MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  angval Unicode version

Theorem angval 20099
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 3749 . 2  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
2 eldifsn 3749 . 2  |-  ( B  e.  ( CC  \  { 0 } )  <-> 
( B  e.  CC  /\  B  =/=  0 ) )
3 oveq12 5867 . . . . . 6  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y  /  x
)  =  ( B  /  A ) )
43ancoms 439 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  /  x
)  =  ( B  /  A ) )
54fveq2d 5529 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( log `  (
y  /  x ) )  =  ( log `  ( B  /  A
) ) )
65fveq2d 5529 . . 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 5539 . . 3  |-  ( Im
`  ( log `  ( B  /  A ) ) )  e.  _V
96, 7, 8ovmpt2a 5978 . 2  |-  ( ( A  e.  ( CC 
\  { 0 } )  /\  B  e.  ( CC  \  {
0 } ) )  ->  ( A F B )  =  ( Im `  ( log `  ( B  /  A
) ) ) )
101, 2, 9syl2anbr 466 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 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   0cc0 8737    / cdiv 9423   Imcim 11583   logclog 19912
This theorem is referenced by:  angcan  20100  angvald  20102  ang180lem4  20110  lawcos  20114  isosctrlem3  20120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
  Copyright terms: Public domain W3C validator