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Theorem angvald 20102
Description: The (signed) angle between two vectors is the argument of their quotient. Deduction form of angval 20099. (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 20099 . 2  |-  ( ( ( X  e.  CC  /\  X  =/=  0 )  /\  ( Y  e.  CC  /\  Y  =/=  0 ) )  -> 
( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
71, 2, 3, 4, 6syl22anc 1183 1  |-  ( ph  ->  ( X F Y )  =  ( Im
`  ( log `  ( Y  /  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = 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:  angcld  20103  angrteqvd  20104  cosangneg2d  20105  angpieqvdlem2  20126
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
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