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Theorem angvald 22200
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 22197. (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 22197 . 2  |-  ( ( ( X  e.  CC  /\  X  =/=  0 )  /\  ( Y  e.  CC  /\  Y  =/=  0 ) )  -> 
( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
71, 2, 3, 4, 6syl22anc 1219 1  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2606    \ cdif 3325   {csn 3877   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   CCcc 9280   0cc0 9282    / cdiv 9993   Imcim 12587   logclog 22006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096
This theorem is referenced by:  angcld  22201  angrteqvd  22202  cosangneg2d  22203  ang180lem4  22208  lawcos  22212  isosctrlem3  22218  angpieqvdlem2  22224
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