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Theorem angval 20512
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 3872 . 2  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
2 eldifsn 3872 . 2  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
3 oveq12 6031 . . . . . 6  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y  /  x
)  =  ( B  /  A ) )
43ancoms 440 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  /  x
)  =  ( B  /  A ) )
54fveq2d 5674 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  (
y  /  x ) )  =  ( log `  ( B  /  A
) ) )
65fveq2d 5674 . . 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 5684 . . 3  |-  ( Im
`  ( log `  ( B  /  A ) ) )  e.  _V
96, 7, 8ovmpt2a 6145 . 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 1649    e. wcel 1717    =/= wne 2552    \ cdif 3262   {csn 3759   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   CCcc 8923   0cc0 8925    / cdiv 9611   Imcim 11832   logclog 20321
This theorem is referenced by:  angcan  20513  angvald  20515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027
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