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Theorem angvald 20648
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20645. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
ang.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
angvald.1  |-  ( ph  ->  X  e.  CC )
angvald.2  |-  ( ph  ->  X  =/=  0 )
angvald.3  |-  ( ph  ->  Y  e.  CC )
angvald.4  |-  ( ph  ->  Y  =/=  0 )
Assertion
Ref Expression
angvald  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Distinct variable groups:    x, y, X    x, Y, y
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem angvald
StepHypRef Expression
1 angvald.1 . 2  |-  ( ph  ->  X  e.  CC )
2 angvald.2 . 2  |-  ( ph  ->  X  =/=  0 )
3 angvald.3 . 2  |-  ( ph  ->  Y  e.  CC )
4 angvald.4 . 2  |-  ( ph  ->  Y  =/=  0 )
5 ang.1 . . 3  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
65angval 20645 . 2  |-  ( ( ( X  e.  CC  /\  X  =/=  0 )  /\  ( Y  e.  CC  /\  Y  =/=  0 ) )  -> 
( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
71, 2, 3, 4, 6syl22anc 1186 1  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319   {csn 3816   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   CCcc 8990   0cc0 8992    / cdiv 9679   Imcim 11905   logclog 20454
This theorem is referenced by:  angcld  20649  angrteqvd  20650  cosangneg2d  20651  ang180lem4  20656  lawcos  20660  isosctrlem3  20666  angpieqvdlem2  20672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088
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