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Theorem angval 20633
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
StepHypRef Expression
1 eldifsn 3919 . 2  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
2 eldifsn 3919 . 2  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
3 oveq12 6082 . . . . . 6  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y  /  x
)  =  ( B  /  A ) )
43ancoms 440 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  /  x
)  =  ( B  /  A ) )
54fveq2d 5724 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  (
y  /  x ) )  =  ( log `  ( B  /  A
) ) )
65fveq2d 5724 . . 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 5734 . . 3  |-  ( Im
`  ( log `  ( B  /  A ) ) )  e.  _V
96, 7, 8ovmpt2a 6196 . 2  |-  ( ( A  e.  ( CC 
\  { 0 } )  /\  B  e.  ( CC  \  {
0 } ) )  ->  ( A F B )  =  ( Im `  ( log `  ( B  /  A
) ) ) )
101, 2, 9syl2anbr 467 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A F B )  =  ( Im
`  ( log `  ( B  /  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   CCcc 8978   0cc0 8980    / cdiv 9667   Imcim 11893   logclog 20442
This theorem is referenced by:  angcan  20634  angvald  20636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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