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Theorem angval 20047
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 3709 . 2  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
2 eldifsn 3709 . 2  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
3 oveq12 5787 . . . . . 6  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y  /  x
)  =  ( B  /  A ) )
43ancoms 441 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  /  x
)  =  ( B  /  A ) )
54fveq2d 5448 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  (
y  /  x ) )  =  ( log `  ( B  /  A
) ) )
65fveq2d 5448 . . 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 5458 . . 3  |-  ( Im
`  ( log `  ( B  /  A ) ) )  e.  _V
96, 7, 8ovmpt2a 5898 . 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 1619    e. wcel 1621    =/= wne 2419    \ cdif 3110   {csn 3600   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   CCcc 8689   0cc0 8691    / cdiv 9377   Imcim 11534   logclog 19860
This theorem is referenced by:  angcan  20048  angvald  20050  ang180lem4  20058  lawcos  20062  isosctrlem3  20068
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783
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