3 |- F = ( x e. ( CC \ { 0 } ) , y e. ( CC \ { 0 } ) |-> ( Im ` ( log ` ( y / x ) ) ) )
8		fvex 6201	. . 3 |- ( Im ` ( log ` ( B / A ) ) ) e. _V
9	6, 7, 8	ovmpt2a 6791	. 2 |- ( ( A e. ( CC \ { 0 } ) /\ B e. ( CC \ { 0 } ) ) -> ( A F B ) = ( Im ` ( log ` ( B / A ) ) ) )
10	1, 2, 9	syl2anbr 497	1 |- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( A F B ) = ( Im ` ( log ` ( B / A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   0cc0 9936   / cdiv 10684   Imcim 13838   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  angcan  24532  angvald  24534