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Theorem angval 20093
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 3750 . 2  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
2 eldifsn 3750 . 2  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
3 oveq12 5828 . . . . . 6  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y  /  x
)  =  ( B  /  A ) )
43ancoms 441 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  /  x
)  =  ( B  /  A ) )
54fveq2d 5489 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  (
y  /  x ) )  =  ( log `  ( B  /  A
) ) )
65fveq2d 5489 . . 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 5499 . . 3  |-  ( Im
`  ( log `  ( B  /  A ) ) )  e.  _V
96, 7, 8ovmpt2a 5939 . 2  |-  ( ( A  e.  ( CC 
\  { 0 } )  /\  B  e.  ( CC  \  {
0 } ) )  ->  ( A F B )  =  ( Im `  ( log `  ( B  /  A
) ) ) )
101, 2, 9syl2anbr 468 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 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447    \ cdif 3150   {csn 3641   ` cfv 5221  (class class class)co 5819    e. cmpt2 5821   CCcc 8730   0cc0 8732    / cdiv 9418   Imcim 11577   logclog 19906
This theorem is referenced by:  angcan  20094  angvald  20096  ang180lem4  20104  lawcos  20108  isosctrlem3  20114
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824
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